0,7

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 411.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..260 (first 100 terms from Vincenzo Librandi)

Élie de Panafieu, Asymptotic expansion of regular and connected regular graphs, arXiv:2408.12459 [math.CO], 2024. See p. 9.

I. P. Goulden, D. M. Jackson, and J. W. Reilly, The Hammond series of a symmetric function and its application to P-recursiveness, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.

Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.

Vaclav Kotesovec, Recurrence (of order 10)

R. C. Read and N. C. Wormald, Number of labeled 4-regular graphs, J. Graph Theory, 4 (1980), 203-212.

From Vladeta Jovovic, Mar 26 2001: (Start)

E.g.f. f(x) = Sum_{n >= 0} a(n)*x^n/(n)! satisfies the differential equation 16*x^2*(x - 1)^2*(x + 2)^2*(x^5 + 2*x^4 + 2*x^2 + 8*x - 4)*(d^2/dx^2)y(x) - 4*(x^13 + 4*x^12 - 16*x^10 - 10*x^9 - 36*x^8 - 220*x^7 - 348*x^6 - 48*x^5 + 200*x^4 - 336*x^3 - 240*x^2 + 416*x - 96)*(d/dx)y(x) - x^4*(x^5 + 2*x^4 + 2*x^2 + 8*x - 4)^2*y(x) = 0.

Recurrence: a(n) = - 1/384*(( - 256*n^2 - 896*n + 1152)*a(n - 1) + (768*n^3 - 3648*n^2 + 5568*n - 2688)*a(n - 2) + ( - 192*n^4 + 3264*n^3 - 14784*n^2 + 24384*n - 12672)*a(n - 3) + (224*n^6 - 4512*n^5 + 36304*n^4 - 148160*n^3 + 320016*n^2 - 341728*n + 137856)*a(n - 5) + ( - 640*n^5 + 8800*n^4 - 46400*n^3 + 116000*n^2 - 135360*n + 57600)*a(n - 4) + ( - 24*n^10 + 1320*n^9 - 31680*n^8 + 435600*n^7 - 3786552*n^6 + 21649320*n^5 - 82006320*n^4 + 201828000*n^3 - 306085824*n^2 + 255087360*n - 87091200)*a(n - 11) + (64*n^10 - 3480*n^9 + 82692*n^8 - 1127232*n^7 + 9726024*n^6 - 55255032*n^5 + 208179908*n^4 - 510068208*n^3 + 770738352*n^2 - 640484928*n + 218211840)*a(n - 9) + (16*n^11 - 992*n^10 + 27256*n^9 - 437160*n^8 + 4536288*n^7 - 31876656*n^6 + 154182488*n^5 - 510784360*n^4 + 1128552896*n^3 - 1570313952*n^2 + 1223830656*n - 397716480)*a(n - 10) + ( - 128*n^8 + 5488*n^7 - 94576*n^6 + 864976*n^5 - 4606672*n^4 + 14604352*n^3 - 26753984*n^2 + 25611264*n - 9630720)*a(n - 7) + (16*n^9 - 576*n^8 + 8704*n^7 - 71680*n^6 + 348880*n^5 - 1013824*n^4 + 1673376*n^3 - 1333120*n^2 + 226944*n + 161280)*a(n - 8) + (128*n^7 - 2192*n^6 + 12048*n^5 - 8240*n^4 - 151248*n^3 + 565312*n^2 - 765248*n + 349440)*a(n - 6) + ( - 4*n^13 + 364*n^12 - 14924*n^11 + 364364*n^10 - 5897892*n^9 + 66678612*n^8 - 540145892*n^7 + 3163772612*n^6 - 13344475144*n^5 + 39830815024*n^4 - 81255012384*n^3 + 106386868224*n^2 - 79211036160*n + 24908083200)*a(n - 14) + ( - 4*n^13 + 360*n^12 - 14612*n^11 + 353496*n^10 - 5674812*n^9 + 63680760*n^8 - 512439356*n^7 + 2983811688*n^6 - 12520194544*n^5 + 37201987680*n^4 - 75598952832*n^3 + 98660630016*n^2 - 73265264640*n + 22992076800)*a(n - 13) + ( - 16*n^12 + 1244*n^11 - 43208*n^10 + 884620*n^9 - 11860728*n^8 + 109396452*n^7 - 709293464*n^6 + 3243764260*n^5 - 10331326456*n^4 + 22203205904*n^3 - 30301280928*n^2 + 23300910720*n - 7504358400)*a(n - 12) + ( - n^14 + 105*n^13 - 5005*n^12 + 143325*n^11 - 2749747*n^10 + 37312275*n^9 - 368411615*n^8 + 2681453775*n^7 - 14409322928*n^6 + 56663366760*n^5 - 159721605680*n^4 + 310989260400*n^3 - 392156797824*n^2 + 283465647360*n - 87178291200)*a(n - 15)). (End)

a(n) = Sum_{d=0..floor(n/2), c=0..floor(n/2-d), b=0..(n-2c-2d), f=0..(n-2c-2d-b), k=0..min(n-b-2c-2d-f, 2n-2f-2b-3c-4d), j=0..floor(k/2+f)} ((-1)^(k+2f-j+d)*n!*(k+2f)!(2(2n-k-2f-2b-3c-4d))!) / (2^(5n-2k-2f-3b-8c-7d) * 3^(n-b-c-2d-k-f)*(2n-k-2f-2b-3c-4d)!*(k+2f-2j)!*j!*b!*c!*d!*k!*f!*(n-b-2c-2d-k-f)!). - Shanzhen Gao, Jun 05 2009

E.g.f.: (1+x-(1/3)*x^2-(1/6)*x^3)^(-1/2)*hypergeom([1/4, 3/4],[],-12*x*(x+2)*(x-1)/(x^3+2*x^2-6*x-6)^2)*exp(-x*(x^2-6)/(8*x+16)). - Mark van Hoeij, Nov 07 2011

a(n) ~ n^(2*n) * 2^(n+1/2) / (3^n * exp(2*n+15/4)). - Vaclav Kotesovec, Mar 11 2014

egf := (1+x-(1/3)*x^2-(1/6)*x^3)^(-1/2)*hypergeom([1/4, 3/4], [], -12*x*(x+2)*(x-1)/(x^3+2*x^2-6*x-6)^2)*exp(-x*(x^2-6)/(8*x+16));

ser := convert(series(egf, x=0, 40), polynom):

seq(coeff(ser, x, i)*i!, i=0..degree(ser)); # Mark van Hoeij, Nov 07 2011

max = 17; f[x_] := HypergeometricPFQ[{1/4, 3/4}, {}, -12*x*(x + 2)*(x - 1)/(x^3 + 2*x^2 - 6*x - 6)^2]*Exp[-x*(x^2 - 6)/(8*x + 16)]/(1 + x - x^2/3 - x^3/6)^ (1/2); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Jun 19 2012, from e.g.f. *)

Cf. A005814, A002829, A005816, A272905 (connected). A diagonal of A059441.

Sequence in context: A177080 A306609 A272905 * A120600 A377498 A377497

Adjacent sequences: A005812 A005813 A005814 * A005816 A005817 A005818

nonn,nice,easy

More terms from Vladeta Jovovic, Mar 26 2001

approved