A375218 - OEIS

A375218

a(n) = Sum_{k=0..floor(n/2)} (k+1) * binomial(k,n-2*k)^2.

1, 0, 2, 2, 3, 12, 7, 36, 41, 84, 186, 230, 612, 852, 1733, 3198, 5112, 10628, 16873, 32562, 57463, 99892, 188103, 319188, 591982, 1040076, 1849352, 3351304, 5854119, 10610416, 18707180, 33370938, 59618393, 105291208, 188572347, 333462928, 593859439, 1055432400, 1870161060

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..38.

FORMULA

G.f.: (1-x^2-x^3)/((1-x^2-x^3)^2 - 4*x^5)^(3/2).

D-finite with recurrence 2*n*(2*n+1)*a(n) +3*(n-1)*(2*n-3)*a(n-1) +4*(-2*n^2-3*n+4)*a(n-2) +2*(-10*n^2+n+27)*a(n-3) +2*(-4*n^2+11*n+27)*a(n-4) +(-2*n^2-27*n-27)*a(n-5) +2*(-4*n^2+7*n+18)*a(n-6) +3*(2*n+3)*(n-1)*a(n-7)=0. - R. J. Mathar, Oct 17 2024

Recurrence (of order 6): n*(2*n - 1)*(2*n + 1)*a(n) = 2*(2*n - 1)*(2*n^2 + 3*n - 4)*a(n-2) + 2*(2*n + 1)*(2*n^2 + 2*n - 9)*a(n-3) - n*(2*n - 1)*(2*n + 5)*a(n-4) + 4*n^2*(2*n + 5)*a(n-5) - n*(2*n + 1)*(2*n + 5)*a(n-6). - Vaclav Kotesovec, Nov 09 2025

PROG

(PARI) a(n) = sum(k=0, n\2, (k+1)*binomial(k, n-2*k)^2);

CROSSREFS

Cf. A182884, A375470.

Cf. A298567.

Sequence in context: A182779 A199673 A377660 * A391488 A335942 A240133

Adjacent sequences: A375215 A375216 A375217 * A375219 A375220 A375221

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 17 2024

STATUS

approved