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%I A386567 #39 Nov 09 2025 10:01:46
%S A386567 0,1,17,268,4129,62955,954392,14417376,217279857,3269099590,
%T A386567 49125066135,737516631908,11064270530632,165889863957065,
%U A386567 2486052264852180,37241727274394640,557707191712371729,8349517132932620730,124971965902300790390,1870139909398530770760
%N A386567 a(n) = Sum_{k=0..n-1} binomial(6*k-1,k) * binomial(6*n-6*k,n-k-1).
%H A386567 Robert Israel, <a href="/A386567/b386567.txt">Table of n, a(n) for n = 0..849</a>
%F A386567 G.f.: g*(g-1)/(6-5*g)^2 where g=1+x*g^6.
%F A386567 G.f.: g/(1-6*g)^2 where g*(1-g)^5 = x.
%F A386567 L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ).
%F A386567 a(n) = Sum_{k=0..n-1} binomial(6*k-1+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
%F A386567 a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n,k).
%F A386567 a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k,k).
%F A386567 D-finite with recurrence (-222263767190910309923662331904*n^5 - 555659417977275774809155829760*n^4 - 524789450311871565097536061440*n^3 - 231524757490531572837148262400*n^2 - 46990950779559741449910091776*n - 3429996407267134412402196480)*a(n) + (7798003586831755283760414720000*n^5 + 66126715547634568734314987520000*n^4 + 219349022707445168726964633600000*n^3 + 356470039962759734013954981888000*n^2 + 284342242397891191738598357729280*n + 89200981138631056629276672000000)*a(n + 1) + (-379640117762749798156247040000*n^5 - 31026809924916272977582080000*n^4 + 20243589425281401374311695360000*n^3 + 95023225942150180310844526080000*n^2 + 167492858867589749900118612480000*n + 105247951334754585073931059200000)*a(n + 2) + (-46924791500353872024000000000*n^5 - 862323823086236519755500000000*n^4 - 5702925280832292498711750000000*n^3 - 17597329354953786117613095000000*n^2 - 25641968805267887570912692200000*n - 14088334615890526562148748800000)*a(n + 3) + (1956455198298711474609375000*n^5 + 25517997495175828710937500000*n^4 + 111555814249354873974609375000*n^3 + 129439020028916245623046875000*n^2 - 288016598578621100812031250000*n - 612999477689785535479921875000)*a(n + 4) + (42837686337833404541015625*n^5 + 1004728712390098571777343750*n^4 + 9475371893480300903320312500*n^3 + 44948833345403132629394531250*n^2 + 107339402026573143035888671875*n + 103309757767241510009765625000)*a(n + 5) + (-397409796714782714843750*n^5 - 11127474308013916015625000*n^4 - 124548230290412902832031250*n^3 - 696579891681671142578125000*n^2 - 1946687408763885498046875000*n - 2174717970840454101562500000)*a(n + 6) = 0. - _Robert Israel_, Nov 09 2025
%F A386567 From _Vaclav Kotesovec_, Nov 09 2025: (Start)
%F A386567 Recurrence: 15625*n*(5*n - 4)*(5*n - 3)*(5*n - 2)*(5*n - 1)*(94500*n^4 - 541170*n^3 + 1150593*n^2 - 1076089*n + 373416)*a(n) = 360*(76545000000*n^9 - 648846450000*n^8 + 2347467480000*n^7 - 4729478017500*n^6 + 5800978755000*n^5 - 4442132674215*n^4 + 2089167372336*n^3 - 567203022503*n^2 + 77091309382*n - 3609375000)*a(n-1) - 3359232*(2*n - 3)*(3*n - 5)*(3*n - 4)*(6*n - 11)*(6*n - 7)*(94500*n^4 - 163170*n^3 + 94083*n^2 - 20413*n + 1250)*a(n-2).
%F A386567 a(n) ~ 2^(6*n-1) * 3^(6*n) / 5^(5*n+1) * (1 - 7/(3*sqrt(15*Pi*n))). (End)
%e A386567 (1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ) = x + 17*x^2/2 + 268*x^3/3 + 4129*x^4/4 + 12591*x^5 + ...
%p A386567 f:= proc(n) local k; add(binomial(6*k-1, k)*binomial(6*n-6*k, n-k-1), k=0..n-1) end proc:
%p A386567 map(f, [$0..40]); # _Robert Israel_, Nov 09 2025
%o A386567 (PARI) a(n) = sum(k=0, n-1, binomial(6*k-1, k)*binomial(6*n-6*k, n-k-1));
%o A386567 (PARI) my(N=20, x='x+O('x^N), g=sum(k=0, N, binomial(6*k, k)/(5*k+1)*x^k)); concat(0, Vec(g*(g-1)/(6-5*g)^2))
%Y A386567 Cf. A000346, A062236, A386565, A386566.
%Y A386567 Cf. A079679, A386368, A386615, A386616.
%Y A386567 Cf. A002295, A130564.
%K A386567 nonn
%O A386567 0,3
%A A386567 _Seiichi Manyama_, Jul 26 2025

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