A383701 - OEIS

%I #17 Mar 12 2026 11:36:59

%S 0,0,0,1,28,730,20460,633619,21740040,823020596,34174098440,

%T 1546855384261,75883563554436,4013184755214414,227719025845257492,

%U 13804358188086757719,890571834923460488784,60933371174617735181160,4407783770975985847999440,336154167664942342604334345

%N Coefficient of x^3 in expansion of (x+1) * (x+5) * ... * (x+4*n-3).

%H Robert Israel, <a href="/A383701/b383701.txt">Table of n, a(n) for n = 0..365</a>

%F a(n) = Sum_{k=3..n} 4^(n-k) * binomial(k,3) * |Stirling1(n,k)|.

%F a(n) = Sum_{k=3..n} (4*n-3)^(k-3) * 4^(n-k) * binomial(k,3) * Stirling1(n,k).

%F E.g.f.: f(x) * log(f(x))^3 / 6, where f(x) = 1/(1 - 4*x)^(1/4).

%F Conjecture D-finite with recurrence a(n) +4*(-4*n+9)*a(n-1) +2*(48*n^2-264*n+371)*a(n-2) -4*(4*n-13)*(16*n^2-104*n+173)*a(n-3) +(4*n-15)^4*a(n-4)=0. - _R. J. Mathar_, May 07 2025

%F Conjecture confirmed using differential equation y + (624*x - 156)*y' + (2656*x^2 - 1328*x + 166)*y'' + (1792*x^3 - 1344*x^2 + 336*x - 28)*y''' + (256*x^4 - 256*x^3 + 96*x^2 - 16*x + 1)*y'''' satisfied by the e.g.f. - _Robert Israel_, Mar 12 2026

%p f:= gfun:-rectoproc({(4*n + 1)^4*a(n) - 4*a(n + 1)*(4*n + 3)*(16*n^2 + 24*n + 13) + 2*(48*n^2 + 120*n + 83)*a(n + 2) - 4*a(n + 3)*(4*n + 7) + a(n + 4), a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1}, a(n), remember):

%p map(f, [$0..20]); # _Robert Israel_, Mar 12 2026

%o (PARI) a(n) = polcoef(prod(k=0, n-1, x+4*k+1), 3);

%Y Column k=3 of A290319.

%K nonn

%O 0,5

%A _Seiichi Manyama_, May 06 2025