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A383838 Expansion of 1/((1-x) * (1-4*x) * (1-9*x) * (1-16*x)).

%I A383838 #25 May 12 2025 11:53:55
%S A383838 1,30,627,11440,196053,3255330,53157079,860181300,13850000505,
%T A383838 222384254950,3565207699131,57106865357880,914281747641757,
%U A383838 14633655168987690,234184807922193183,3747373855152257980,59961734043737254209,959421515974412698350,15351048197153778821635
%N A383838 Expansion of 1/((1-x) * (1-4*x) * (1-9*x) * (1-16*x)).
%H A383838 Seiichi Manyama, <a href="/A383838/b383838.txt">Table of n, a(n) for n = 0..830</a>
%H A383838 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (30,-273,820,-576).
%F A383838 a(n) = A269945(n+4,4).
%F A383838 a(n) = 30*a(n-1) - 273*a(n-2) + 820*a(n-3) - 576*a(n-4).
%F A383838 a(n) = (2*16^(n+3) - 9^(n+4) + 14*4^(n+3) - 7)/2520.
%F A383838 sinh(x)^8/8! = Sum_{k>=0} 4^k * a(k) * x^(2*k+8)/(2*k+8)!.
%F A383838 a(n) = (1/8!) * Sum_{k=0..8} (-1)^k * (4-k)^(2*n+8) * binomial(8,k).
%F A383838 a(n) = Sum_{k=0..2*n} (-4)^k * binomial(2*n+8,k) * Stirling2(2*n-k+8,8).
%F A383838 a(n) = Sum_{k=0..2*n} (-1)^k * Stirling2(k+4,4) * Stirling2(2*n-k+4,4).
%o A383838 (PARI) a(n) = (2*16^(n+3)-9^(n+4)+14*4^(n+3)-7)/2520;
%Y A383838 Cf. A002451, A269945.
%K A383838 nonn,easy
%O A383838 0,2
%A A383838 _Seiichi Manyama_, May 11 2025