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A383703 Coefficient of x^3 in expansion of (x+3) * (x+7) * ... * (x+4*n-1).

%I A383703 #15 May 07 2025 11:54:40
%S A383703 0,0,0,1,36,1130,36660,1280419,48644344,2011398164,90267003960,
%T A383703 4379275249701,228707424551100,12804721289403966,765571832220427596,
%U A383703 48704512002823186119,3286171504510664002992,234445313277315235203624,17637135196532479070107824,1395584859384468591633567945
%N A383703 Coefficient of x^3 in expansion of (x+3) * (x+7) * ... * (x+4*n-1).
%H A383703 Vincenzo Librandi, <a href="/A383703/b383703.txt">Table of n, a(n) for n = 0..200</a>
%F A383703 a(n) = Sum_{k=3..n} 3^(k-3) * 4^(n-k) * binomial(k,3) * |Stirling1(n,k)|.
%F A383703 a(n) = Sum_{k=3..n} (4*n-1)^(k-3) * 4^(n-k) * binomial(k,3) * Stirling1(n,k).
%F A383703 E.g.f.: f(x)^3 * log(f(x))^3 / 6, where f(x) = 1/(1 - 4*x)^(1/4).
%t A383703 Table[Sum[(4*n-1)^(k-3)*4^(n-k)*Binomial[k,3]*StirlingS1[n,k],{k,3,n}],{n,0,25}] (* _Vincenzo Librandi_, May 07 2025 *)
%o A383703 (PARI) a(n) = polcoef(prod(k=0, n-1, x+4*k+3), 3);
%o A383703 (Magma) [&+[(4*n-1)^(k-3) * 4^(n-k) * Binomial(k,3) * StirlingFirst(n,k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, May 07 2025
%Y A383703 Column k=3 of A225471.
%K A383703 nonn
%O A383703 0,5
%A A383703 _Seiichi Manyama_, May 06 2025