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A383702 Coefficient of x^2 in expansion of (x+3) * (x+7) * ... * (x+4*n-1).

%I A383702 #20 May 08 2025 08:58:42
%S A383702 0,0,1,21,446,10670,290599,8951355,308846124,11822475564,497794079421,
%T A383702 22881487815153,1140642637297866,61312161303209466,
%U A383702 3535773901817957955,217787248332803277495,14271822475100747003160,991517953843097370650520,72799719644532661375481145
%N A383702 Coefficient of x^2 in expansion of (x+3) * (x+7) * ... * (x+4*n-1).
%H A383702 Vincenzo Librandi, <a href="/A383702/b383702.txt">Table of n, a(n) for n = 0..200</a>
%F A383702 a(n) = Sum_{k=2..n} 3^(k-2) * 4^(n-k) * binomial(k,2) * |Stirling1(n,k)|.
%F A383702 a(n) = Sum_{k=2..n} (4*n-1)^(k-2) * 4^(n-k) * binomial(k,2) * Stirling1(n,k).
%F A383702 E.g.f.: f(x)^3 * log(f(x))^2 / 2, where f(x) = 1/(1 - 4*x)^(1/4).
%F A383702 Conjecture D-finite with recurrence a(n) +3*(-4*n+5)*a(n-1) +(48*n^2-168*n+151)*a(n-2) -(4*n-9)^3*a(n-3)=0. - _R. J. Mathar_, May 07 2025
%t A383702 Array[Residue[Product[(x + 4*k - 1), {k, 1, #}]/x^3, {x, 0}] &, 19, 0] (* _Shenghui Yang_, May 06 2025 *)
%t A383702 Table[Sum[(4*n-1)^(k-2)*4^(n-k)*Binomial[k,2]*StirlingS1[n,k],{k,0,n}],{n,0,18}] (* _Vincenzo Librandi_, May 07 2025 *)
%o A383702 (PARI) a(n) = polcoef(prod(k=0, n-1, x+4*k+3), 2);
%o A383702 (Magma) [&+[(4*n-1)^(k-2) * 4^(n-k) * Binomial(k,2) * StirlingFirst(n,k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, May 07 2025
%Y A383702 Column k=2 of A225471.
%K A383702 nonn
%O A383702 0,4
%A A383702 _Seiichi Manyama_, May 06 2025