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A129256 Central coefficient of Product_{k=0..n} (1+k*x)^2.

%I A129256 #32 May 19 2025 04:45:39
%S A129256 1,2,13,144,2273,46710,1184153,35733376,1251320145,49893169050,
%T A129256 2232012515445,110722046632560,6032418472347265,358103844593876654,
%U A129256 23007314730623658225,1590611390957425536000,117745011140615270168865
%N A129256 Central coefficient of Product_{k=0..n} (1+k*x)^2.
%H A129256 Vaclav Kotesovec, <a href="/A129256/b129256.txt">Table of n, a(n) for n = 0..354</a>
%F A129256 a(n) = (-1)^n*Sum_{k=0..n} Stirling1(n+1,k+1)*Stirling1(n+1,n-k+1). - _Paul D. Hanna_, Jul 16 2009
%F A129256 a(n) ~ c * d^n * (n-1)!, where d = A238261 = -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)) = 4.910814964568255..., c = (-LambertW(-1, -exp(-1/2)/2))^(3/2)/(sqrt(-1 - LambertW(-1, -exp(-1/2)/2))*Pi) = 0.851946112888790982829578047527831525434714038256... . - _Vaclav Kotesovec_, Feb 10 2015, updated May 14 2025
%e A129256 This sequence equals the central terms of the triangle in which the g.f. of row n is (1+x)^2*(1+2x)^2*(1+3x)^2*...*(1+n*x)^2, as illustrated by:
%e A129256   (1);
%e A129256    1, (2),  1;
%e A129256    1,  6, (13),  12,     4;
%e A129256    1, 12,  58, (144),  193,    132,      36;
%e A129256    1, 20, 170,  800, (2273),  3980,    4180,   2400,    576;
%e A129256    1, 30, 395, 3000, 14523, (46710), 100805, 143700, 129076, 65760, 14400;
%e A129256   ...
%t A129256 Flatten[{1,Table[Coefficient[Expand[Product[(1+k*x),{k,0,n}]^2],x^n],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 10 2015 *)
%o A129256 (PARI) a(n)=polcoeff(prod(k=0,n,1+k*x)^2,n)
%o A129256 (PARI) {a(n)=(-1)^n*sum(k=0,n,stirling(n+1,k+1,1)*stirling(n+1,n-k+1,1))} \\ _Paul D. Hanna_, Jul 16 2009
%Y A129256 Cf. A008275 (Stirling1 numbers), A187235, A238261, A246117, A254882, A350376.
%Y A129256 Cf. A384012, A384031, A384017.
%K A129256 nonn
%O A129256 0,2
%A A129256 _Paul D. Hanna_, Apr 06 2007