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Coefficient of x^(n-1) in Product_{k=1..n} (k + x + (n+1-k)*x^2), for n >= 1.
3

%I #11 Dec 30 2018 01:14:49

%S 1,3,32,210,3594,37457,896412,12899520,398293065,7333821066,

%T 277832923686,6242376401170,280450554344037,7444127625398565,

%U 387040500799449256,11856077367530958400,700385265181462012083,24325550963621400151002,1609526297187139688483020,62510661428443938909212244,4579921704394565828797270344,196706096648058100149910675962,15809764480004991957201666377340

%N Coefficient of x^(n-1) in Product_{k=1..n} (k + x + (n+1-k)*x^2), for n >= 1.

%C A diagonal of triangle A322229, where, for n >= 0:

%C (1) A322229(n,0) = A322229(n,2*n) = n!,

%C (2) Sum_{k=0..2*n} A322229(n,k) = (n+2)^n,

%C (3) Sum_{k=0..2*n} A322229(n,k)*(-1)^k = n^n.

%H Paul D. Hanna, <a href="/A322234/b322234.txt">Table of n, a(n) for n = 1..300</a>

%F a(n) ~ (exp(2) - (-1)^n) * sqrt(3/Pi) * n^(n - 1/2) / 2. - _Vaclav Kotesovec_, Dec 27 2018

%e Triangle A322229, where row n gives coefficients in Product_{k=1..n} (k + x + (n+1-k)*x^2), begins

%e 1;

%e 1, 1, 1;

%e 2, 3, 6, 3, 2;

%e 6, 11, 32, 27, 32, 11, 6;

%e 24, 50, 189, 210, 350, 210, 189, 50, 24;

%e 120, 274, 1269, 1689, 3594, 2915, 3594, 1689, 1269, 274, 120;

%e 720, 1764, 9652, 14651, 37750, 37457, 58156, 37457, 37750, 14651, 9652, 1764, 720;

%e 5040, 13068, 82396, 138473, 417780, 481074, 896412, 714483, 896412, 481074, 417780, 138473, 82396, 13068, 5040; ...

%e in which the coefficient of x^(n-1) in row n for n>=1 yields this sequence.

%t a[n_] := SeriesCoefficient[Product[k+x+(n-k+1)x^2, {k, 1, n}], {x, 0, n-1}];

%t Array[a, 23] (* _Jean-François Alcover_, Dec 29 2018 *)

%o (PARI) {A322229(n, k) = polcoeff( prod(m=1, n, m + x + (n+1-m)*x^2) +x*O(x^k), k)}

%o for(n=1, 30, print1( A322229(n, n-1), ", "));

%Y Cf. A322229, A322233.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Dec 18 2018