A322234 Coefficient of x^(n-1) in Product_{k=1..n} (k + x + (n+1-k)*x^2), for n >= 1.

1, 3, 32, 210, 3594, 37457, 896412, 12899520, 398293065, 7333821066, 277832923686, 6242376401170, 280450554344037, 7444127625398565, 387040500799449256, 11856077367530958400, 700385265181462012083, 24325550963621400151002, 1609526297187139688483020, 62510661428443938909212244, 4579921704394565828797270344, 196706096648058100149910675962, 15809764480004991957201666377340

Offset

1,2

Comments

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

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

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

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

a(n) ~ (exp(2) - (-1)^n) * sqrt(3/Pi) * n^(n - 1/2) / 2. - Vaclav Kotesovec, Dec 27 2018

Example

Triangle A322229, where row n gives coefficients in Product_{k=1..n} (k + x + (n+1-k)*x^2), begins
1;
1, 1, 1;
2, 3, 6, 3, 2;
6, 11, 32, 27, 32, 11, 6;
24, 50, 189, 210, 350, 210, 189, 50, 24;
120, 274, 1269, 1689, 3594, 2915, 3594, 1689, 1269, 274, 120;
720, 1764, 9652, 14651, 37750, 37457, 58156, 37457, 37750, 14651, 9652, 1764, 720;
5040, 13068, 82396, 138473, 417780, 481074, 896412, 714483, 896412, 481074, 417780, 138473, 82396, 13068, 5040; ...
in which the coefficient of x^(n-1) in row n for n>=1 yields this sequence.

Mathematica

a[n_] := SeriesCoefficient[Product[k+x+(n-k+1)x^2, {k, 1, n}], {x, 0, n-1}];
Array[a, 23] (* Jean-François Alcover, Dec 29 2018 *)

Prog

(PARI) {A322229(n, k) = polcoeff( prod(m=1, n, m + x + (n+1-m)*x^2) +x*O(x^k), k)}
for(n=1, 30, print1( A322229(n, n-1), ", "));

Crossrefs

Cf. A322229, A322233.

Sequence in context: A119940 A004256 A183457 * A264574 A002059 A028447

Adjacent sequences: A322231 A322232 A322233 * A322235 A322236 A322237

Keyword

nonn

Author

Paul D. Hanna, Dec 18 2018

Status

approved