A322233 Coefficient of x^n in Product_{k=1..n} (k + x + (n+1-k)*x^2), for n >= 0.

1, 1, 6, 27, 350, 2915, 58156, 714483, 19279494, 314352219, 10702577028, 217898279709, 8961716977044, 218984398436089, 10559918611732824, 301232711294645459, 16665825788607694854, 543748938748390962083, 33949005044435560537540, 1247096314671704716427281, 86735021741507120417172516, 3542923311990891744041871249, 271628142382739329884065326824, 12213792889354508458626454059325, 1023380788552996903255162728503100

Offset

0,3

Comments

Main 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 = 0..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 in row n yields this sequence.

Prog

(PARI) {A322229(n, k) = polcoeff( prod(m=1, n, m + x + (n+1-m)*x^2) +x*O(x^k), k)}
for(n=0, 30, print1( A322229(n, n), ", "))

Crossrefs

Cf. A322229, A322234.

Sequence in context: A360754 A367886 A351735 * A164985 A367884 A048436

Adjacent sequences: A322230 A322231 A322232 * A322234 A322235 A322236

Keyword

nonn

Author

Paul D. Hanna, Dec 18 2018

Status

approved