A324304 a(n) = [y^(n-1)] Product_{k=0..n-2} (n + k*y + n*y^2) for n > 1 with a(1) = 1.

1, 0, 18, 96, 4300, 81360, 3604342, 128389632, 6704335980, 346778956800, 21896347260084, 1459386186255360, 110117675704707190, 8898156449299703040, 786739773441598071750, 74406732202318884372480, 7565016269351818379826372, 818338704493281924572946432, 94154670956813022045927404464, 11458715042302170139584184320000, 1472412964588453156024745207931636

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..100

Formula

a(n) = A324305(n, n-1) for n >= 1.

a(n) ~ c * n! * (27/4)^n / n^2, where c = 1/(6*Pi*sqrt(3*log(3/2))) = 0.04810181967270783985882272373499905248047631331... - Vaclav Kotesovec, Mar 13 2019, updated Mar 17 2024

Example

E.g.f.: A(x) = x + 18*x^3/3! + 96*x^4/4! + 4300*x^5/5! + 81360*x^6/6! + 3604342*x^7/7! + 128389632*x^8/8! + 6704335980*x^9/9! + 346778956800*x^10/10! + 21896347260084*x^11/11! + 1459386186255360*x^12/12! + ...
RELATED TRIANGLE.
Triangle A324305 of coefficients in Product_{k=0..n-2} (n + k*y + n*y^2), n >= 1, begins
1;
2, 0, 2;
9, 3, 18, 3, 9;
64, 48, 200, 96, 200, 48, 64;
625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625;
7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776;
117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649; ...
in which the central terms, A324305(n, n-1) for n >= 1, form this sequence.

Mathematica

Flatten[{1, Table[Coefficient[Expand[Product[(n + k*y + n*y^2), {k, 0, n-2}]], y^(n-1)], {n, 2, 20}]}] (* Vaclav Kotesovec, Mar 13 2019 *)

Prog

(PARI) {A324305(n, k) = polcoeff( prod(j=0, n-2,  n + j*y + n*y^2), k, y)}
{a(n) = A324305(n, n-1)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A324305.

Cf. A201950 (variant).

Sequence in context: A243995 A264202 A338783 * A118864 A118606 A318063

Adjacent sequences: A324301 A324302 A324303 * A324305 A324306 A324307

Keyword

nonn

Author

Paul D. Hanna, Feb 28 2019

Status

approved