A324304 - OEIS

%I #21 Mar 17 2024 06:08:36

%S 1,0,18,96,4300,81360,3604342,128389632,6704335980,346778956800,

%T 21896347260084,1459386186255360,110117675704707190,

%U 8898156449299703040,786739773441598071750,74406732202318884372480,7565016269351818379826372,818338704493281924572946432,94154670956813022045927404464,11458715042302170139584184320000,1472412964588453156024745207931636

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

%H Paul D. Hanna, <a href="/A324304/b324304.txt">Table of n, a(n) for n = 1..100</a>

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

%F 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

%e 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! + ...

%e RELATED TRIANGLE.

%e Triangle A324305 of coefficients in Product_{k=0..n-2} (n + k*y + n*y^2), n >= 1, begins

%e 1;

%e 2, 0, 2;

%e 9, 3, 18, 3, 9;

%e 64, 48, 200, 96, 200, 48, 64;

%e 625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625;

%e 7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776;

%e 117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649; ...

%e in which the central terms, A324305(n, n-1) for n >= 1, form this sequence.

%t 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 *)

%o (PARI) {A324305(n, k) = polcoeff( prod(j=0, n-2,  n + j*y + n*y^2), k, y)}

%o {a(n) = A324305(n, n-1)}

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A324305.

%Y Cf. A201950 (variant).

%K nonn

%O 1,3

%A _Paul D. Hanna_, Feb 28 2019