A298695 G.f.: Sum_{n>=0} binomial(n^2, n) * x^n / (1 + x)^(n^2).

1, 1, 5, 61, 1123, 27671, 853411, 31603447, 1365807689, 67469763889, 3749935785301, 231591200859701, 15733654527061483, 1166102347943957815, 93629607937879486019, 8096167402408961507311, 750088483178476669111441, 74127049788588758257392161, 7783440821906363883725443813, 865349148215025766722403077229, 101553078711812924877087765912371

Offset: 0

Author

Paul D. Hanna, Feb 04 2018

Keywords

• nonn

Comments

Compare g.f. to: Sum_{n>=0} binomial(m*n, n) * x^n / (1+x)^(m*n) = (1+x)/(1 - (m-1)*x) holds for fixed m.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 61*x^3 + 1123*x^4 + 27671*x^5 + 853411*x^6 + 31603447*x^7 + 1365807689*x^8 + 67469763889*x^9 + 3749935785301*x^10 + ...
such that
A(x) = 1 + C(1,1)*x/(1+x) + C(4,2)*x^2/(1+x)^4 + C(9,3)*x^3/(1+x)^9 + C(16,4)*x^4/(1+x)^16 + C(25,5)*x^5/(1+x)^25 + C(36,6)*x^6/(1+x)^36 + ...
more explicitly,
A(x) = 1 + x/(1+x) + 6*x^2/(1+x)^4 + 84*x^3/(1+x)^9 + 1820*x^4/(1+x)^16 + 53130*x^5/(1+x)^25 + 1947792*x^6/(1+x)^36 + ... + A014062(n)*x^n/(1+x)^(n^2) + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A298696, A014062.

Programs

Mathematica

terms = 21; s = Sum[Binomial[n^2, n]*x^n/(1 + x)^(n^2), {n, 0, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Feb 06 2018 *)

PARI

{a(n) = my(A = sum(m=0,n,binomial(m^2,m)*x^m/(1+x +x*O(x^n))^(m^2) ) ); polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))

Formula

a(n) ~ c * d^n * (n-1)!, , where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.176554609483480358231680164050876553672889794... and c = 0.127903391767990118250352331247574466909912463001514793015830303493876... - Vaclav Kotesovec, Oct 10 2020

c = exp(LambertW(-2*exp(-2))^2/8 - 1/2) / (2*Pi*sqrt(1 + LambertW(-2*exp(-2)))). - Vaclav Kotesovec, Mar 18 2022