|
%I #13 Mar 18 2022 07:31:25
%S 1,1,5,61,1123,27671,853411,31603447,1365807689,67469763889,
%T 3749935785301,231591200859701,15733654527061483,1166102347943957815,
%U 93629607937879486019,8096167402408961507311,750088483178476669111441,74127049788588758257392161,7783440821906363883725443813,865349148215025766722403077229,101553078711812924877087765912371
%N G.f.: Sum_{n>=0} binomial(n^2, n) * x^n / (1 + x)^(n^2).
%C 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.
%H Paul D. Hanna, <a href="/A298695/b298695.txt">Table of n, a(n) for n = 0..300</a>
%F 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
%F c = exp(LambertW(-2*exp(-2))^2/8 - 1/2) / (2*Pi*sqrt(1 + LambertW(-2*exp(-2)))). - _Vaclav Kotesovec_, Mar 18 2022
%e 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 + ...
%e such that
%e 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 + ...
%e more explicitly,
%e 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) + ...
%t 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 *)
%o (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)}
%o for(n=0,25, print1(a(n),", "))
%Y Cf. A298696, A014062.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 04 2018
|
