A298696 - OEIS

%I #14 Mar 18 2022 07:43:47

%S 1,1,3,28,410,8386,220962,7140736,273712896,12146997564,612813677300,

%T 34647736132384,2170381958609592,149223874286440552,

%U 11173356309069883320,905099760309260722560,78870011549256151244288,7356892186010414244194704,731435433368215011644979504,77216368897429504869064200256,8626428901029156775683110378400,1016792561657783042048699052986016

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

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

%H Paul D. Hanna, <a href="/A298696/b298696.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) ~ c * d^n * n! / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.176554609483480358231680164050876553672889... and c = 0.226094037474708064867716267720651240574569526310006844420310030408773601638... - _Vaclav Kotesovec_, Feb 07 2018

%F c = exp(LambertW(-2*exp(-2))^2/8 - 1/2) * sqrt(2) * (2 + LambertW(-2*exp(-2))) / (4*Pi*sqrt(-LambertW(-2*exp(-2))) * sqrt(1 + LambertW(-2*exp(-2)))). - _Vaclav Kotesovec_, Mar 18 2022

%e G.f.: A(x) = 1 + x + 3*x^2 + 28*x^3 + 410*x^4 + 8386*x^5 + 220962*x^6 + 7140736*x^7 + 273712896*x^8 + 12146997564*x^9 + 612813677300*x^10 + ...

%e such that

%e A(x) = 1 + C(2,1)/2*x/(1+x)^2 + C(6,2)/3*x^2/(1+x)^6 + C(12,3)/4*x^3/(1+x)^12 + C(20,4)/5*x^4/(1+x)^20 + C(30,5)/6*x^5/(1+x)^30 + ...

%e more explicitly,

%e A(x) = 1 + x/(1+x)^2 + 5*x^2/(1+x)^6 + 55*x^3/(1+x)^12 + 969*x^4/(1+x)^20 + 23751*x^5/(1+x)^30 + ... + A135861(n)*x^n/(1+x)^(n*(n+1)) + ...

%t terms = 22; s = Sum[Binomial[n*(n + 1), n]/(n + 1)*x^n/(1 + x)^(n*(n + 1)), {n, 0, terms}] + O[x]^terms; CoefficientList[s, x] (* _Jean-François Alcover_, Feb 05 2018 *)

%o (PARI) {a(n) = my(A = sum(m=0,n,binomial(m*(m+1),m)/(m+1)*x^m/(1+x +x*O(x^n))^(m*(m+1)) ) ); polcoeff(A,n)}

%o for(n=0,25, print1(a(n),", "))

%Y Cf. A298695, A135861.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 04 2018