A359917 - OEIS

%I #13 Mar 21 2024 04:47:04

%S 1,1,3,28,413,9216,268327,9831424,432251577,22259307520,1313366140331,

%T 87431498993664,6482838033725077,529958491541291008,

%U 47356678577690489295,4592761099982656823296,480465410003489098874993,53933291626260492656050176,6466413087139041540884403667

%N E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).

%C Related limits:

%C (C1) exp(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).

%C (C2) W(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n)^n * (x/N)^n/n! ]^(1/N), where W(x) = LambertW(-x)/(-x).

%H Paul D. Hanna, <a href="/A359917/b359917.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.

%F (1) A(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).

%F (2) A(x) = exp( Sum_{n>=0} A359918(n)*x^n/n! ), where A359918(n) = (1/2) * [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! ).

%F a(n) ~ c * d^n * n! / n^(5/2), where d = 7.68892218919697462312... and c = 0.155267010681833... - _Vaclav Kotesovec_, Mar 21 2024

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 28*x^3/3! + 413*x^4/4! + 9216*x^5/5! + 268327*x^6/6! + 9831424*x^7/7! + 432251577*x^8/8! + 22259307520*x^9/9! + 1313366140331*x^10/10! + ...

%e where A(x) equals the limit, as N -> oo, of the series

%e [1 + (N^2+N+2)*(x/N) + (N^2+2*N+2*2^2)^2*(x/N)^2/2! + (N^2+3*N+2*3^2)^3*(x/N)^3/3! + (N^2+4*N+2*4^2)^4*(x/N)^4/4! + (N^2+5*N+2*5^2)^5*(x/N)^5/5! + (N^2+6*N+2*6^2)^6*(x/N)^6/6! + ...]^(1/N).

%e RELATED SERIES.

%e The logarithm of the g.f. A(x) begins:

%e (a) log(A(x)) = x + 2*x^2/2! + 21*x^3/3! + 304*x^4/4! + 6985*x^5/5! + 205056*x^6/6! + 7607509*x^7/7! + ... + A359918(n)*x^n/n! + ...

%e where A359918(n) = [x^n*y^(n+1)/n!] (1/2) * log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! );

%e that is, the coefficients in the logarithm of e.g.f A(x) equals the coefficients of y^(n+1)*x^n/n! in the series given by

%e (b) (1/2) * log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! ) = (y^2 + 1/2*y + 1/2)*x + (2*y^3 + 15/2*y^2 + 7*y + 15/2)*x^2/2! + (21*y^4 + 197/2*y^3 + 543/2*y^2 + 639/2*y + 683/2)*x^3/3! + (304*y^5 + 2495*y^4 + 8984*y^3 + 22246*y^2 + 29360*y + 31019)*x^4/4! + (6985*y^6 + 150489/2*y^5 + 817005/2*y^4 + 1335885*y^3 + 3162830*y^2 + 8940045/2*y + 9342629/2)*x^5/5! + (205056*y^7 + 2946228*y^6 + 20587128*y^5 + 94146240*y^4 + 294518400*y^3 + 684700836*y^2 + 1013688168*y + 1050241608)*x^6/6! + ...

%o (PARI) /* Using formula for the logarithm of g.f. A(x) */

%o {L(n) = (1/2) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m^2 + m*y + 2*y^2)^m *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}

%o {a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) /* Using limit formula */

%o \p100

%o P(n) = sum(k=0, 31, (n^2 + n*k + 2*k^2)^k * x^k/k! +O(x^31))

%o Vec(round( serlaplace( subst(P(10^100)^(1/10^100), x, x/10^100) )*1.) )

%Y Cf. A359918, A266485, A359927, A319147, A266481, A318633, A319834.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 21 2023