cp's OEIS Frontend

A359927 E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + 3*N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).

%I A359927 #17 Mar 14 2023 04:34:02
%S A359927 1,1,7,112,2965,111856,5528419,339433984,24965493865,2142654088960,
%T A359927 210377086601311,23269631260880896,2864038963868253373,
%U A359927 388330717110688399360,57521524729462484086075,9242821569458332441378816,1601434996324769244061560529
%N A359927 E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + 3*N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).
%C A359927 Related limits:
%C A359927 (C1) exp(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
%C A359927 (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 A359927 Paul D. Hanna, <a href="/A359927/b359927.txt">Table of n, a(n) for n = 0..200</a>
%F A359927 E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.
%F A359927 (1) A(x) = lim_{N->oo} [ Sum_{n>=0} (N + n)^n*(N + 2*n)^n * (x/N)^n/n! ]^(1/N).
%F A359927 (2) A(x) = exp( Sum_{n>=0} A359928(n)*x^n/n! ), where A359928(n) = (1/2) * [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n + y)^n*(n + 2*y)^n *x^n/n! ).
%F A359927 a(n) ~ c * n! * d^n / n^(5/2), where d = 12.7029497597456784744445675253711147535742245945208995646... and c = 0.17380315134029681101563539591890111670852050181568... - _Vaclav Kotesovec_, Mar 14 2023
%e A359927 E.g.f.: A(x) = 1 + x + 7*x^2/2! + 112*x^3/3! + 2965*x^4/4! + 111856*x^5/5! + 5528419*x^6/6! + 339433984*x^7/7! + 24965493865*x^8/8! + 2142654088960*x^9/9! + 210377086601311*x^10/10! + 23269631260880896*x^11/11! + 2864038963868253373*x^12/12! + ...
%e A359927 where A(x) equals the limit, as N -> oo, of the series
%e A359927 [1 + (N^2+3*N+2)*(x/N) + (N^2+3*2*N+2*2^2)^2*(x/N)^2/2! + (N^2+3*3*N+2*3^2)^3*(x/N)^3/3! + (N^2+3*4*N+2*4^2)^4*(x/N)^4/4! + (N^2+3*5*N+2*5^2)^5*(x/N)^5/5! + (N^2+3*6*N+2*6^2)^6*(x/N)^6/6! + ...]^(1/N).
%e A359927 RELATED SERIES.
%e A359927 The logarithm of the g.f. A(x) begins:
%e A359927 (a) log(A(x)) = x + 6*x^2/2! + 93*x^3/3! + 2448*x^4/4! + 92505*x^5/5! + 4589568*x^6/6! + 283008621*x^7/7! + 20903023872*x^8/8! + ... + A359928(n)*x^n/n! + ...
%e A359927 where A359928(n) = [x^n*y^(n+1)/n!] (1/2) * log( Sum_{n>=0} (n^2 + 3*n*y + 2*y^2)^n * x^n/n! );
%e A359927 that is, the coefficient of x^n/n! in the logarithm of e.g.f A(x) equals the coefficient of y^(n+1)*x^n/n!, n >= 1, in the series given by
%e A359927 (b) (1/2) * log( Sum_{n>=0} (n^2 + 3*n*y + 2*y^2)^n * x^n/n! ) = x*(y^2 + 3/2*y + 1/2) + x^2/2!*(6*y^3 + 39/2*y^2 + 21*y + 15/2) + x^3/3!*(93*y^4 + 999/2*y^3 + 2055/2*y^2 + 1917/2*y + 683/2) + x^4/4!*(2448*y^5 + 19119*y^4 + 61704*y^3 + 102742*y^2 + 88080*y + 31019) + x^5/5!*(92505*y^6 + 1948347/2*y^5 + 8887325/2*y^4 + 11224575*y^3 + 16525750*y^2 + 26820135/2*y + 9342629/2) + x^6/6!*(4589568*y^7 + 61994772*y^6 + 374546664*y^5 + 1310466240*y^4 + 2862046080*y^3 + 3891543876*y^2 + 3041064504*y + 1050241608) + ...
%o A359927 (PARI) /* Using formula for the logarithm of g.f. A(x) */
%o A359927 {L(n) = (1/2) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^m*(m + 2*y)^m *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}
%o A359927 {a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
%o A359927 for(n=0, 30, print1(a(n), ", "))
%o A359927 (PARI) /* Using limit formula */
%o A359927 \p100
%o A359927 P(n) = sum(k=0, 31, ((n + k)*(n + 2*k))^k * x^k/k! +O(x^31))
%o A359927 Vec(round( serlaplace( subst(P(10^100)^(1/10^100), x, x/10^100) )*1.) )
%Y A359927 Cf. A359928, A319147, A266481, A318633.
%K A359927 nonn
%O A359927 0,3
%A A359927 _Paul D. Hanna_, Jan 20 2023