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A360339 a(n) = coefficient of x^n*y^(2*n+1)/n! in log( Sum_{n>=0} (n + y)^(3*n) * x^n/n! ).

%I A360339 #13 Mar 20 2024 07:55:17
%S A360339 1,6,99,2832,117405,6423408,438143391,35869775616,3430351996569,
%T A360339 375544727136000,46333978359977979,6362713275564589056,
%U A360339 962689133095843525749,159139760744994666835968,28539360163037720058960375,5518961894002049077780611072,1144859158421455331276272257201
%N A360339 a(n) = coefficient of x^n*y^(2*n+1)/n! in log( Sum_{n>=0} (n + y)^(3*n) * x^n/n! ).
%H A360339 Paul D. Hanna, <a href="/A360339/b360339.txt">Table of n, a(n) for n = 1..100</a>
%F A360339 E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! may be defined as follows.
%F A360339 (1) A(x) = Limit_{N->oo} (1/N) * log( Sum_{n>=0} (N + n)^(3*n) * (x/N^2)^n/n! ).
%F A360339 (2) a(n) = [x^n*y^(2*n+1)/n!] log( Sum_{n>=0} (n + y)^(3*n) * x^n/n! ).
%F A360339 a(n) ~ c * d^n * n! / n^(5/2), where d = (3/2) * (3 + sqrt(6)) * exp(3 - sqrt(6)) = 14.175247991325192557234088913125084764719990898660219459... and c = sqrt((3 - sqrt(6))/Pi)/4 = 0.1046520596183180437324097699670683850916674939335504... - _Vaclav Kotesovec_, Feb 12 2023, updated Mar 20 2024
%e A360339 E.g.f.: A(x) = x + 6*x^2/2! + 99*x^3/3! + 2832*x^4/4! + 117405*x^5/5! + 6423408*x^6/6! + 438143391*x^7/7! + 35869775616*x^8/8! + ... + a(n)*x^n/n! + ...
%e A360339 where a(n) equals the coefficient of y^(2*n+1)*x^n/n! in the series given by
%e A360339 log( Sum_{n>=0} (n + y)^(3*n) * x^n/n! ) = (y^3 + 3*y^2 + 3*y + 1)*x + (6*y^5 + 45*y^4 + 140*y^3 + 225*y^2 + 186*y + 63)*x^2/2! + (99*y^7 + 1305*y^6 + 7722*y^5 + 26514*y^4 + 56844*y^3 + 75780*y^2 + 57915*y + 19493)*x^3/3! + (2832*y^9 + 56214*y^8 + 521784*y^7 + 2965716*y^6 + 11339280*y^5 + 30131946*y^4 + 55424512*y^3 + 67771380*y^2 + 49792368*y + 16686958)*x^4/4! + (117405*y^11 + 3214647*y^10 + 42201705*y^9 + 349928235*y^8 + 2030468625*y^7 + 8627152275*y^6 + 27284511927*y^5 + 63980788365*y^4 + 108602299435*y^3 + 126629082945*y^2 + 90978438315*y + 30421607649)*x^5/5! + ...
%e A360339 Exponentiation yields the e.g.f. of A266482:
%e A360339 exp(A(x)) = 1 + x + 7*x^2/2! + 118*x^3/3! + 3373*x^4/4! + 139096*x^5/5! + 7565779*x^6/6! + 513277024*x^7/7! + 41820455065*x^8/8! + 3982842285184*x^9/9! + 434457816912991*x^10/10! + ... + A266483(n)*x^n/n! + ...
%e A360339 which equals
%e A360339 lim_{N->oo} [ Sum_{n>=0} (N + n)^(3*n) * (x/N^2)^n/n! ]^(1/N).
%e A360339 RELATED SEQUENCES.
%e A360339 a(n) is divisible by n where a(n)/n begins:
%e A360339 [1, 3, 33, 708, 23481, 1070568, 62591913, 4483721952, ...].
%o A360339 (PARI) /* Using logarithmic formula */
%o A360339 {a(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(3*m) *x^m/m! ) +x*O(x^n) ), n, x), 2*n+1, y)}
%o A360339 for(n=1, 20, print1(a(n), ", "))
%Y A360339 Cf. A266482, A360340, A360341, A359926.
%K A360339 nonn
%O A360339 1,2
%A A360339 _Paul D. Hanna_, Feb 10 2023