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%I #9 Mar 19 2024 04:58:09
%S 1,2,15,184,3325,79056,2345539,83505920,3472829721,165321395200,
%T 8868765212791,529513463098368,34831327847918485,2503184803456354304,
%U 195151614670701520875,16405316791445973139456,1479333355684885588136881,142443466217414911148359680,14587416733382035646737882591,1583199811285962289889116160000
%N a(n) = coefficient of x^n*y^(n+1)/n! in log( Sum_{n>=0} (n^2 + n*y + y^2)^n * x^n/n! ).
%C E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! equals the logarithm of the e.g.f. of A319147.
%H Paul D. Hanna, <a href="/A319834/b319834.txt">Table of n, a(n) for n = 1..100</a>
%F a(n) ~ c * d^n * n! / n^(5/2), where d = 6.1601834100761946... (same as for A319147) and c = 0.193396776821391327... - _Vaclav Kotesovec_, Mar 19 2024
%e E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 184*x^4/4! + 3325*x^5/5! + 79056*x^6/6! + 2345539*x^7/7! + 83505920*x^8/8! + 3472829721*x^9/9! + ...
%e Exponentiation yields the e.g.f. of A319147:
%e exp(A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4776*x^5/5! + 111967*x^6/6! + 3280264*x^7/7! + 115550073*x^8/8! +...+ A319147(n)*x^n/n! + ...
%e which equals
%e Limit_{N->oo} [ Sum_{n>=0} (N^2 + N*n + n^2)^n * (x/N)^n/n! ]^(1/N).
%e RELATED SEQUENCES.
%e a(n) is divisible by n where a(n)/n begins:
%e [1, 1, 5, 46, 665, 13176, 335077, 10438240, 385869969, 16532139520, ...].
%o (PARI) {a(n) = n! * polcoeff( polcoeff( log( sum(m=0, 2*n, (m^2 + m*y + y^2)^m *x^m/m! ) +x*O(x^(2*n)) ), n, x), n+1, y)}
%o for(n=1, 20, print1(a(n), ", "))
%Y Cf. A319147, A318634.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Sep 30 2018
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