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A359928
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a(n) = coefficient of x^n*y^(n+1)/n! in (1/2) * log( Sum_{n>=0} (n^2 + 3*n*y + 2*y^2)^n * x^n/n! ).
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4
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1, 6, 93, 2448, 92505, 4589568, 283008621, 20903023872, 1800986581521, 177455695795200, 19690717755237309, 2430478269127673856, 330392930155527272553, 49053029845102480576512, 7898602773992589665290125, 1371137549213022697047785472, 255275516636592894833768588961
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OFFSET
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1,2
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LINKS
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FORMULA
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E.g.f. A(x) = Sum_{n>=1} a(n) * x^n/n! may be defined by the following.
(1) a(n) = [x^n*y^(n+1)/n!] (1/2)*log( Sum_{n>=0} (n + y)^n*(n + 2*y)^n *x^n/n! ).
(2) A(x) = lim_{N->oo} (1/N)*log( Sum_{n>=0} (N + n)^n*(N + 2*n)^n * (x/N)^n/n! ).
a(n) ~ c * d^n * n! / n^(5/2), where d = 12.7029497597456784744445675253711147535742245945208995646083627... and c = 0.15440395598650604464793307483290467035754174771895993579108... - Vaclav Kotesovec, Mar 21 2024
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EXAMPLE
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E.g.f.: 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! + 1800986581521*x^9/9! + 177455695795200*x^10/10! + ...
Exponentiation yields the e.g.f. of A319147:
exp(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! + ... + A319147(n)*x^n/n! + ...
which equals the limit
exp(A(x)) = lim_{N->oo} [ Sum_{n>=0} (N^2 + 3*N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).
RELATED SEQUENCES.
a(n) is divisible by n where a(n)/n begins:
[1, 3, 31, 612, 18501, 764928, 40429803, 2612877984, 200109620169, ...].
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PROG
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(PARI) {a(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)}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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