A305598 -id:A305598 - cp's OEIS frontend

A305596 O.g.f. A(x) satisfies: 0 = [x^n] exp( n(n-1) x * A(x) ) / A(x), for n > 0, with A'(0) = 0.

1, 0, 2, 36, 1012, 39344, 1999736, 128430272, 10191730992, 983072197248, 113716916603648, 15586891405986048, 2503750145139262912, 466531385595202181888, 99898407773515906674688, 24374095428098168225056256, 6724465905018382760077058816, 2083282714601993506101791682560, 720279202970620106946642875741696, 276363182440771615371629345051272192, 117079396081246222639524111231517394944

Offset: 0

Paul D. Hanna, Jun 05 2018

nonn

It is remarkable that this sequence should consist entirely of integers.

			O.g.f.: A(x) = 1 + 2*x^2 + 36*x^3 + 1012*x^4 + 39344*x^5 + 1999736*x^6 + 128430272*x^7 + 10191730992*x^8 + 983072197248*x^9 + ...
RELATED SERIES.
A'(x)/A(x) = 4*x + 108*x^2 + 4040*x^3 + 196360*x^4 + 11982400*x^5 + 898207072*x^6 + 81486477600*x^7 + 8844334636032*x^8 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A305597, A305598, A304861.

{a(n) = my(A=[1,0], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m*(m-1)*x*Ser(A)) / Ser(A) )[m+1] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * n^(2*n + 2) / exp(2*n), where c = 6.9180696045148043278035608619439... - Vaclav Kotesovec, Aug 11 2021

A305597 O.g.f. A(x) satisfies: 0 = [x^n] exp( n(n-1) x * A(x) ) / A(x), for n > 1, with A'(0) = 1.

Original entry on oeis.org

1, 1, 3, 59, 1775, 71511, 3735265, 245211865, 19803108233, 1936950231585, 226553844824131, 31331105054010115, 5069552336811706983, 950370245531340684983, 204551803567400710962529, 50129834142929585060592433, 13883636379729966042468837937, 4315912911594085891292265635265, 1496719856496801168910452641821123, 575834703501821334832940981183532907

Offset: 0

Paul D. Hanna, Jun 05 2018

nonn

It is remarkable that this sequence should consist entirely of integers.

			O.g.f.: A(x) = 1 + x + 3*x^2 + 59*x^3 + 1775*x^4 + 71511*x^5 + 3735265*x^6 + 245211865*x^7 + 19803108233*x^8 + 1936950231585*x^9 + ...
RELATED SERIES.
A'(x)/A(x) = 1 + 5*x + 169*x^2 + 6857*x^3 + 348121*x^4 + 21952541*x^5 + 1688688793*x^6 + 156361635585*x^7 + 17247060489337*x^8 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A305596, A305598.

{a(n) = my(A=[1,1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m*(m-1)*x*Ser(A)) / Ser(A) )[m+1] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * n^(2*n + 2) / exp(2*n), where c = 15.159858073932625358279885219043... - Vaclav Kotesovec, Aug 11 2021

cp's OEIS Frontend

A305596 O.g.f. A(x) satisfies: 0 = [x^n] exp( n(n-1) x * A(x) ) / A(x), for n > 0, with A'(0) = 0.

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A305597 O.g.f. A(x) satisfies: 0 = [x^n] exp( n(n-1) x * A(x) ) / A(x), for n > 1, with A'(0) = 1.

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cp's OEIS Frontend

A305596 O.g.f. A(x) satisfies: 0 = [x^n] exp( n*(n-1) * x * A(x) ) / A(x), for n > 0, with A'(0) = 0.

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A305597 O.g.f. A(x) satisfies: 0 = [x^n] exp( n*(n-1) * x * A(x) ) / A(x), for n > 1, with A'(0) = 1.

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A305596 O.g.f. A(x) satisfies: 0 = [x^n] exp( n(n-1) x * A(x) ) / A(x), for n > 0, with A'(0) = 0.

A305597 O.g.f. A(x) satisfies: 0 = [x^n] exp( n(n-1) x * A(x) ) / A(x), for n > 1, with A'(0) = 1.