A305597 -id:A305597 - 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

A305598 O.g.f. A(x) satisfies: 0 = [x^n] exp( n(n-1) x * A(x) ) / A(x), for n > 1, with A'(0) = 2.

Original entry on oeis.org

1, 2, 6, 88, 2780, 116664, 6267112, 420608240, 34572479504, 3429310382112, 405606535944256, 56597546528049792, 9224898378261096640, 1739878267027277640320, 376408403518123974467840, 92655494325433865126295552, 25760341142562165894390054656, 8035192189219613566404655052288, 2794962730829750601921056190097920, 1078222968162147343121712468145172480

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 + 6*x^2 + 88*x^3 + 2780*x^4 + 116664*x^5 + 6267112*x^6 + 420608240*x^7 + 34572479504*x^8 + 3429310382112*x^9 + ...
RELATED SERIES.
A'(x)/A(x) = 2 + 8*x + 236*x^2 + 10424*x^3 + 554792*x^4 + 36154208*x^5 + 2853579584*x^6 + 269659065120*x^7 + 30227424970304*x^8 + ...

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

Cf. A305596, A305597.

{a(n) = my(A=[1,2], 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 = 29.852509905235658503543295008026... - 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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A305598 O.g.f. A(x) satisfies: 0 = [x^n] exp( n(n-1) x * A(x) ) / A(x), for n > 1, with A'(0) = 2.

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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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A305598 O.g.f. A(x) satisfies: 0 = [x^n] exp( n*(n-1) * x * A(x) ) / A(x), for n > 1, with A'(0) = 2.

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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.

A305598 O.g.f. A(x) satisfies: 0 = [x^n] exp( n(n-1) x * A(x) ) / A(x), for n > 1, with A'(0) = 2.