A305596 -id:A305596 - cp's OEIS frontend

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.

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

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

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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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

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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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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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.

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.