A137961 -id:A137961 - cp's OEIS frontend

A137960 G.f. satisfies A(x) = 1 + x(1 + xA(x)^5)^2.

1, 1, 2, 11, 50, 275, 1560, 9212, 56082, 348675, 2207120, 14171155, 92075064, 604266000, 3999688050, 26670727220, 178997024610, 1208160130227, 8195828345756, 55849242272130, 382119958804520, 2624041637846210

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A137961, A137959; A137952, A137955, A137967.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5)^2);polcoeff(A,n)}

a(n)=if(n==0,1,sum(k=0,n-1,binomial(2*(n-k),k)/(n-k)*binomial(5*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

G.f.: A(x) = 1 + x*B(x)^2 where B(x) is the g.f. of A137961.
a(n) = Sum_{k=0..n-1} C(2*(n-k),k)/(n-k) * C(5*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(2*s*(1-s)*(5-6*s) / ((90*s - 80)*Pi)) / (n^(3/2) * r^n), where r = 0.1354712934479194768810666044866029126617104117352... and s = 1.354660923650925199331121807468321286698258863972... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^2, 10 * r^2 * s^4 * (1 + r*s^5) = 1. - Vaclav Kotesovec, Nov 22 2017

A137962 G.f. satisfies A(x) = 1 + x(1 + xA(x)^5)^3.

Original entry on oeis.org

1, 1, 3, 18, 106, 720, 5085, 37493, 284331, 2204973, 17404720, 139369905, 1129411314, 9244823986, 76326154857, 634847759955, 5314684735045, 44746683774474, 378652035541761, 3218705637379698, 27471657413667780

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Cf. A137963, A137961; A137953, A137957, A137969.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5)^3);polcoeff(A,n)}

a(n)=if(n==0,1,sum(k=0,n-1,binomial(3*(n-k),k)/(n-k)*binomial(5*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

G.f.: A(x) = 1 + x*B(x)^3 where B(x) is the g.f. of A137963.
a(n) = Sum_{k=0..n-1} C(3*(n-k),k)/(n-k) * C(5*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(3*s*(1-s)*(5-6*s) / ((140*s - 120)*Pi)) / (n^(3/2) * r^n), where r = 0.1085884782751570249717333800652227343328635496829... and s = 1.301018963559115613510052458264916439485131890857... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^3, 15 * r^2 * s^4 * (1 + r*s^5)^2 = 1. - Vaclav Kotesovec, Nov 22 2017

A137963 G.f. satisfies A(x) = 1 + x(1 + xA(x)^3)^5.

Original entry on oeis.org

1, 1, 5, 25, 160, 1075, 7671, 56760, 431865, 3357790, 26558520, 213032988, 1728808700, 14168337265, 117096909495, 974842628790, 8167462511193, 68813778610350, 582675107162175, 4955767502292960, 42318868510894860

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Cf. A137962, A137964; A137961, A137965, A137973.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^3)^5);polcoeff(A,n)}

a(n)=if(n==0,1,sum(k=0,n-1,binomial(5*(n-k),k)/(n-k)*binomial(3*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

G.f.: A(x) = 1 + x*B(x)^5 where B(x) is the g.f. of A137962.
a(n) = Sum_{k=0..n-1} C(5*(n-k),k)/(n-k) * C(3*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(5*s*(1-s)*(3-4*s) / ((84*s - 60)*Pi)) / (n^(3/2) * r^n), where r = 0.1085884782751570249717333800652227343328635496829... and s = 1.404764002126311415321709718173984955120001713401... are real roots of the system of equations s = 1 + r*(1 + r*s^3)^5, 15 * r^2 * s^2 * (1 + r*s^3)^4 = 1. - Vaclav Kotesovec, Nov 22 2017

A137965 G.f. satisfies A(x) = 1 + x(1 + xA(x)^4)^5.

Original entry on oeis.org

1, 1, 5, 30, 220, 1725, 14356, 124020, 1102770, 10023680, 92722620, 870039474, 8261024380, 79225392830, 766302511445, 7466883915800, 73227699088806, 722228333119200, 7159117292177840, 71284856957207030, 712673042497177450

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Cf. A137964, A137966; A137961, A137963, A137973.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^4)^5);polcoeff(A,n)}

a(n)=if(n==0,1,sum(k=0,n-1,binomial(5*(n-k),k)/(n-k)*binomial(4*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

G.f.: A(x) = 1 + x*B(x)^5 where B(x) is the g.f. of A137964.
a(n) = Sum_{k=0..n-1} C(5*(n-k),k)/(n-k) * C(4*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(5*s*(1-s)*(4-5*s) / ((152*s - 120)*Pi)) / (n^(3/2) * r^n), where r = 0.0927175295193852172913829423030505161354091369581... and s = 1.306924048536121339538817141295744998528778296640... are real roots of the system of equations s = 1 + r*(1 + r*s^4)^5, 20 * r^2 * s^3 * (1 + r*s^4)^4 = 1. - Vaclav Kotesovec, Nov 22 2017

A137973 G.f. satisfies A(x) = 1 + x(1 + xA(x)^6)^5.

Original entry on oeis.org

1, 1, 5, 40, 355, 3495, 36251, 391650, 4355810, 49550130, 573811635, 6742112506, 80175836395, 963137138105, 11670425726255, 142471372540290, 1750641388279500, 21634966222174020, 268734270298502640

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A137974, A137972; A137961, A137963, A137965.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^6)^5);polcoeff(A,n)}

a(n)=if(n==0,1,sum(k=0,n-1,binomial(5*(n-k),k)/(n-k)*binomial(6*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

G.f.: A(x) = 1 + x*B(x)^5 where B(x) is the g.f. of A137974.
a(n) = Sum_{k=0..n-1} C(5*(n-k),k)/(n-k) * C(6*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(5*s*(1-s)*(6-7*s) / ((348*s - 300)*Pi)) / (n^(3/2) * r^n), where r = 0.0739607593319208338998816978154858830062403258604... and s = 1.212436147090690045831533523759068212147683922018... are real roots of the system of equations s = 1 + r*(1 + r*s^6)^5, 30 * r^2 * s^5 * (1 + r*s^6)^4 = 1. - Vaclav Kotesovec, Nov 22 2017

cp's OEIS Frontend

A137960 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^2.

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A137962 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^3.

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A137963 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^3)^5.

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A137965 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^4)^5.

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A137973 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^6)^5.

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Formula

A137960 G.f. satisfies A(x) = 1 + x(1 + xA(x)^5)^2.

A137962 G.f. satisfies A(x) = 1 + x(1 + xA(x)^5)^3.

A137963 G.f. satisfies A(x) = 1 + x(1 + xA(x)^3)^5.

A137965 G.f. satisfies A(x) = 1 + x(1 + xA(x)^4)^5.

A137973 G.f. satisfies A(x) = 1 + x(1 + xA(x)^6)^5.