A137963 -id:A137963 - cp's OEIS frontend

A137961 G.f. satisfies A(x) = 1 + x(1 + xA(x)^2)^5.

1, 1, 5, 20, 105, 575, 3306, 19760, 121035, 757230, 4815530, 31039402, 202335855, 1331569725, 8834918160, 59035907480, 396937508816, 2683518356850, 18230570856710, 124390574548960, 852074529347120, 5857453791938135

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

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

Cf. A137960, A137962; A137963, A137965, A137973.

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

a(n)=if(n==0,1,sum(k=0,n-1,binomial(5*(n-k),k)/(n-k)*binomial(2*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 A137960.
a(n) = Sum_{k=0..n-1} C(5*(n-k),k)/(n-k) * C(2*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(5*s*(1-s)*(2-3*s) / ((36*s - 20)*Pi)) / (n^(3/2) * r^n), where r = 0.1354712934479194768810666044866029126617104117352... and s = 1.618016818966151244027202981456410137451426090894... are real roots of the system of equations s = 1 + r*(1 + r*s^2)^5, 10 * r^2 * s * (1 + r*s^2)^4 = 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

A137964 G.f. satisfies A(x) = 1 + x(1 + xA(x)^5)^4.

Original entry on oeis.org

1, 1, 4, 26, 184, 1451, 12020, 103734, 921132, 8364877, 77317704, 725029730, 6880482816, 65955731874, 637703938860, 6211709281162, 60900108419200, 600486291654444, 5950951929703520, 59242473406384472, 592166933647780576

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

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

Cf. A137965, A137963; A137956, A137958, A137971.

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

a(n)=if(n==0,1,sum(k=0,n-1,binomial(4*(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)^4 where B(x) is the g.f. of A137965.
a(n) = Sum_{k=0..n-1} C(4*(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(4*s*(1-s)*(5-6*s) / ((190*s - 160)*Pi)) / (n^(3/2) * r^n), where r = 0.0927175295193852172913829423030505161354091369581... and s = 1.270497495855793662015513509713357933752729700697... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^4, 20 * r^2 * s^4 * (1 + r*s^5)^3 = 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

A137961 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^5.

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

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A137964 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^4.

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

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PARI

PARI

Formula

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

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Formula

A137961 G.f. satisfies A(x) = 1 + x(1 + xA(x)^2)^5.

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

A137964 G.f. satisfies A(x) = 1 + x(1 + xA(x)^5)^4.

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.