A137974 -id:A137974 - cp's OEIS frontend

A137968 G.f. satisfies A(x) = 1 + x(1 + xA(x)^2)^6.

1, 1, 6, 27, 158, 981, 6342, 42728, 295008, 2079882, 14908740, 108312873, 795836544, 5903472999, 44151306690, 332552305818, 2520416719368, 19207222744326, 147086508325056, 1131292622149352, 8735383810590486

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

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

Cf. A137967, A137969; A137970, A137972, A137974.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^2)^6);polcoeff(A,n)}

a(n)=if(n==0,1,sum(k=0,n-1,binomial(6*(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)^6 where B(x) is the g.f. of A137967.
a(n) = Sum_{k=0..n-1} C(6*(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(6*s*(1-s)*(2-3*s) / ((44*s - 24)*Pi)) / (n^(3/2) * r^n), where r = 0.1201742080825038015263858974579392344239858277873... and s = 1.572098697306844482137442690518486437859864764710... are real roots of the system of equations s = 1 + r*(1 + r*s^2)^6, 12 * r^2 * s * (1 + r*s^2)^5 = 1. - Vaclav Kotesovec, Nov 22 2017

A137970 G.f. satisfies A(x) = 1 + x(1 + xA(x)^3)^6.

Original entry on oeis.org

1, 1, 6, 33, 236, 1776, 14148, 117070, 995568, 8653068, 76508562, 686035674, 6223653276, 57018806567, 526802616954, 4902775644477, 45919926029588, 432511043009679, 4094087001128088, 38927025591433926, 371607779425490280

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

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

Cf. A137969, A137971; A137968, A137972, A137974.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^3)^6);polcoeff(A,n)}

a(n)=if(n==0,1,sum(k=0,n-1,binomial(6*(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)^6 where B(x) is the g.f. of A137969.
a(n) = Sum_{k=0..n-1} C(6*(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(6*s*(1-s)*(3-4*s) / ((102*s - 72)*Pi)) / (n^(3/2) * r^n), where r = 0.0971328555591006631243189792661187629516513365080... and s = 1.377827066365760014851094517875193622070040930150... are real roots of the system of equations s = 1 + r*(1 + r*s^3)^6, 18 * r^2 * s^2 * (1 + r*s^3)^5 = 1. - Vaclav Kotesovec, Nov 22 2017

A137972 G.f. satisfies A(x) = 1 + x(1 + xA(x)^4)^6.

Original entry on oeis.org

1, 1, 6, 39, 320, 2787, 25788, 247731, 2449188, 24753960, 254610962, 2656496133, 28046838948, 299085697722, 3216723340218, 34852657892685, 380063012970680, 4168108473073596, 45941874232280862, 508664757809869052

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

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

Cf. A137971, A137973; A137968, A137970, A137974.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^4)^6);polcoeff(A,n)}

a(n)=if(n==0,1,sum(k=0,n-1,binomial(6*(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)^6 where B(x) is the g.f. of A137971.
a(n) = Sum_{k=0..n-1} C(6*(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(6*s*(1-s)*(4-5*s) / ((184*s - 144)*Pi)) / (n^(3/2) * r^n), where r = 0.0833821738312503523008482260558417829257343369560... and s = 1.287689442730957770948767878255357456556632139740... are real roots of the system of equations s = 1 + r*(1 + r*s^4)^6, 24 * r^2 * s^3 * (1 + r*s^4)^5 = 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

A137968 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^6.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A137970 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^3)^6.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A137972 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^4)^6.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A137968 G.f. satisfies A(x) = 1 + x(1 + xA(x)^2)^6.

A137970 G.f. satisfies A(x) = 1 + x(1 + xA(x)^3)^6.

A137972 G.f. satisfies A(x) = 1 + x(1 + xA(x)^4)^6.

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