A137968 -id:A137968 - cp's OEIS frontend

A137967 G.f. satisfies A(x) = 1 + x(1 + xA(x)^6)^2.

1, 1, 2, 13, 66, 406, 2602, 17271, 118444, 829514, 5914980, 42791085, 313277294, 2316793170, 17281455882, 129867946828, 982293317064, 7472406051744, 57132051350160, 438797394096378, 3383870656327576, 26191385476141936

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

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

Cf. A137968, A137966; A137952, A137955, A137960.

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

a(n)=if(n==0,1,sum(k=0,n-1,binomial(2*(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)^2 where B(x) is the g.f. of A137968.
a(n) = Sum_{k=0..n-1} C(2*(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(2*s*(1-s)*(6-7*s) / ((132*s - 120)*Pi)) / (n^(3/2) * r^n), where r = 0.1201742080825038015263858974579392344239858277873... and s = 1.297009871974239150024579315539982910111693413337... are real roots of the system of equations s = 1 + r*(1 + r*s^6)^2, 12 * r^2 * s^5 * (1 + r*s^6) = 1. - Vaclav Kotesovec, Nov 22 2017

A137969 G.f. satisfies A(x) = 1 + x(1 + xA(x)^6)^3.

Original entry on oeis.org

1, 1, 3, 21, 136, 1032, 8139, 66975, 567417, 4915386, 43350639, 387889254, 3512655498, 32133132074, 296496163113, 2756279003712, 25790064341592, 242699145598212, 2295564345035100, 21811226043019788, 208084639385653938

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

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

Cf. A137970, A137968; A137953, A137957, A137962.

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

a(n)=if(n==0,1,sum(k=0,n-1,binomial(3*(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)^3 where B(x) is the g.f. of A137970.
a(n) = Sum_{k=0..n-1} C(3*(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(3*s*(1-s)*(6-7*s) / ((204*s - 180)*Pi)) / (n^(3/2) * r^n), where r = 0.0971328555591006631243189792661187629516513365080... and s = 1.254068189138542668013320901661524162625316815207... are real roots of the system of equations s = 1 + r*(1 + r*s^6)^3, 18 * r^2 * s^5 * (1 + r*s^6)^2 = 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

A137974 G.f. satisfies A(x) = 1 + x(1 + xA(x)^5)^6.

Original entry on oeis.org

1, 1, 6, 45, 410, 4020, 41826, 452207, 5033910, 57300285, 663912420, 7804131660, 92838682242, 1115595461915, 13521340799310, 165104951405235, 2029162664033790, 25081468301798301, 311593507408597920

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

In general, if g.f. satisfies A(x) = 1 + x*(1 + x*A(x)^p)^q, p >= 1, q >= 1, p + q > 2, then a(n) ~ sqrt(q*s*(1-s)*(p*(1-s)-s) / (2*Pi*p*(q-s-p*q*(1-s)))) / (n^(3/2) * r^n), where r and s are real roots of the system of equations s = 1 + r*(1 + r*s^p)^q, p*q * r^2 * s^(p-1) * (1 + r*s^p)^(q-1) = 1. - Vaclav Kotesovec, Nov 22 2017

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

Cf. A137973; A137968, A137970, A137972.

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

a(n)=if(n==0,1,sum(k=0,n-1,binomial(6*(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)^6 where B(x) is the g.f. of A137973.
a(n) = Sum_{k=0..n-1} C(6*(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(6*s*(1-s)*(5-6*s) / ((290*s - 240)*Pi)) / (n^(3/2) * r^n), where r = 0.0739607593319208338998816978154858830062403258604... and s = 1.234938729532398384561936758596402363403570701060... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^6, 30 * r^2 * s^4 * (1 + r*s^5)^5 = 1. - Vaclav Kotesovec, Nov 22 2017

cp's OEIS Frontend

A137967 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^6)^2.

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A137969 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^6)^3.

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

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

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A137974 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^6.

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A137967 G.f. satisfies A(x) = 1 + x(1 + xA(x)^6)^2.

A137969 G.f. satisfies A(x) = 1 + x(1 + xA(x)^6)^3.

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.

A137974 G.f. satisfies A(x) = 1 + x(1 + xA(x)^5)^6.