A137970 -id:A137970 - 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

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

A137971 G.f. satisfies A(x) = 1 + x(1 + xA(x)^6)^4.

Original entry on oeis.org

1, 1, 4, 30, 232, 2037, 18720, 179454, 1770380, 17864490, 183510672, 1912621814, 20175123732, 214980182783, 2310645275932, 25021270486830, 272717638241172, 2989549949264304, 32938634975109864, 364566094737276708

Offset: 0

Paul D. Hanna, Feb 26 2008

nonn

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

Cf. A137972, A137970; A137956, A137958, A137964.

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

a(n)=if(n==0,1,sum(k=0,n-1,binomial(4*(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)^4 where B(x) is the g.f. of A137972.
a(n) = Sum_{k=0..n-1} C(4*(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(4*s*(1-s)*(6-7*s) / ((276*s - 240)*Pi)) / (n^(3/2) * r^n), where r = 0.0833821738312503523008482260558417829257343369560... and s = 1.229254439060935443156800948762443928645579909446... are real roots of the system of equations s = 1 + r*(1 + r*s^6)^4, 24 * r^2 * s^5 * (1 + r*s^6)^3 = 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

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

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

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A137971 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^6)^4.

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

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

A137971 G.f. satisfies A(x) = 1 + x(1 + xA(x)^6)^4.

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.