A107595 -id:A107595 - cp's OEIS frontend

A157134 G.f. satisfies: A(x) = Sum_{n>=0} x^(n^2) * A(x)^n.

1, 1, 1, 1, 2, 4, 7, 11, 18, 33, 63, 117, 211, 383, 713, 1348, 2547, 4793, 9039, 17165, 32785, 62761, 120243, 230768, 444119, 857015, 1656931, 3207990, 6219994, 12079544, 23496417, 45767352, 89256038, 174269488, 340646238, 666604642

Offset: 0

Paul D. Hanna, Feb 24 2009

nonn

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 11*x^7 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 27*x^6 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 10*x^3 + 18*x^4 + 36*x^5 + 73*x^6 +...
A(x)^4 = 1 + 4*x + 10*x^2 + 20*x^3 + 39*x^4 + 80*x^5 + 168*x^6 +...
where
A(x) = 1 + x*A(x) + x^4*A(x)^2 + x^9*A(x)^3 + x^16*A(x)^4 +...

Cf. A157135, A157133, A157136.

Cf. A107595. [From Paul D. Hanna, Apr 25 2010]

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

G.f. satisfies: A(x) = B(x/A(x)) where B(x) = A(x*B(x)) = g.f. of A157135,
where A157135(n) = [x^n] A(x)^(n+1)/(n+1) for n>=0,
and a(n) = [x^n] -1/B(x)^(n-1)/(n-1) for n>1.
From Paul D. Hanna, Apr 25 2010: (Start)
G.f. A(x) satisfies the continued fraction:
A(x) = 1/(1- x*A(x)/(1- (x^3-x)*A(x)/(1- x^5*A(x)/(1- (x^7-x^3)*A(x)/(1- x^9*A(x)/(1- (x^11-x^5)*A(x)/(1- x^13*A(x)/(1- (x^15-x^7)*A(x)/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
(End)
From Paul D. Hanna, May 05 2010: (Start)
Let A = g.f. A(x) at x=q, then A satisfies the q-series:
A = Sum_{n>=0} q^n*A^n*Product_{k=1..n} (1-q^(4k-3)*A)/(1-q^(4k-1)*A).
(End)

Typo in data corrected by D. S. McNeil, Aug 17 2010

A176719 G.f. satisfies: A(x) = 1 + Sum_{n>=1} 2x^n A(x)^(n^2).

Original entry on oeis.org

1, 2, 6, 30, 194, 1442, 11782, 103102, 951554, 9173186, 91780614, 948985822, 10110931650, 110794764642, 1247186300934, 14412811665278, 170949340705794, 2081185257723778, 26012832364535814, 333929563132811678, 4404347475363755714, 59705917899701420834, 832080588205468939782

Offset: 0

Paul D. Hanna, Apr 25 2010

nonn

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 30*x^3 + 194*x^4 + 1442*x^5 +...
A(x) = 1 + 2*x*A(x) + 2*x^2*A(x)^4 + 2*x^3*A(x)^9 + 2*x^4*A(x)^16 + ...
Contribution from _Paul D. Hanna_, May 11 2010: (Start)
Given g.f. A(x), then Q = A(-x^2) satisfies the q-series:
Q/(1-x) = 1 + x*(xQ;Q)_1/(-xQ;Q)_1 + x^2*(xQ;Q)_2/(-xQ;Q)_2 +...
where the q-Pochhammer symbol (x;q)_n = Product_{k=0..n-1} (1-x*q^k). (End)

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A107595, A176720.

{a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,2*x^m*(A+x*O(x^n))^(m^2)));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Contribution from Paul D. Hanna, May 11 2010: (Start)
Given g.f. A(x), then Q = A(-x^2) satisfies:
Q = (1-x)*Sum_{n>=0} x^n*Product_{k=1..n} (1 - x*Q^k)/(1 + x*Q^k)
due to a q-series expansion for the Jacobi theta_4 function. (End)

Edited by Paul D. Hanna, Apr 27 2010

A191803 G.f. satisfies: A(x) = Sum_{n>=0} x^nA(x)^(5n^2).

Original entry on oeis.org

1, 1, 6, 61, 791, 11701, 188462, 3225915, 57840755, 1076423857, 20666351126, 407645638428, 8237858879315, 170229866493435, 3592746391559133, 77393340642273491, 1701286171473636404, 38169860244429063080

Offset: 0

Paul D. Hanna, Jun 16 2011

nonn

			G.f.: A(x) = 1 + x + 6*x^2 + 61*x^3 + 791*x^4 + 11701*x^5 + 188462*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x)^5 + x^2*A(x)^20 + x^3*A(x)^45 + x^4*A(x)^80 +...+ x^n*A(x)^(5*n^2) +...

Cf. A107595, A191800, A191801, A191802, A191804.

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

Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*A^(5*n)*Product_{k=1..n} (1-x*A^(20*k-15))/(1-x*A^(20*k-5));
(2) A = 1/(1- A^5*x/(1- A^5*(A^10-1)*x/(1- A^25*x/(1- A^15*(A^20-1)*x/(1- A^45*x/(1- A^25*(A^30-1)*x/(1- A^65*x/(1- A^35*(A^40-1)*x/(1- ...))))))))) (continued fraction);
due to a q-series identity and an identity of a partial elliptic theta function, respectively.

A191800 G.f. satisfies: A(x) = Sum_{n>=0} x^nA(x)^(2n^2).

Original entry on oeis.org

1, 1, 3, 16, 109, 851, 7275, 66393, 637239, 6371848, 65961782, 703953599, 7722738071, 86924392498, 1002603956938, 11842465020207, 143208130730229, 1773099186411938, 22483740028949531, 292129222113885503, 3891268435685371911

Offset: 0

Paul D. Hanna, Jun 16 2011

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 109*x^4 + 851*x^5 + 7275*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x)^2 + x^2*A(x)^8 + x^3*A(x)^18 + x^4*A(x)^32 +...+ x^n*A(x)^(2*n^2) +...

Cf. A107595, A191801, A191802, A191803, A191804.

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

Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*A^(2*n)*Product_{k=1..n} (1-x*A^(8*k-6))/(1-x*A^(8*k-2));
(2) A = 1/(1- A^2*x/(1- A^2*(A^4-1)*x/(1- A^10*x/(1- A^6*(A^8-1)*x/(1- A^18*x/(1- A^10*(A^12-1)*x/(1- A^26*x/(1- A^14*(A^16-1)*x/(1- ...))))))))) (continued fraction);
due to a q-series identity and an identity of a partial elliptic theta function, respectively.

A191801 G.f. satisfies: A(x) = Sum_{n>=0} x^nA(x)^(3n^2).

Original entry on oeis.org

1, 1, 4, 28, 251, 2573, 28813, 343833, 4308210, 56154805, 756731761, 10499096630, 149551069156, 2182935186698, 32613646656198, 498420592612153, 7790219357236805, 124545937719356873, 2037614647316548891, 34134979366157116560

Offset: 0

Paul D. Hanna, Jun 16 2011

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 251*x^4 + 2573*x^5 + 28813*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x)^3 + x^2*A(x)^12 + x^3*A(x)^27 + x^4*A(x)^48 +...+ x^n*A(x)^(3*n^2) +...

Cf. A107595, A191800, A191802, A191803, A191804.

{a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^(3*m^2)));polcoeff(A,n)}

Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*A^(3*n)*Product_{k=1..n} (1-x*A^(12*k-9))/(1-x*A^(12*k-3));
(2) A = 1/(1- A^3*x/(1- A^3*(A^6-1)*x/(1- A^15*x/(1- A^9*(A^12-1)*x/(1- A^27*x/(1- A^15*(A^18-1)*x/(1- A^39*x/(1- A^21*(A^24-1)*x/(1- ...))))))))) (continued fraction);
due to a q-series identity and an identity of a partial elliptic theta function, respectively.

A191802 G.f. satisfies: A(x) = Sum_{n>=0} x^nA(x)^(4n^2).

Original entry on oeis.org

1, 1, 5, 43, 473, 5942, 81393, 1186342, 18132473, 287948903, 4722077279, 79636530163, 1377304530677, 24382127678100, 441294262119031, 8160739579770316, 154169018332135841, 2975846752734820345, 58718914018159811186

Offset: 0

Paul D. Hanna, Jun 16 2011

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 43*x^3 + 473*x^4 + 5942*x^5 + 81393*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x)^4 + x^2*A(x)^16 + x^3*A(x)^36 + x^4*A(x)^64 +...+ x^n*A(x)^(4*n^2) +...

Cf. A107595, A191800, A191801, A191803, A191804.

{a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^(4*m^2)));polcoeff(A,n)}

Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*A^(4*n)*Product_{k=1..n} (1-x*A^(16*k-12))/(1-x*A^(16*k-4));
(2) A = 1/(1- A^4*x/(1- A^4*(A^8-1)*x/(1- A^20*x/(1- A^12*(A^16-1)*x/(1- A^36*x/(1- A^20*(A^24-1)*x/(1- A^52*x/(1- A^28*(A^32-1)*x/(1- ...))))))))) (continued fraction);
due to a q-series identity and an identity of a partial elliptic theta function, respectively.

A191804 G.f. satisfies: A(x) = Sum_{n>=0} x^nA(x)^(6n^2).

Original entry on oeis.org

1, 1, 7, 82, 1221, 20718, 382315, 7489683, 153551487, 3264643144, 71545452946, 1609541143713, 37065029428453, 872037022019930, 20935244357544798, 512498682139660135, 12790021472251565047, 325439165493879484025

Offset: 0

Paul D. Hanna, Jun 16 2011

nonn

			G.f.: A(x) = 1 + x + 7*x^2 + 82*x^3 + 1221*x^4 + 20718*x^5 + 382315*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x)^6 + x^2*A(x)^24 + x^3*A(x)^54 + x^4*A(x)^96 +...+ x^n*A(x)^(6*n^2) +...

Cf. A107595, A191800, A191801, A191802, A191803.

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

Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*A^(6*n)*Product_{k=1..n} (1-x*A^(24*k-18))/(1-x*A^(24*k-6));
(2) A = 1/(1- A^6*x/(1- A^6*(A^12-1)*x/(1- A^30*x/(1- A^18*(A^24-1)*x/(1- A^54*x/(1- A^30*(A^36-1)*x/(1- A^78*x/(1- A^42*(A^48-1)*x/(1- ...))))))))) (continued fraction);
due to a q-series identity and an identity of a partial elliptic theta function, respectively.

A301929 G.f. A(x) satisfies: x = Sum_{n>=1} (1+x)^(n^2) * x^n / A(x)^n.

Original entry on oeis.org

1, 2, 3, 5, 12, 37, 138, 595, 2843, 14844, 83212, 496473, 3128584, 20707672, 143342216, 1034075244, 7752274237, 60251286521, 484483164365, 4023459643530, 34455215830001, 303839675537827, 2755675307738286, 25675275100067189, 245502965520844801, 2406797239543382867, 24170220195274548727, 248441483165679473094, 2611787614440970964621

Offset: 0

Paul D. Hanna, May 04 2018

nonn

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 5*x^3 + 12*x^4 + 37*x^5 + 138*x^6 + 595*x^7 + 2843*x^8 + 14844*x^9 + 83212*x^10 + 496473*x^11 + 3128584*x^12 + ...
such that
x = (1+x)*x/A(x) + (1+x)^4*x^2/A(x)^2 + (1+x)^9*x^3/A(x)^3 + (1+x)^16*x^4/A(x)^4 + (1+x)^25*x^5/A(x)^5 + (1+x)^36*x^6/A(x)^6 + (1+x)^49*x^7/A(x)^7 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A301927, A303058, A107595.

{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(n=0, #A, ((1+x)^n +x*O(x^#A))^n * x^n/Ser(A)^n ) )[#A+1] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f.: x = Sum_{n>=1} x^n/A(x)^n * (1+x)^n * Product_{k=1..n} (A(x) - x*(1+x)^(4*k-3)) / (A(x) - x*(1+x)^(4*k-1)), due to a q-series identity.

G.f.: 1+x = 1/(1 - q*x/(A(x) - q*(q^2-1)*x/(1 - q^5*x/(A(x) - q^3*(q^4-1)*x/(1 - q^9*x/(A(x) - q^5*(q^6-1)*x/(1 - q^13*x/(A(x) - q^7*(q^8-1)*x/(1 - ...))))))))), where q = (1+x), a continued fraction due to a partial elliptic theta function identity.

A191805 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(n^3).

Original entry on oeis.org

1, 1, 2, 11, 83, 809, 9503, 130107, 2056768, 37137351, 761543233, 17637050418, 458225405825, 13265643258608, 424971569670808, 14963974979730138, 575619170223815326, 24056794697473352763, 1087014831018615150024

Offset: 0

Paul D. Hanna, Jun 16 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 83*x^4 + 809*x^5 + 9503*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^8 + x^3*A(x)^27 + x^4*A(x)^64 + x^5*A(x)^125 + x^6*A(x)^216 +...+ x^n*A(x)^(n^3) +...

Cf. A107595, A191806, A191807, A191808.

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

A191806 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(n^4).

Original entry on oeis.org

1, 1, 2, 19, 253, 5256, 153121, 5793349, 292530822, 18658710139, 1476004466687, 143228682526672, 16603062548806759, 2272210780577578355, 363396388117576899042, 67028665570181029621005, 14142153576677394736652147

Offset: 0

Paul D. Hanna, Jun 16 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 19*x^3 + 253*x^4 + 5256*x^5 + 153121*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^16 + x^3*A(x)^81 + x^4*A(x)^256 + x^5*A(x)^625 + x^6*A(x)^1296 +...+ x^n*A(x)^(n^4) +...

Cf. A107595, A191805, A191807, A191808.

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

cp's OEIS Frontend

A157134 G.f. satisfies: A(x) = Sum_{n>=0} x^(n^2) * A(x)^n.

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A176719 G.f. satisfies: A(x) = 1 + Sum_{n>=1} 2*x^n * A(x)^(n^2).

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A191803 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(5*n^2).

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A191800 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(2*n^2).

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A191801 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(3*n^2).

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A191802 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(4*n^2).

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A191804 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(6*n^2).

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A301929 G.f. A(x) satisfies: x = Sum_{n>=1} (1+x)^(n^2) * x^n / A(x)^n.

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A191805 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(n^3).

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A176719 G.f. satisfies: A(x) = 1 + Sum_{n>=1} 2x^n A(x)^(n^2).

A191803 G.f. satisfies: A(x) = Sum_{n>=0} x^nA(x)^(5n^2).

A191800 G.f. satisfies: A(x) = Sum_{n>=0} x^nA(x)^(2n^2).

A191801 G.f. satisfies: A(x) = Sum_{n>=0} x^nA(x)^(3n^2).

A191802 G.f. satisfies: A(x) = Sum_{n>=0} x^nA(x)^(4n^2).

A191804 G.f. satisfies: A(x) = Sum_{n>=0} x^nA(x)^(6n^2).