cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A107590 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n*(n-1)/2).

Original entry on oeis.org

1, 1, 1, 2, 5, 15, 50, 181, 698, 2837, 12062, 53374, 244923, 1162536, 5697119, 28786266, 149814059, 802436166, 4420515689, 25031466730, 145616087486, 869760092469, 5330945435272, 33508699787635, 215863606818041
Offset: 0

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Author

Paul D. Hanna, May 17 2005

Keywords

Examples

			A = 1 + x + x^2*A^1 + x^3*A^3 + x^4*A^6 + x^5*A^10 +...
= 1 + x + (x^2 + x^3 + x^4 + 2*x^5 + 5*x^6 + 15*x^7 +...)
+ (x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 33*x^7 +...)
+ (x^4 + 6*x^5 + 21*x^6 + 62*x^7 +...)
+ (x^5 + 10*x^6 + 55*x^7 +...) +...
= 1 + x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 50*x^6 + 181*x^7 +...

Links

Crossrefs

Cf. A107591, A107592. A155804, A219358.

Programs

  • PARI
    {a(n) = my(A=1+x+x*O(x^n)); for(k=1,n, A = 1 + sum(j=1,n, x^j * A^(j*(j-1)/2) + x*O(x^n)) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = x/series-reversion(x*F(x)) and thus A(x) = F(x/A(x)) where F(x) = A(x*F(x)) is the g.f. of A107591.
G.f. A(x)^2 = x/series-reversion(x*G(x)^2) and thus A(x) = G(x/A(x)^2) where G(x) = A(x*G(x)^2) is the g.f. of A107592.
Contribution from Paul D. Hanna, Apr 25 2010: (Start)
Let A = g.f. A(x), then A satisfies the continued fraction:
A = 1/(1- x/(1- (A-1)*x/(1- A^2*x/(1- A*(A^2-1)*x/(1- A^4*x/(1- A^2*(A^3-1)*x/(1- A^6*x/(1- A^3*(A^4-1)*x/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
(End)

A155805 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * A(x)^(n(n+1)/2).

Original entry on oeis.org

1, 1, 3, 19, 191, 2656, 47392, 1034335, 26721781, 798018616, 27058991246, 1027237384009, 43172232488959, 1990253576425960, 99871804451808040, 5419775866582473211, 316301430225674131433, 19756213549154356027408
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 191*x^4/4! + 2656*x^5/5! +...
where e.g.f. A(x) satisfies:
A(x) = 1 + x*A(x) + x^2/2!*A(x)^3 + x^3/3!*A(x)^6 + x^4/4!*A(x)^10 +...
Let B(x) = A(x/B(x)) be the e.g.f. of A155804 then:
B(x) = 1 + x + x^2/2!*B(x) + x^3/3!*B(x)^3 + x^4/4!*B(x)^6 + x^5/5!*B(x)^10 +...
B(x) = 1 + x + x^2/2! + 4*x^3/3! + 19*x^4/4! + 161*x^5/5! + 1606*x^6/6! +...

Crossrefs

Cf. A155804, A155806, A155807, A107591, A219359.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+sum(k=1,n,x^k*A^(k*(k+1)/2)/k!+x*O(x^n))); n!*polcoeff(A,n)}

Formula

E.g.f. satisfies: A(x) = B(x*A(x)) and A(x/B(x)) = B(x) where B(x) satisfies:
B(x) = Sum_{n>=0} x^n/n! * B(x)^(n*(n-1)/2) and is the e.g.f. of A155804.

A107592 G.f. satisfies: A(x)^2 = Sum_{n>=0} x^n * A(x)^((n+1)*(n+2)/2).

Original entry on oeis.org

1, 1, 3, 13, 67, 382, 2327, 14855, 98208, 667180, 4632647, 32751382, 235072482, 1709232902, 12568852562, 93348649555, 699485096637, 5283685539096, 40205412111227, 308020225286402, 2374795521493354, 18419175004781334
Offset: 0

Views

Author

Paul D. Hanna, May 17 2005

Keywords

Examples

			A^2 = A + x*A^3 + x^2*A^6 + x^3*A^10 + x^4*A^15 + x^5*A^21 ...
= (1 + x + 3*x^2 + 13*x^3 + 67*x^4 + 382*x^5 + 2327*x^6 +...)
+ (x + 3*x^2 + 12*x^3 + 58*x^4 + 315*x^5 + 1848*x^6 +...)
+ (x^2 + 6*x^3 + 33*x^4 + 188*x^5 + 1122*x^6 +...)
+ (x^3 + 10*x^4 + 75*x^5 + 520*x^6 +...)
+ (x^4 + 15*x^5 + 150*x^6 +...) +...
= 1 + 2*x + 7*x^2 + 32*x^3 + 169*x^4 + 976*x^5 + 5989*x^6 +...

Crossrefs

Cf. A107590, A107591, A107593 (self-convolution).

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(k=1,n,A=1+sum(j=1,n,x^j*A^((j+1)*(j+2)/2-1)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. A(x)^2 = (1/x)*series-reversion(x/F(x)^2) and thus A(x) = F(x*A(x)^2) where F(x) is the g.f. of A107590.
G.f. A(x) = (1/x)*series-reversion(x/G(x)) and thus A(x) = G(x*A(x)) where G(x) is the g.f. of A107591.

A191809 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^binomial(n+2,3).

Original entry on oeis.org

1, 1, 2, 7, 32, 174, 1071, 7281, 53943, 432555, 3743146, 34934853, 351853883, 3827477399, 44985837602, 570985992828, 7814212692498, 115024461077654, 1815588345261996, 30628743324667923, 550414603283527315, 10503650627005928698
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 32*x^4 + 174*x^5 + 1071*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^4 + x^3*A(x)^10 + x^4*A(x)^20 + x^5*A(x)^35 + x^6*A(x)^56 + x^7*A(x)^84 +...+ x^n*A(x)^(n*(n+1)*(n+2)/3!) +...

Crossrefs

Cf. A107591, A191810, A191811, A191812.

Programs

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

A191810 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^binomial(n+3,4).

Original entry on oeis.org

1, 1, 2, 8, 44, 305, 2521, 24389, 273990, 3569531, 53944055, 944215131, 19065096323, 441174226355, 11609627641798, 344702951590401, 11463058468995522, 424180616752269732, 17366249924363207650, 782666399665891947949
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 44*x^4 + 305*x^5 + 2521*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^5 + x^3*A(x)^15 + x^4*A(x)^35 + x^5*A(x)^70 + x^6*A(x)^126 + x^7*A(x)^210 +...+ x^n*A(x)^(n*(n+1)*(n+2)*(n+3)/4!) +...

Crossrefs

Cf. A107591, A191809, A191811, A191812.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^binomial(m+3,4)));polcoeff(A,n)}

A191811 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^binomial(n+4,5).

Original entry on oeis.org

1, 1, 2, 9, 58, 501, 5452, 74211, 1257414, 26480393, 689598502, 21957924255, 844532153323, 38719749230469, 2091808065954023, 131835936103587004, 9607988537163939224, 803620426590302536069, 76622443259122023510169
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 58*x^4 + 501*x^5 + 5452*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^6 + x^3*A(x)^21 + x^4*A(x)^56 + x^5*A(x)^126 + x^6*A(x)^252 + x^7*A(x)^462 +...+ x^n*A(x)^(n*(n+1)*(n+2)*(n+3)*(n+4)/5!) +...

Crossrefs

Cf. A107591, A191809, A191810, A191812.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^binomial(m+4,5)));polcoeff(A,n)}

A191812 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^binomial(n+5,6).

Original entry on oeis.org

1, 1, 2, 10, 74, 782, 10982, 206346, 5142544, 168789842, 7201158787, 391194813232, 26651992683511, 2239377066821882, 229147222941318059, 28241058833042859637, 4149246030879282392144, 720738467750916348374860, 146838784937226592635807695
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 74*x^4 + 782*x^5 + 10982*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^7 + x^3*A(x)^28 + x^4*A(x)^84 + x^5*A(x)^210 + x^6*A(x)^462 + x^7*A(x)^924 +...+ x^n*A(x)^(n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)/6!) +...

Crossrefs

Cf. A107591, A191809, A191810, A191811.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^binomial(m+5,6)));polcoeff(A,n)}

A219359 G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^(n*(n+1)/2).

Original entry on oeis.org

1, 1, 3, 15, 99, 785, 7151, 73097, 827501, 10295549, 140100323, 2077648871, 33466994639, 583535328293, 10971899911703, 221571516903233, 4786126571515961, 110142864494603313, 2690257065904106571, 69500954207702580927, 1893141166438848208747, 54217997416999007587201
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 99*x^4 + 785*x^5 + 7151*x^6 +...
where
A(x) = 1 + 1!*x*A(x) + 2!*x^2*A(x)^3 + 3!*x^3*A(x)^6 + 4!*x^4*A(x)^10 + 5!*x^5*A(x)^15 +...

Crossrefs

Cf. A155805, A107591, A219358.

Programs

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

Formula

G.f. satisfies: A(x) = B(x*A(x)) and A(x/B(x)) = B(x) where B(x) satisfies:
B(x) = Sum_{n>=0} n!*x^n * B(x)^(n*(n-1)/2) and is the g.f. of A219358.

A107593 Self-convolution of A107592.

Original entry on oeis.org

1, 2, 7, 32, 169, 976, 5989, 38398, 254509, 1731596, 12032874, 85092944, 610714311, 4439136084, 32626373027, 242153129074, 1813069499846, 13682961621602, 104014376985334, 796004610604094, 6129901459731357, 47484532009772272
Offset: 0

Views

Author

Paul D. Hanna, May 17 2005

Keywords

Examples

			A = A^(1/2) + x*A^(3/2) + x^2*A^(6/2) + x^3*A^(10/2) +...
= 1 + 2*x + 7*x^2 + 32*x^3 + 169*x^4 + 976*x^5 + 5989*x^6 +...

Crossrefs

Cf. A107590, A107591, A107592.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(k=1,n,A=1+sum(j=1,n,x^j*A^((j+1)*(j+2)/2-1)+x*O(x^n)));polcoeff(A^2,n)}

Formula

G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^((n+1)*(n+2)/4).
G.f. A(x) = (1/x)*series-reversion(x/G(x)^2) and thus A(x) = G(x*A(x))^2 where G(x) is the g.f. of A107590.

A192259 G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 + A(x))^n * A(x)^(n*(n+1)/2).

Original entry on oeis.org

1, 2, 10, 74, 658, 6514, 69210, 773306, 8974114, 107288162, 1314003882, 16420439978, 208754062258, 2693915486418, 35228738082298, 466239274517274, 6238546207411778, 84330947396776642, 1150982783030893386, 15854319075541606666, 220344302315492953298, 3089322686040279975474, 43693043476823499717018, 63085549664634453982706, 6423320378114329801258421518738
Offset: 0

Views

Author

Paul D. Hanna, Jun 26 2011

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 74*x^3 + 658*x^4 + 6514*x^5 +...
Let A = g.f. A(x), then A satisfies:
A = 1 + x*(1+A)*A + x^2*(1+A)^2*A^3 + x^3*(1+A)^3*A^6 + x^4*(1+A)^4*A^10 +...
Equivalently,
A = 1 + x*(A + A^2) + x^2*(A^3 + 2*A^4 + A^5) + x^3*(A^6 + 3*A^7 + 3*A^8 + A^9) +...

Crossrefs

Cf. A107591, A192260.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m*(1+A)^m*(A+x*O(x^n))^(m*(m+1)/2)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*(1+A)^n*A^n * Product_{k=1..n} (1 - x*(1+A)*A^(2*k-1))/(1 - x*(1+A)*A^(2*k))
(2) A = 1/(1- A*(1+A)*x/(1- A*(A-1)*(1+A)*x/(1- A^3*(1+A)*x/(1- A^2*(A^2-1)*(1+A)*x/(1- A^5*(1+A)*x/(1- A^3*(A^3-1)*(1+A)*x/(1- A^7*(1+A)*x/(1- A^4*(A^4-1)*(1+A)*x/(1- ...))))))))) (continued fraction)
The above formulas are due to (1) a q-series identity and (2) a partial elliptic theta function expression.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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