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 1-4 of 4 results.

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.

A107591 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n*(n+1)/2).

Original entry on oeis.org

1, 1, 2, 6, 22, 91, 408, 1939, 9635, 49614, 263140, 1431301, 7959568, 45152340, 260847526, 1532825675, 9154581802, 55537885743, 342147577227, 2140251570508, 13594688301758, 87702596534110, 574815620158265, 3829029514213952
Offset: 0

Views

Author

Paul D. Hanna, May 17 2005, May 05 2010

Keywords

Examples

			A = 1 + x*A^1 + x^2*A^3 + x^3*A^6 + x^4*A^10 + x^5*A^15 ...
= 1 + (x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 91*x^6 +...)
+ (x^2 + 3*x^3 + 9*x^4 + 31*x^5 + 120*x^6 +...)
+ (x^3 + 6*x^4 + 27*x^5 + 116*x^6 +...)
+ (x^4 + 10*x^5 + 65*x^6 +...) +...
= 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 91*x^5 + 408*x^6 +...

Crossrefs

Cf. A107590, A107592, A155805, A219359.

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*(j+1)/2)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. A(x) = (1/x)*series-reversion(x/F(x)) and A(x) = F(x*A(x)) where F(x) = A(x/F(x)) is the g.f. of A107590.
G.f. A(x) = x/series-reversion(x*G(x)) and A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A107592.
Contribution from Paul D. Hanna, Apr 24 2010: (Start)
Let A = g.f. A(x), then A satisfies the continued fraction:
A = 1/(1- A*x/(1- A*(A-1)*x/(1- A^3*x/(1- A^2*(A^2-1)*x/(1- A^5*x/(1- A^3*(A^3-1)*x/(1- A^7*x/(1- A^4*(A^4-1)*x/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
(End)
Contribution from Paul D. Hanna, May 05 2010: (Start)
Let A = g.f. A(x), then A satisfies:
A = Sum_{n>=0} x^n*A^n*Product_{k=1..n} (1-x*A^(2k-1))/(1-x*A^(2k))
due to a q-series identity.
(End)

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

Views

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)

A107596 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n^2+n).

Original entry on oeis.org

1, 1, 3, 14, 80, 514, 3567, 26153, 199900, 1579107, 12816020, 106421359, 901430144, 7771535382, 68085001080, 605420138920, 5459655601753, 49904765136264, 462228258349278, 4337787743946224, 41249375376404380, 397572319756235577
Offset: 0

Views

Author

Paul D. Hanna, May 17 2005

Keywords

Examples

			A = 1 + x*A^2 + x^2*A^6 + x^3*A^12 + x^4*A^20 + x^5*A^30 ...
= 1 + (x + 2*x^2 + 7*x^3 + 34*x^4 + 197*x^5 + 1272*x^6 +...)
+ (x^2 + 6*x^3 + 33*x^4 + 194*x^5 + 1230*x^6 +...)
+ (x^3 + 12*x^4 + 102*x^5 + 784*x^6 +...)
+ (x^4 + 20*x^5 + 250*x^6 +...) +...
= 1 + x + 3*x^2 + 14*x^3 + 80*x^4 + 514*x^5 + 3567*x^6 +...

Crossrefs

Cf. A107592, A107594, A107595.

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^2+j)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. A(x)^2 = (1/x)*series-reversion(x/G107594(x)^2) and thus A(x) = G107594(x*A(x)^2) where G107594(x) is the g.f. of A107594. G.f. A(x) = (1/x)*series-reversion(x/G107595(x)) and thus A(x) = G107595(x*A(x)) where G107595(x) is the g.f. of A107595.
Contribution from Paul D. Hanna, Apr 25 2010: (Start)
Let A = g.f. A(x), then A satisfies the continued fraction:
A = 1/(1- A^2*x/(1- (A^4-A^2)*x/(1- A^6*x/(1- (A^8-A^4)*x/(1- A^10*x/(1- (A^12-A^6)*x/(1- A^14*x/(1- (A^16-A^8)*x/(1- A^18*x)))))))))
due to an identity of a partial elliptic theta function.
(End)
Showing 1-4 of 4 results.

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