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 1-7 of 7 results.

A107595 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n^2).

Original entry on oeis.org

1, 1, 2, 7, 31, 158, 884, 5292, 33385, 219797, 1500449, 10573815, 76688602, 571232869, 4363912280, 34161879247, 273906591562, 2248935278231, 18909284838057, 162842178607893, 1436660527685476, 12988076148036405, 120345643023918566, 1143054910071718088, 11129160383826078389
Offset: 0

Views

Author

Paul D. Hanna, May 17 2005

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 158*x^5 + 884*x^6 + 5292*x^7 +...
Let A = g.f. A(x) then
A = 1 + x*A^1 + x^2*A^4 + x^3*A^9 + x^4*A^16 + x^5*A^25 ...
= 1 + x*(1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 158*x^5 + 884*x^6 +...)
+ x^2*(1 + 4*x + 14*x^2 + 56*x^3 + 257*x^4 + 1312*x^5 +...)
+ x^3*(1 + 9*x + 54*x^2 + 291*x^3 + 1557*x^4 + 8568*x^5 +..)
+ x^4*(1 + 16*x + 152*x^2 + 1152*x^3 + 7836*x^4 +...)
+ x^5*(1 + 25*x + 350*x^2 + 3675*x^3 + 32625*x^4 +...)
+ x^6*(1 + 36*x + 702*x^2 + 9912*x^3 + 114201*x^4 +...) +...
= 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 158*x^5 + 884*x^6 +...

Links

Crossrefs

Cf. A176719, A107590, A107594, A107596.

Programs

  • Mathematica
    m = 25; A[_] = 0;
Do[A[x_] = 1 + Sum[x^k A[x]^(k^2) + O[x]^j, {k, 1, j}], {j, m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 05 2019 *)
  • 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)+x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = (1/x)*Series_Reversion(x/F(x)) and thus A(x) = F(x*A(x)) where F(x) is the g.f. of A107594.
G.f. A(x) = x/Series_Reversion(x*G(x)) and thus A(x) = G(x/A(x)) where G(x) is the g.f. of A107596.
From Paul D. Hanna, Apr 23 2010: (Start)
Let A = g.f. A(x), then A satisfies the continued fraction:
A = 1/(1 - A*x/(1 - (A^3-A)*x/(1 - A^5*x/(1 - (A^7-A^3)*x/(1 - A^9*x/(1- (A^11-A^5)*x/(1 - A^13*x/(1 - (A^15-A^7)*x/(1 - ...)))))))))
due to an identity of a partial elliptic theta function. (End)
From Paul D. Hanna, May 04 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^(4k-3)) / (1 - x*A^(4k-1))
due to a q-series identity. (End)

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)

A155804 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, 1, 4, 19, 161, 1606, 21022, 323485, 5874913, 122077756, 2871573596, 75437801539, 2193468714373, 70020045331510, 2437979768144026, 92073099488632441, 3753886179551636513, 164556499026975482008
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 19*x^4/4! + 161*x^5/5! +...
where e.g.f. A(x) satisfies:
A(x) = 1 + x + x^2/2!*A(x) + x^3/3!*A(x)^3 + x^4/4!*A(x)^6 + x^5/5!*A(x)^10 +...
Let B(x) = A(x*B(x)) be the e.g.f. of A155805 then:
B(x) = 1 + x*B(x) + x^2/2!*B(x)^3 + x^3/3!*B(x)^6 + x^4/4!*B(x)^10 +...
B(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 191*x^4/4! + 2656*x^5/5! + 47392*x^6/6! +...

Crossrefs

Cf. A155805, A155806, A155807, A107590, A219358.

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 A155805.

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.

A107594 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n^2-n).

Original entry on oeis.org

1, 1, 1, 3, 10, 42, 194, 979, 5274, 30037, 179527, 1120612, 7280750, 49120810, 343547469, 2487670468, 18631824735, 144215785791, 1152745117570, 9508011730755, 80861962283808, 708502494881786, 6390084112199801, 59272034375915217, 564899767969587670
Offset: 0

Views

Author

Paul D. Hanna, May 17 2005

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 42*x^5 + 194*x^6 + 979*x^7 +...
Let A = A(x) then
A = 1 + x*A^0 + x^2*A^2 + x^3*A^6 + x^4*A^12 + x^5*A^20 + x^6*A^30 +...
= 1 + x + (x^2 + 2*x^3 + 3*x^4 + 8*x^5 + 27*x^6 + 110*x^7 +...)
+ (x^3 + 6*x^4 + 21*x^5 + 68*x^6 + 240*x^7 + 948*x^8 + 4140*x^9 +...)
+ (x^4 + 12*x^5 + 78*x^6 + 388*x^7 + 1737*x^8 + 7632*x^9 +...)
+ (x^5 + 20*x^6 + 210*x^7 + 1580*x^8 + 9795*x^9 +...)
+ (x^6 + 30*x^7 + 465*x^8 + 5020*x^9 +...) +...

Crossrefs

Cf. A107590, A107595, A107596.

Programs

  • Mathematica
    m = 25; A[_] = 0;
Do[A[x_] = 1 + x + Sum[x^k A[x]^(k^2 - k) + O[x]^j, {k, 2, j}], {j, m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 05 2019 *)
  • 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)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = x/series-reversion(x*G107595(x)) and thus A(x) = G107595(x/A(x)) where G107595(x) is the g.f. of A107595.
G.f. A(x)^2 = x/series-reversion(x*G107596(x)^2) and thus A(x) = G107596(x/A(x)^2) where G107596(x) is the g.f. of A107596.
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^2-1)*x/(1- A^4*x/(1- (A^6-A^2)*x/(1- A^8*x/(1- (A^10-A^4)*x/(1- A^12*x/(1- (A^14-A^6)*x/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
(End)

A357233 a(n) = coefficient of x^n in power series A(x) such that: 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).

Original entry on oeis.org

1, 1, 3, 11, 46, 207, 980, 4810, 24258, 124951, 654587, 3476985, 18682885, 101372340, 554655435, 3056823864, 16953795008, 94555853982, 529986289496, 2983788539017, 16865736120654, 95677703975144, 544554485912572, 3108656601838926, 17794927199793895
Offset: 0

Views

Author

Paul D. Hanna, Oct 17 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 46*x^4 + 207*x^5 + 980*x^6 + 4810*x^7 + 24258*x^8 + 124951*x^9 + 654587*x^10 + 3476985*x^11 + 18682885*x^12 + ...
such that
0 = 1 - A(x) + x*A(x)^3 - x^3*A(x)^6 + x^6*A(x)^10 - x^10*A(x)^15 + x^15*A(x)^21 - x^21*A(x)^28 + ... + (-1)^n*x^(n*(n-1)/2)*A(x)^(n*(n+1)/2) + ...
SPECIFIC VALUES.
A(1/7) = 1.2997111125331190764482142994969231...
A(1/8) = 1.221202992288263902503896694281250380662689...
CONTINUED FRACTION.
The continued fraction in formula (2) may be seen to converge to zero as a limit of successive steps that begin as follows:
[2] 1/(1 + A/(1 - A*(1 - x*A)))
[3] 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3)))
[4] 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3/(1 - x*A^2*(1 - x^2*A^2)))))
[5] 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3/(1 - x*A^2*(1 - x^2*A^2)/(1 + x^4*A^5)))))
[6] 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3/(1 - x*A^2*(1 - x^2*A^2)/(1 + x^4*A^5/(1 - x^2*A^3*(1 - x^3*A^3)))))))
...
substituting A = A(x), the resulting power series in x are:
[2] x^2 - 3*x^3 - 13*x^4 - 58*x^5 - 275*x^6 - 1350*x^7 + ...
[3] x^3 - 5*x^4 - 23*x^5 - 111*x^6 - 553*x^7 - 2820*x^8 + ...
[4] x^7 + 11*x^8 + 87*x^9 + 602*x^10 + 3894*x^11 + 24245*x^12 + ...
[5] x^9 + 14*x^10 + 132*x^11 + 1046*x^12 + 7538*x^13 + ...
[6] -x^15 - 21*x^16 - 273*x^17 - 2821*x^18 - 25432*x^19 + ...
[7] -x^18 - 25*x^19 - 375*x^20 - 4375*x^21 - 43800*x^22 + ...
[8] x^26 + 34*x^27 + 663*x^28 + 9725*x^29 + 119226*x^30 + ...
...
the limit of these series converges to zero for |x| < r < 1 where r is the radius of convergence of g.f. A(x).

Links

Crossrefs

Cf. A195980, A193111, A107590.

Programs

  • PARI
    {a(n) = my(A=[1],M=1); for(i=1,n, A = concat(A,0); M = ceil(sqrt(2*(#A)+1));
A[#A] = polcoeff( sum(n=0,M, (-1)^n * x^(n*(n-1)/2) * Ser(A)^(n*(n+1)/2) ), #A-1) ); A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas, some of which may use A = A(x) for brevity.
(1) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) 0 = 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3/(1 - x*A^2*(1 - x^2*A^2)/(1 + x^4*A^5/(1 - x^2*A^3*(1 - x^3*A^3)/(1 + x^6*A^7/(1 - x^3*A^4*(1 - x^4*A^4)/(1 + ...))))))))), a continued fraction due to an identity of a partial elliptic theta function.
(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A195980. - Paul D. Hanna, Jul 13 2023

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.
Showing 1-7 of 7 results.

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