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 1-6 of 6 results.

A193114 G.f. A(x) satisfies 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(n+1).

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 111, 345, 1103, 3604, 11977, 40356, 137543, 473317, 1642258, 5738828, 20179338, 71346433, 253485527, 904536366, 3240418665, 11649734335, 42017535527, 151992797355, 551298507620, 2004602732825, 7305747551718, 26682235709115
Offset: 0

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Author

Paul D. Hanna, Jul 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 37*x^5 + 111*x^6 + ...
which satisfies:
1 = A(x) - x*A(x)^2 + x^4*A(x)^3 - x^9*A(x)^4 + x^16*A(x)^5 -+ ...
Related expansions.
A(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 40*x^4 + 120*x^5 + 373*x^6 + ...
A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 87*x^4 + 276*x^5 + 893*x^6 + ...

Links

Crossrefs

Cf. A193111, A193112, A193113, A193115, A193116.

Programs

  • Maple
    e36:= 1 - add((-x)^(n^2)*a^(n+1),n=0..6):
S:= series(RootOf(e36,a),x,37):
seq(coeff(S,x,i),i=0..36); # Robert Israel, Apr 10 2023
  • PARI
    {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(#A)+1, (-x)^(m^2)*Ser(A)^(m+1)), #A-1)); if(n<0, 0, A[n+1])}

Formula

The g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n>=0} (-x)^n*A(x)^(n+1) * Product_{k=1..n} (1 + x^(4*k-3)*A(x))/(1 + x^(4*k-1)*A(x));
(2) 1 = A(x)/(1 + x*A(x)/(1 - x*(1-x^2)*A(x)/(1 + x^5*A(x)/(1 - x^3*(1-x^4)*A(x)/(1 + x^9*A(x)/(1 - x^5*(1-x^6)*A(x)/(1 + x^13*A(x)/(1 - x^7*(1-x^8)*A(x)/(1- ...))))))))) (continued fraction);
due to identities of a partial elliptic theta function.

A193112 G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^(2*n+1).

Original entry on oeis.org

1, 1, 3, 13, 63, 328, 1796, 10200, 59529, 354837, 2151079, 13221261, 82200739, 516053099, 3266812048, 20829635112, 133651716406, 862342656359, 5591505085491, 36416212224801, 238114435569354, 1562560513492974, 10287406857203911
Offset: 0

Views

Author

Paul D. Hanna, Jul 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 63*x^4 + 328*x^5 + 1796*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^3 - x^3*A(x)^5 + x^6*A(x)^7 + x^10*A(x)^9 - x^15*A(x)^11 - x^21*A(x)^13 ++--...
Related expansions.
A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 303*x^4 + 1662*x^5 + 9447*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 760*x^4 + 4401*x^5 +...

Crossrefs

Cf. A193111, A193113, A193114, A193115, A193116.

Programs

  • PARI
    {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(2*(#A))+1, (-x)^(m*(m+1)/2)*Ser(A)^(2*m+1)), #A-1)); if(n<0, 0, A[n+1])}

Formula

G.f. A(x) satisfies the continued fraction:
1 = A(x)/(1+ x*A(x)^2/(1- x*(1+x)*A(x)^2/(1+ x^3*A(x)^2/(1+ x^2*(1-x^2)*A(x)^2/(1+ x^5*A(x)^2/(1- x^3*(1+x^3)*A(x)^2/(1+ x^7*A(x)^2/(1+ x^4*(1-x^4)*A(x)^2/(1- ...)))))))))
due to an identity of a partial elliptic theta function.

A193113 G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^(3*n+1).

Original entry on oeis.org

1, 1, 4, 23, 151, 1074, 8059, 62814, 503619, 4126954, 34411602, 291025337, 2490377810, 21523367553, 187603609077, 1647252368595, 14556722879278, 129366008725176, 1155458240271571, 10366549508487178, 93382085749705066, 844255894224907354
Offset: 0

Views

Author

Paul D. Hanna, Jul 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 151*x^4 + 1074*x^5 + 8059*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^4 - x^3*A(x)^7 + x^6*A(x)^10 + x^10*A(x)^13 - x^15*A(x)^16 - x^21*A(x)^19 ++--...
Related expansions.
A(x)^4 = 1 + 4*x + 22*x^2 + 144*x^3 + 1025*x^4 + 7696*x^5 +...
A(x)^7 = 1 + 7*x + 49*x^2 + 364*x^3 + 2814*x^4 + 22400*x^5 +...

Crossrefs

Cf. A193111, A193112, A193114, A193115, A193116.

Programs

  • PARI
    {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(2*(#A))+1, (-x)^(m*(m+1)/2)*Ser(A)^(3*m+1)), #A-1)); if(n<0, 0, A[n+1])}

Formula

G.f. A(x) satisfies the continued fraction:
1 = A(x)/(1+ x*A(x)^3/(1- x*(1+x)*A(x)^3/(1+ x^3*A(x)^3/(1+ x^2*(1-x^2)*A(x)^3/(1+ x^5*A(x)^3/(1- x^3*(1+x^3)*A(x)^3/(1+ x^7*A(x)^3/(1+ x^4*(1-x^4)*A(x)^3/(1- ...)))))))))
due to an identity of a partial elliptic theta function.

A193115 G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(2*n+1).

Original entry on oeis.org

1, 1, 3, 12, 54, 265, 1373, 7388, 40888, 231250, 1330618, 7764670, 45841323, 273316120, 1643345418, 9953021248, 60665811025, 371850104167, 2290623433302, 14173331572490, 88049709138896, 548978010516319, 3434070688405887, 21545961024510032
Offset: 0

Views

Author

Paul D. Hanna, Jul 16 2011

Keywords

Examples

			 G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 54*x^4 + 265*x^5 + 1373*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^3 + x^4*A(x)^5 - x^9*A(x)^7 + x^16*A(x)^9 -+...
Related expansions.
A(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 270*x^4 + 1398*x^5 + 7518*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 130*x^3 + 695*x^4 + 3816*x^5 +...

Crossrefs

Cf. A193111, A193112, A193113, A193114, A193116.

Formula

The g.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} (-x)^n*A(x)^(2*n+1) * Product_{k=1..n} (1 + x^(4*k-3)*A(x)^2)/(1 + x^(4*k-1)*A(x)^2);
(2) 1 = A(x)/(1+ x*A(x)^2/(1- x*(1-x^2)*A(x)^2/(1+ x^5*A(x)^2/(1- x^3*(1-x^4)*A(x)^2/(1+ x^9*A(x)^2/(1- x^5*(1-x^6)*A(x)^2/(1+ x^13*A(x)^2/(1- x^7*(1-x^8)*A(x)^2/(1- ...))))))))) (continued fraction);
due to identities of a partial elliptic theta function.

A193116 G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(3*n+1).

Original entry on oeis.org

1, 1, 4, 22, 139, 958, 6979, 52851, 411884, 3281684, 26609931, 218874331, 1821767351, 15315464340, 129859965329, 1109239893974, 9536166375605, 82449167265098, 716449009997437, 6253709697731562, 54808237437608982, 482103739329417219
Offset: 0

Views

Author

Paul D. Hanna, Jul 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 139*x^4 + 958*x^5 + 6979*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^4 + x^4*A(x)^7 - x^9*A(x)^10 + x^16*A(x)^13 -+...
Related expansions.
A(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 965*x^4 + 7028*x^5 +...
A(x)^7 = 1 + 7*x + 49*x^2 + 357*x^3 + 2688*x^4 + 20811*x^5 +...

Crossrefs

Cf. A193111, A193112, A193113, A193114, A193115.

Programs

  • PARI
    {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(#A)+1, (-x)^(m^2)*Ser(A)^(3*m+1) ), #A-1)); if(n<0, 0, A[n+1])}

Formula

The g.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} (-x)^n*A(x)^(3*n+1) * Product_{k=1..n} (1 + x^(4*k-3)*A(x)^3)/(1 + x^(4*k-1)*A(x)^3);
(2) 1 = A(x)/(1+ x*A(x)^3/(1- x*(1-x^2)*A(x)^3/(1+ x^5*A(x)^3/(1- x^3*(1-x^4)*A(x)^3/(1+ x^9*A(x)^3/(1- x^5*(1-x^6)*A(x)^3/(1+ x^13*A(x)^3/(1- x^7*(1-x^8)*A(x)^3/(1- ...))))))))) (continued fraction);
due to identities of a partial elliptic theta function.

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
Showing 1-6 of 6 results.

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