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.

A356783 Coefficients in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 2, 6, 17, 50, 163, 525, 1770, 6066, 21154, 74787, 267371, 965233, 3513029, 12877687, 47499333, 176167086, 656568385, 2457710598, 9236079055, 34832753818, 131792634266, 500121476517, 1902979982421, 7258942377746, 27752992782498, 106333425162358, 408213503595652
Offset: 0

Views

Author

Paul D. Hanna, Sep 15 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 17*x^4 + 50*x^5 + 163*x^6 + 525*x^7 + 1770*x^8 + 6066*x^9 + 21154*x^10 + 74787*x^11 + 267371*x^12 + ...
such that
1 = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x)^2 + x^(-1)/A(x) + x*0 + x^3*(1 - x)^2*A(x) + x^5*(1 - x^2)^3*A(x)^2 + x^7*(1 - x^3)^4*A(x)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x)^n + ...
also
-A(x)^3 = ... + x^(-3)*(A(x) - x^(-2))^(-1)*A(x)^2 + x^(-1)*A(x) + x*(A(x) - 1) + x^3*(A(x) - x)^2/A(x) + x^5*(1 - x^2)^3/A(x)^2 + x^7*(A(x) - x^3)^4/A(x)^3 + ... + x^(2*n+1)*(A(x) - x^n)^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A357151, A357152, A357153, A357154, A357155.
Cf. A357200, A357400, A357402, A357403, A357404, A357405.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
(1) 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
(2) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
(3) -x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
(4) -A(x)^3 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.
a(n) ~ c * d^n / n^(3/2), where d = 4.04962821886295599791727073173857... and c = 0.613483546803830745310382482744... - Vaclav Kotesovec, Mar 22 2025

A357151 Coefficients in the power series A(x) such that: A(x) = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 3, 13, 60, 299, 1586, 8697, 49117, 283437, 1664128, 9908903, 59694494, 363179981, 2228272706, 13771458148, 85655772108, 535759514193, 3367801361510, 21264574306632, 134804893426581, 857682458939905, 5474890014327326, 35053167752718368, 225046818744827456
Offset: 0

Views

Author

Paul D. Hanna, Sep 16 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 60*x^4 + 299*x^5 + 1586*x^6 + 8697*x^7 + 49117*x^8 + 283437*x^9 + 1664128*x^10 + 9908903*x^11 + 59694494*x^12 + ...
such that
A(x) = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x)^2 + x^(-1)/A(x) + x*0 + x^3*(1 - x)^2*A(x) + x^5*(1 - x^2)^3*A(x)^2 + x^7*(1 - x^3)^4*A(x)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x)^n + ...
also
-A(x)^4 = ... + x^(-3)*(A(x) - x^(-2))^(-1)*A(x)^2 + x^(-1)*A(x) + x*(A(x) - 1) + x^3*(A(x) - x)^2/A(x) + x^5*(1 - x^2)^3/A(x)^2 + x^7*(A(x) - x^3)^4/A(x)^3 + ... + x^(2*n+1)*(A(x) - x^n)^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A356783, A357152, A357153, A357154, A357155.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
(1) A(x) = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
(2) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
(3) -x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
(4) -A(x)^4 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.

A357152 Coefficients in the power series A(x) such that: A(x)^2 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 4, 23, 147, 1022, 7529, 57605, 453691, 3653149, 29937140, 248865368, 2093488837, 17787701638, 152433293056, 1315973808843, 11434434212115, 99918928175263, 877543565096334, 7741838176253076, 68576621373325887, 609670801860847612, 5438211584097291663
Offset: 0

Views

Author

Paul D. Hanna, Sep 16 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 147*x^4 + 1022*x^5 + 7529*x^6 + 57605*x^7 + 453691*x^8 + 3653149*x^9 + 29937140*x^10 + 248865368*x^11 + 2093488837*x^12 + ...
such that
A(x)^2 = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x)^2 + x^(-1)/A(x) + x*0 + x^3*(1 - x)^2*A(x) + x^5*(1 - x^2)^3*A(x)^2 + x^7*(1 - x^3)^4*A(x)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x)^n + ...
also
-A(x)^5 = ... + x^(-3)*(A(x) - x^(-2))^(-1)*A(x)^2 + x^(-1)*A(x) + x*(A(x) - 1) + x^3*(A(x) - x)^2/A(x) + x^5*(1 - x^2)^3/A(x)^2 + x^7*(A(x) - x^3)^4/A(x)^3 + ... + x^(2*n+1)*(A(x) - x^n)^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A356783, A357151, A357153, A357154, A357155.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
(1) A(x)^2 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
(2) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
(3) -x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
(4) -A(x)^5 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.

A357154 Coefficients in the power series A(x) such that: A(x)^4 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 6, 52, 517, 5615, 64587, 772961, 9526304, 120084968, 1541062520, 20066028177, 264441631790, 3520463590183, 47274535397701, 639587090815124, 8709694025888081, 119288137354977880, 1642104576551818747, 22707897424654348214, 315300786621008803900
Offset: 0

Views

Author

Paul D. Hanna, Sep 16 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 52*x^3 + 517*x^4 + 5615*x^5 + 64587*x^6 + 772961*x^7 + 9526304*x^8 + 120084968*x^9 + 1541062520*x^10 + 20066028177*x^11 + 264441631790*x^12 + ...
such that
A(x)^4 = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x)^2 + x^(-1)/A(x) + x*0 + x^3*(1 - x)^2*A(x) + x^5*(1 - x^2)^3*A(x)^2 + x^7*(1 - x^3)^4*A(x)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x)^n + ...
also
-A(x)^7 = ... + x^(-3)*(A(x) - x^(-2))^(-1)*A(x)^2 + x^(-1)*A(x) + x*(A(x) - 1) + x^3*(A(x) - x)^2/A(x) + x^5*(1 - x^2)^3/A(x)^2 + x^7*(A(x) - x^3)^4/A(x)^3 + ... + x^(2*n+1)*(A(x) - x^n)^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A356783, A357151, A357152, A357153, A357155.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(Ser(A)^4 - sum(n=-#A\2-1, #A\2+1, x^(2*n+1) * (1 - x^n +x*O(x^#A))^(n+1) * Ser(A)^n  ), #A-2); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
(1) A(x)^4 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
(2) x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
(3) -x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
(4) -A(x)^7 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.

A357155 Coefficients in the power series A(x) such that: A(x)^5 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 7, 71, 832, 10660, 144684, 2043814, 29736131, 442562703, 6706068107, 103109044005, 1604621459651, 25226987525340, 400062373648799, 6392118111706099, 102801779216363982, 1662854341556813731, 27034758217304814579, 441537893821034707720, 7240848432876171585800
Offset: 0

Views

Author

Paul D. Hanna, Sep 16 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 71*x^3 + 832*x^4 + 10660*x^5 + 144684*x^6 + 2043814*x^7 + 29736131*x^8 + 442562703*x^9 + 6706068107*x^10 + 103109044005*x^11 + 1604621459651*x^12 + ...
such that
A(x)^5 = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x)^2 + x^(-1)/A(x) + x*0 + x^3*(1 - x)^2*A(x) + x^5*(1 - x^2)^3*A(x)^2 + x^7*(1 - x^3)^4*A(x)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x)^n + ...
also
-A(x)^8 = ... + x^(-3)*(A(x) - x^(-2))^(-1)*A(x)^2 + x^(-1)*A(x) + x*(A(x) - 1) + x^3*(A(x) - x)^2/A(x) + x^5*(1 - x^2)^3/A(x)^2 + x^7*(A(x) - x^3)^4/A(x)^3 + ... + x^(2*n+1)*(A(x) - x^n)^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A356783, A357151, A357152, A357153, A357154.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(Ser(A)^5 - sum(n=-#A\2-1, #A\2+1, x^(2*n+1) * (1 - x^n +x*O(x^#A))^(n+1) * Ser(A)^n  ), #A-2); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
(1) A(x)^5 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
(2) x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
(3) -x*A(x)^7 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
(4) -A(x)^8 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.

A357163 Coefficients in the power series A(x) such that: A(x)^3 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 5, 38, 313, 2834, 27088, 269380, 2757797, 28872568, 307696566, 3326835855, 36403128996, 402370063992, 4485931975701, 50386112677647, 569624341701738, 6476615022560400, 74013180802610161, 849642206122063571, 9793310961240979983, 113297108937174512275
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2022

Keywords

Comments

Compare to A357153.
Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1).
Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 313*x^4 + 2834*x^5 + 27088*x^6 + 269380*x^7 + 2757797*x^8 + 28872568*x^9 + 307696566*x^10 + ...
such that
A(x)^3 = ... + x^(-4)*(1 - 1/x^3)^(-1)/A(x)^2 + x^(-1)/A(x) + x^2*(1 - 1/x) + x^5*0*A(x) + x^8*(1 - x)^3*A(x)^2 + x^11*(1 - x^2)^4*A(x)^3 + ... + x^(3*n+2)*(1 - x^(n-1))^(n+1)*A(x)^n + ...
also
-A(x)^6 = ... + x^(-4)*(A(x) - 1/x^3)^(-1)*A(x)^2 + x^(-1)*A(x) + x^2*(A(x) - 1/x) + x^5*(A(x) - 1)^2/A(x) + x^8*(A(x) - x)^3/A(x)^2 + x^11*(A(x) - x^2)^4/A(x)^3 + ... + x^(3*n+2)*(A(x) - x^(n-1))^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A357153, A357160, A357161, A357162, A357164, A357165.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
A[#A] = polcoeff(Ser(A)^3 - sum(n=-#A\3-2,#A\3+2, x^(3*n+2) * (1 - x^(n-1) +x*O(x^#A))^(n+1) * Ser(A)^n  ),#A-2); );A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
(1) A(x)^3 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
(2) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ).
(3) -x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n.
(4) -A(x)^6 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1)*A(x))^(n+1) / A(x)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+2))^n.

A357203 Coefficients in the power series A(x) such that: A(x)^3 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.

Original entry on oeis.org

1, 1, 3, 18, 111, 800, 5990, 46995, 379090, 3129713, 26301576, 224282112, 1935668344, 16876028036, 148410725830, 1314933853171, 11726585616205, 105178923513494, 948185788906100, 8586757756571261, 78079244607685021, 712592590813142079, 6525273550226573555
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2022

Keywords

Comments

Compare to A357153 and A357163.
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1).
Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 111*x^4 + 800*x^5 + 5990*x^6 + 46995*x^7 + 379090*x^8 + 3129713*x^9 + 26301576*x^10 + ...
such that
A(x)^3 = ... + x^(-2)*(1 - 1/x)^(-1)/A(x)^2 + x^(-1)/A(x) + (1 - x) + x*(1 - x^2)*A(x) + x^2*(1 - x^3)^3*A(x)^2 + x^3*(1 - x^4)^4*A(x)^3 + ... + x^n*(1 - x^(n+1))^(n+1)*A(x)^n + ...
also
-A(x)^6 = ... + x^(-2)*(A(x) - 1/x)^(-1)*A(x)^2 + x^(-1)*A(x) + (A(x) - x) + x*(A(x) - x^2)^2/A(x) + x^2*(A(x) - x^3)^3/A(x)^2 + x^3*(A(x) - x^4)^4/A(x)^3 + ... + x^n*(A(x) - x^(n+1))^(n+1)/A(x)^n + ...

Links

Crossrefs

Cf. A357153, A357163, A357200, A357201, A357202, A357204, A357205.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(Ser(A)^3 - sum(n=-#A-2, #A+2, x^(n) * (1 - x^(n+1) +x*O(x^#A))^(n+1) * Ser(A)^n  ), #A-2); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
(1) A(x)^3 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.
(2) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^n)^n * A(x)^n ).
(3) -x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^n*A(x))^n.
(4) -A(x)^6 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(n+1))^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1)*A(x))^(n+1) / A(x)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^n)^n.
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