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.

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

A357402 Coefficients in the power series A(x) such that: 2 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 2, 8, 42, 236, 1420, 8976, 58644, 393200, 2689522, 18694164, 131658910, 937490780, 6737990172, 48816739048, 356142597586, 2614103310384, 19291118713324, 143044431901580, 1065237986700788, 7963426677825000, 59741019702076168, 449601401992383464, 3393484429948103486
Offset: 0

Views

Author

Paul D. Hanna, Sep 26 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
a(n) = Sum_{k=0..n} A357400(n,k) * 2^k, for n >= 0.

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 42*x^3 + 236*x^4 + 1420*x^5 + 8976*x^6 + 58644*x^7 + 393200*x^8 + 2689522*x^9 + 18694164*x^10 + ...
such that
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
-2*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. A357400, A356783, A357403, A357404, A357405.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(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) 2 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
(2) 2*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
(3) -2*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) -2*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.

A357403 Coefficients in the power series A(x) such that: 3 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 3, 18, 138, 1161, 10470, 98979, 967719, 9705378, 99290130, 1032123366, 10870453785, 115749660723, 1244016993747, 13477172250201, 147021521096445, 1613619363015645, 17805435511256394, 197414608524234453, 2198189145649419426, 24571174933256703567, 275615684936993421462
Offset: 0

Views

Author

Paul D. Hanna, Sep 26 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
a(n) = Sum_{k=0..n} A357400(n,k) * 3^k, for n >= 0.

Examples

			G.f.: A(x) = 1 + 3*x + 18*x^2 + 138*x^3 + 1161*x^4 + 10470*x^5 + 98979*x^6 + 967719*x^7 + 9705378*x^8 + 99290130*x^9 + 1032123366*x^10 + ...
such that
3 = ... + 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
-3*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. A357400, A356783, A357402, A357404, A357405.

Programs

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

A357404 Coefficients in the power series A(x) such that: 4 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 4, 32, 324, 3632, 43640, 549472, 7154952, 95563392, 1301943972, 18022506736, 252768034908, 3584103003152, 51294399688504, 739984677348512, 10749373940462452, 157101410692820448, 2308378616597302488, 34080671255517914992, 505321131709023383016, 7521442675843527317728
Offset: 0

Views

Author

Paul D. Hanna, Sep 26 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
a(n) = Sum_{k=0..n} A357400(n,k) * 4^k, for n >= 0.

Examples

			G.f.: A(x) = 1 + 4*x + 32*x^2 + 324*x^3 + 3632*x^4 + 43640*x^5 + 549472*x^6 + 7154952*x^7 + 95563392*x^8 + 1301943972*x^9 + 18022506736*x^10 + ...
such that
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
-4*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. A357400, A356783, A357402, A357403, A357405.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(4 - sum(m=-#A\2-1, #A\2+1, x^(2*m+1) * (1 - x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #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) 4 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
(2) 4*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
(3) -4*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) -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.

A357405 Coefficients in the power series A(x) such that: 5 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

Original entry on oeis.org

1, 5, 50, 630, 8825, 132490, 2084115, 33903705, 565697930, 9627904690, 166493454330, 2917050253615, 51670197054515, 923774673549045, 16647699155752645, 302098954307654995, 5515438344643031325, 101237254225602624790, 1867129260849076888865, 34583287418814030368150
Offset: 0

Views

Author

Paul D. Hanna, Sep 26 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
a(n) = Sum_{k=0..n} A357400(n,k) * 5^k, for n >= 0.

Examples

			G.f.: A(x) = 1 + 5*x + 50*x^2 + 630*x^3 + 8825*x^4 + 132490*x^5 + 2084115*x^6 + 33903705*x^7 + 565697930*x^8 + 9627904690*x^9 + 166493454330*x^10 + ...
such that
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
-5*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. A357400, A356783, A357402, A357403, A357404.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(5 - sum(m=-#A\2-1, #A\2+1, x^(2*m+1) * (1 - x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #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) 5 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
(2) 5*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
(3) -5*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) -5*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.

A357401 Coefficients in the power series expansion of 1/Sum_{n=-oo..+oo} n * x^(2*n+1) * (1 - x^n)^(n+1).

Original entry on oeis.org

1, 0, 1, 0, -2, 8, -14, 16, -7, -24, 103, -232, 334, -256, -211, 1400, -3562, 6048, -6470, 512, 17788, -53720, 102983, -134832, 76147, 187960, -776169, 1690880, -2558499, 2270952, 1214672, -10443024, 26674201, -45822896, 51953043, -11147384, -126256811, 401311496
Offset: 1

Views

Author

Paul D. Hanna, Sep 26 2022

Keywords

Comments

Equals column 1 of triangle A357400. The g.f. G(x,y) of triangle A357400 satisfies: y = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * G(x,y)^n; this sequence gives the coefficients of x^n*y in G(x,y) for n >= 1.

Examples

			G.f.: A(x) = x + x^3 - 2*x^5 + 8*x^6 - 14*x^7 + 16*x^8 - 7*x^9 - 24*x^10 + 103*x^11 - 232*x^12 + 334*x^13 - 256*x^14 - 211*x^15 + 1400*x^16 - 3562*x^17 + 6048*x^18 - 6470*x^19 + 512*x^20 + ...
Related series.
x/A(x) = 1 - x^2 + 3*x^4 - 8*x^5 + 9*x^6 - 10*x^8 + 24*x^10 - 24*x^11 + 15*x^14 + 9*x^16 - 80*x^17 + 90*x^18 - 43*x^20 + 57*x^22 + ... + A357406(n)*x^n + ...
which equals x*Sum_{n=-oo..+oo} n * x^(2*n+1) * (1 - x^n)^(n+1).

Links

Crossrefs

Cf. A357400, A357406.

Programs

  • PARI
    {a(n) = my(A = 1/sum(m=-n-1,n+1, m * x^(2*m+1) * (1 - x^m +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
for(n=1,40, print1(a(n), ", "))
  • PARI
    /* As Column 1 of triangle A357400 (slow) */
{A357400(n,k) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(y - sum(m=-#A\2-1, #A\2+1, x^(2*m+1) * (1 - x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); polcoeff(A[n+1],k,y)}
for(n=1, 40, print1(A357400(n,1), ", "))

Formula

G.f. A(x,y) = Sum_{n>=1} a(n) * x^n satisfies the following relations.
(1) A(x) = 1/Sum_{n=-oo..+oo} n * x^(2*n+1) * (1 - x^n)^(n+1).
(2) A(X) = -x/Sum_{n=-oo..+oo} n * (-1)^n * x^(n*(n-1)) / (1 - x^(n+1))^n.
Showing 1-6 of 6 results.

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