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.

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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

A357400 Coefficients T(n,k) of x^n*y^k in the function A(x,y) that satisfies: y = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x,y)^n, as a triangle read by rows with k = 0..n for each row index n >= 0.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 1, 0, 5, 0, 0, 3, 0, 14, 0, -2, 0, 10, 0, 42, 0, 8, -12, 0, 35, 0, 132, 0, -14, 36, -52, 0, 126, 0, 429, 0, 16, -76, 148, -210, 0, 462, 0, 1430, 0, -7, 84, -354, 590, -825, 0, 1716, 0, 4862, 0, -24, -27, 416, -1565, 2322, -3199, 0, 6435, 0, 16796, 0, 103, -276, -120, 1950, -6732, 9086, -12320, 0, 24310, 0, 58786, 0, -232, 987, -1752, -560, 8832, -28490, 35464, -47268, 0, 92378, 0, 208012
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).
T(n,n) = binomial(2*n+1, n+1)/(2*n+1) = A000108(n) for n >= 0.
T(n+1,n) = 0 for n>= 0.
T(n+2,n) = binomial(2*n-1, n-1) = A001700(n-1) for n >= 1.
T(n+3,n) = 0 for n>= 0.
T(n+1,1) = A357401(n) for n >= 0.
A356783(n) = Sum_{k=0..n} T(n,k), for n >= 0.
A357402(n) = Sum_{k=0..n} T(n,k) * 2^k, for n >= 0.
A357403(n) = Sum_{k=0..n} T(n,k) * 3^k, for n >= 0.
A357404(n) = Sum_{k=0..n} T(n,k) * 4^k, for n >= 0.
A357405(n) = Sum_{k=0..n} T(n,k) * 5^k, for n >= 0.

Examples

			G.f. A(x,y) = 1 + x*y + x^2*(2*y^2) + x^3*(y + 5*y^3) + x^4*(3*y^2 + 14*y^4) + x^5*(-2*y + 10*y^3 + 42*y^5) + x^6*(8*y - 12*y^2 + 35*y^4 + 132*y^6) + x^7*(-14*y + 36*y^2 - 52*y^3 + 126*y^5 + 429*y^7) + x^8*(16*y - 76*y^2 + 148*y^3 - 210*y^4 + 462*y^6 + 1430*y^8) + x^9*(-7*y + 84*y^2 - 354*y^3 + 590*y^4 - 825*y^5 + 1716*y^7 + 4862*y^9) + x^10*(-24*y - 27*y^2 + 416*y^3 - 1565*y^4 + 2322*y^5 - 3199*y^6 + 6435*y^8 + 16796*y^10) + ...
such that
y = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x,y)^2 + x^(-1)/A(x,y) + x*0 + x^3*(1 - x)^2*A(x,y) + x^5*(1 - x^2)^3*A(x,y)^2 + x^7*(1 - x^3)^4*A(x,y)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x,y)^n + ...
also
-y*A(x,y)^3 = ... + x^(-3)*(A(x,y) - x^(-2))^(-1)*A(x,y)^2 + x^(-1)*A(x,y) + x*(A(x,y) - 1) + x^3*(A(x,y) - x)^2/A(x,y) + x^5*(1 - x^2)^3/A(x,y)^2 + x^7*(A(x,y) - x^3)^4/A(x,y)^3 + ... + x^(2*n+1)*(A(x,y) - x^n)^(n+1)/A(x,y)^n + ...
This triangle of coefficients T(n,k) of x^n*y^k, k = 0..n, in g.f. A(x,y) begins:
n = 0: [1],
n = 1: [0, 1],
n = 2: [0, 0, 2],
n = 3: [0, 1, 0, 5],
n = 4: [0, 0, 3, 0, 14],
n = 5: [0, -2, 0, 10, 0, 42],
n = 6: [0, 8, -12, 0, 35, 0, 132],
n = 7: [0, -14, 36, -52, 0, 126, 0, 429],
n = 8: [0, 16, -76, 148, -210, 0, 462, 0, 1430],
n = 9: [0, -7, 84, -354, 590, -825, 0, 1716, 0, 4862],
n = 10: [0, -24, -27, 416, -1565, 2322, -3199, 0, 6435, 0, 16796],
n = 11: [0, 103, -276, -120, 1950, -6732, 9086, -12320, 0, 24310, 0, 58786],
n = 12: [0, -232, 987, -1752, -560, 8832, -28490, 35464, -47268, 0, 92378, 0, 208012],
n = 13: [0, 334, -2160, 6436, -9460, -2673, 39102, -119296, 138294, -180960, 0, 352716, 0, 742900],
n = 14: [0, -256, 3002, -14484, 36218, -46902, -12929, 170368, -495846, 539240, -691900, 0, 1352078, 0, 2674440], ...
in which the main diagonal equals the Catalan numbers (A000108).

Links

Crossrefs

Cf. A356783 (row sums), A357402 (y=2), A357403 (y=3), A357404 (y=4), A357405 (y=5).
Cf. A357401 (column 1), A357151, A000108, A001700.

Programs

  • PARI
    {T(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=0, 15, for(k=0,n, print1(T(n,k), ", "));print(""))

Formula

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

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.
Showing 1-5 of 5 results.

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