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 1-4 of 4 results.

A364626 G.f. satisfies A(x) = 1/(1-x)^3 + x^2*A(x)^3.

Original entry on oeis.org

1, 3, 7, 19, 63, 231, 895, 3615, 15055, 64111, 277791, 1220767, 5427775, 24371199, 110350335, 503289727, 2309992959, 10661634303, 49452179455, 230391918591, 1077644520703, 5058766156543, 23824929459711, 112541456498175, 533063457631231, 2531252417738751
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2023

Keywords

Crossrefs

Cf. A364625, A364627.
Cf. A086631.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n+4*k+2, 6*k+2)*binomial(3*k, k)/(2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n+4*k+2,6*k+2) * binomial(3*k,k) / (2*k+1).

A086629 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/[(1-x)(1-y)] + xy*f(x,y)^3.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 13, 7, 1, 1, 11, 34, 34, 11, 1, 1, 16, 76, 124, 76, 16, 1, 1, 22, 151, 370, 370, 151, 22, 1, 1, 29, 274, 952, 1419, 952, 274, 29, 1, 1, 37, 463, 2185, 4573, 4573, 2185, 463, 37, 1, 1, 46, 739, 4579, 12892, 18037, 12892, 4579, 739, 46, 1
Offset: 0

Views

Author

Paul D. Hanna, Jul 27 2003

Keywords

Comments

If 1 is subtracted from every element of the table, the resulting table forms the coefficients of f(x,y)^3, where f(x,y) = 1/[(1-x)(1-y)] + xy*f(x,y)^3.

Crossrefs

Cf. A086630 (diagonal), A086631 (antidiagonal sums).

Programs

  • Mathematica
    m = 11; f[, ] = 0;
Do[f[x_, y_] = 1/((1 - x)(1 - y)) + x y f[x, y]^3 + O[x]^m, {m}];
T =CoefficientList[# + O[y]^m, y]& /@ CoefficientList[f[x, y], x];
Table[T[[n-k+1, k]], {n, 1, m}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 15 2019 *)

A086630 Main diagonal of square table A086629; coefficients of x^n*y^n in f(x,y) that satisfies f(x,y) = 1/[(1-x)(1-y)] + xy*f(x,y)^3.

Original entry on oeis.org

1, 2, 13, 124, 1419, 18037, 245650, 3513260, 52114339, 795230788, 12411836882, 197327486617, 3185686181794, 52101500060794, 861628197679360
Offset: 0

Views

Author

Paul D. Hanna, Jul 24 2003

Keywords

Crossrefs

Cf. A086629 (table), A086631 (antidiagonal sums).

A364622 G.f. satisfies A(x) = 1/(1-x)^2 + x^2*A(x)^4.

Original entry on oeis.org

1, 2, 4, 12, 45, 182, 779, 3480, 16005, 75234, 359893, 1746268, 8573477, 42511646, 212587561, 1070897000, 5429174465, 27679933778, 141829437174, 729972918876, 3772160853821, 19563615260102, 101797930474515, 531293155760840, 2780515192595481, 14588670579665882
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2023

Keywords

Crossrefs

Cf. A086615, A086631.

Programs

  • Mathematica
    Table[Sum[Binomial[n + 4 k + 1, 6 k + 1]*Binomial[4 k, k]/(3 k + 1), {k, 0, Floor[n/2]}], {n, 0, 30}] (* Wesley Ivan Hurt, Jan 20 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, binomial(n+4*k+1, 6*k+1)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n+4*k+1,6*k+1) * binomial(4*k,k) / (3*k+1).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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