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 1-3 of 3 results.

A386413 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^2)^(2/3).

Original entry on oeis.org

1, 6, 63, 792, 10935, 160056, 2438667, 38263752, 614014830, 10029572280, 166203389781, 2787232297680, 47213065271268, 806618756189736, 13883029872725475, 240491818267745760, 4189678646994012501, 73357895462268102840, 1290223574267814268290, 22784365638084466567800
Offset: 0

Views

Author

Seiichi Manyama, Jul 21 2025

Keywords

Links

Crossrefs

Cf. A004989, A377269, A386414, A386415.
Cf. A078532, A376636.

Programs

  • Mathematica
    A386413[n_] := 9^n*Binomial[(4*n + 2)/3, n]/(2*n + 1);
Array[A386413, 25, 0] (* Paolo Xausa, Aug 01 2025 *)
  • PARI
    apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = 9^n*apr(n, 4/3, 2/3);

Formula

a(n) = 9^n * binomial((4*n+2)/3,n)/(2*n+1).
G.f.: B(x)^2, where B(x) is the g.f. of A078532.
D-finite with recurrence n*(n-2)*(n+2)*a(n) -216*(2*n-5)*(4*n-7)*(4*n-1)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A386414 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^3)^(2/3).

Original entry on oeis.org

1, 6, 99, 2142, 52785, 1404702, 39331656, 1141839504, 34057559052, 1037385419400, 32133013365915, 1009060082062110, 32050934711814915, 1027914968037080970, 33240367148212098900, 1082645830435810233960, 35483717092533680418039, 1169426742892003447650666
Offset: 0

Views

Author

Seiichi Manyama, Jul 21 2025

Keywords

Links

Crossrefs

Cf. A004989, A377269, A386413, A386415.
Cf. A004987, A008931.

Programs

  • Mathematica
    A386414[n_] := 9^n*Binomial[(6*n + 2)/3, n]/(3*n + 1);
Array[A386414, 20, 0] (* Paolo Xausa, Aug 01 2025 *)
  • PARI
    apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = 9^n*apr(n, 2, 2/3);

Formula

a(n) = 9^n * binomial((6*n+2)/3,n)/(3*n+1).
G.f.: B(x)^2, where B(x) is the g.f. of A008931.
D-finite with recurrence +n*(3*n+2)*a(n) -6*(6*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Jul 30 2025
G.f.: 2F1(1/3,5/6 ; 5/3 ; 36*x). - R. J. Mathar, Jul 30 2025

A386416 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^8)^(1/3).

Original entry on oeis.org

1, 3, 63, 1881, 65610, 2499336, 100777122, 4228144596, 182674383705, 8072369224920, 363154406671485, 16576444298006658, 765806677899249168, 35739548618003938440, 1682429522012566325460, 79793991407758199002740, 3809208342822290233767522, 182890356905449116974950200
Offset: 0

Views

Author

Seiichi Manyama, Jul 21 2025

Keywords

Links

Crossrefs

Cf. A004990, A377268, A376636, A004987, A078532, A245114, A008931, A376282.
Cf. A386415.

Programs

  • Mathematica
    A386416[n_] := 9^n*Binomial[(8*n + 1)/3, n]/(8*n + 1);
Array[A386416, 20, 0] (* Paolo Xausa, Aug 01 2025 *)
  • PARI
    apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = 9^n*apr(n, 8/3, 1/3);

Formula

a(n) = 9^n * binomial((8*n+1)/3,n)/(8*n+1).
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^13).
D-finite with recurrence 5*n*(n-1)*(n-2)*(5*n-8)*(5*n-11)*(5*n+1)*(5*n-2)*a(n) -3456*(8*n-11)*(8*n-5)*(4*n-1)*(8*n-23)*(2*n-5)*(8*n-17)*(4*n-7)*a(n-3)=0. - R. J. Mathar, Jul 30 2025
Showing 1-3 of 3 results.

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