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.

A352066 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * a(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 6, 16, 36, 72, 136, 256, 496, 992, 2016, 4096, 8256, 16512, 32896, 65536, 130816, 261633, 523797, 1048807, 2099947, 4206983, 8443911, 17009071, 34452991, 70311167, 144818751, 301455871, 634774911, 1352698367, 2917079551, 6362776831, 14025038591
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 02 2022

Keywords

Links

Crossrefs

Cf. A351971, A352044, A352045.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 4 k] a[k], {k, 0, Floor[(n - 1)/4]}]; Table[a[n], {n, 0, 36}]
nmax = 36; A[] = 0; Do[A[x] = 1 + x A[x^4/(1 - x)^4]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + x * A(x^4/(1 - x)^4) / (1 - x).
E.g.f.: Integral exp(x) * Sum_{n>=0} a(n) * x^(4*n) / (4*n)! dx.

A351970 a(0) = 1; a(n) = Sum_{k=0..floor(n/3)} binomial(n,3*k) * a(k).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 22, 43, 85, 171, 351, 738, 1590, 3498, 7827, 17757, 40725, 94113, 218439, 507852, 1180395, 2739312, 6342084, 14642100, 33703182, 77344686, 176984406, 403901722, 919533655, 2089000936, 4737194867, 10726221713, 24257223605, 54805216567
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 26 2022

Keywords

Links

Crossrefs

Cf. A027826, A351971.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 3 k] a[k], {k, 0, Floor[n/3]}]; Table[a[n], {n, 0, 33}]
nmax = 33; A[] = 1; Do[A[x] = A[x^3/(1 - x)^3]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = A(x^3/(1 - x)^3) / (1 - x).
E.g.f.: exp(x) * Sum_{n>=0} a(n) * x^(3*n) / (3*n)!.

A352902 a(0) = 1; a(n) = Sum_{k=0..floor(n/4)} binomial(n+1,4*k+1) * a(k).

Original entry on oeis.org

1, 2, 3, 4, 7, 18, 49, 120, 264, 544, 1100, 2256, 4736, 10080, 21456, 45312, 94850, 197412, 410134, 852968, 1778382, 3719364, 7806338, 16451280, 34835184, 74164800, 158854536, 342478432, 743432288, 1625079936, 3576582592, 7922065408, 17648409603
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 07 2022

Keywords

Comments

Self-convolution of A351971.

Crossrefs

Cf. A040027, A051163, A351971, A352901.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 1, 4 k + 1] a[k], {k, 0, Floor[n/4]}]; Table[a[n], {n, 0, 32}]
nmax = 32; A[] = 1; Do[A[x] = A[x^4/(1 - x)^4]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = A(x^4/(1 - x)^4) / (1 - x)^2.
E.g.f.: d/dx ( exp(x) * Sum_{n>=0} a(n) * x^(4*n+1) / (4*n+1)! ).
Showing 1-3 of 3 results.

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

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