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.

A351971 a(0) = 1; a(n) = Sum_{k=0..floor(n/4)} binomial(n,4*k) * a(k).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 36, 72, 136, 256, 496, 992, 2016, 4096, 8256, 16513, 32913, 65689, 131785, 266482, 544230, 1124344, 2352188, 4984968, 10699656, 23244976, 51072256, 113370112, 253930816, 573005056, 1300390016, 2962852353, 6766967329, 15472840497
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 26 2022

Keywords

Links

Crossrefs

Cf. A027826, A351970, A352066.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 4 k] a[k], {k, 0, Floor[n/4]}]; Table[a[n], {n, 0, 34}]
nmax = 34; A[] = 1; Do[A[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) = A(x^4/(1 - x)^4) / (1 - x).
E.g.f.: exp(x) * Sum_{n>=0} a(n) * x^(4*n) / (4*n)!.

A352044 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(k).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 33, 71, 159, 367, 864, 2058, 4934, 11870, 28608, 69020, 166652, 402748, 974464, 2361416, 5733656, 13954488, 34054016, 83351472, 204659440, 504162416, 1246065920, 3089752480, 7685498976, 19174255584, 47971104769, 120326714335
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 01 2022

Keywords

Links

Crossrefs

Cf. A027826, A352045, A352066.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] a[k], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 32}]
nmax = 32; A[] = 0; Do[A[x] = 1 + x A[x^2/(1 - x)^2]/(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^2/(1 - x)^2) / (1 - x).
E.g.f.: Integral exp(x) * Sum_{n>=0} a(n) * x^(2*n) / (2*n)! dx.

A352904 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n,4*k+1) * a(k).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 12, 28, 64, 137, 282, 583, 1244, 2733, 6062, 13343, 28944, 61969, 131602, 278483, 588564, 1242646, 2618924, 5505556, 11542528, 24142217, 50409898, 105154719, 219278860, 457362189, 954629598, 1994940799, 4175986720, 8760742945, 18428667938
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 07 2022

Keywords

Crossrefs

Cf. A040027, A119685, A352066, A352879.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 4 k + 1] a[k], {k, 0, Floor[(n - 1)/4]}]; Table[a[n], {n, 0, 34}]
nmax = 34; A[] = 0; Do[A[x] = 1 + 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) = 1 + x * A(x^4/(1 - x)^4) / (1 - x)^2.
E.g.f.: 1 + exp(x) * Sum_{n>=0} a(n) * x^(4*n+1) / (4*n+1)!.

A354696 G.f. A(x) satisfies: A(x) = 1 + x * A(x^4/(1 - x)^4) / (1 - x)^4.

Original entry on oeis.org

1, 1, 4, 10, 20, 36, 64, 120, 240, 499, 1060, 2314, 5252, 12360, 29632, 70992, 168096, 392465, 905940, 2075314, 4730052, 10735516, 24258688, 54553000, 122076240, 271914499, 603183508, 1333268098, 2937818900, 6455143760, 14146816640, 30929336736, 67473335104
Offset: 0

Views

Author

Seiichi Manyama, Jun 03 2022

Keywords

Crossrefs

Cf. A119685, A354695.
Cf. A351817, A352066.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, (i-1)\4, binomial(i+2, 4*j+3)*v[j+1])); v;

Formula

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

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

Content is available under The OEIS End-User License Agreement.