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 1-5 of 5 results.

A346074 a(n) = 1 + Sum_{k=0..n-5} a(k) * a(n-k-5).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 14, 21, 30, 41, 59, 89, 136, 205, 301, 443, 664, 1011, 1545, 2341, 3530, 5341, 8143, 12487, 19148, 29299, 44817, 68721, 105742, 163025, 251392, 387595, 597988, 924047, 1430167, 2215595, 3433788, 5323915, 8260652, 12829849
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2021

Keywords

Links

Crossrefs

Cf. A007317, A090344, A216604, A307972, A346073.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 5], {k, 0, n - 5}]; Table[a[n], {n, 0, 44}]
nmax = 44; A[] = 0; Do[A[x] = 1/(1 - x) + x^5 A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    a(n) = sum(k=0, n\5, binomial(n-4*k, k)*binomial(2*k, k)/(k+1)); \\ Seiichi Manyama, Jan 22 2023

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^5 * A(x)^2.
Conjecture D-finite with recurrence (n+5)*a(n) +2*(-n-4)*a(n-1) +(n+3)*a(n-2) +2*(-2*n+5)*a(n-5) +4*(n-3)*a(n-6)=0. - R. J. Mathar, Feb 17 2022
a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k,k) * Catalan(k). - Seiichi Manyama, Jan 22 2023

A360026 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * Catalan(k).

Original entry on oeis.org

1, 1, 1, 1, 0, -1, -2, -3, -2, 1, 6, 13, 17, 13, -4, -39, -83, -113, -92, 31, 279, 605, 850, 701, -219, -2129, -4736, -6749, -5690, 1569, 17114, 38713, 55957, 48249, -11498, -142163, -326860, -478957, -421262, 84015, 1210831, 2829363, 4197670, 3762583, -601732
Offset: 0

Views

Author

Seiichi Manyama, Jan 22 2023

Keywords

Links

Crossrefs

Cf. A360024, A360025, A360027.
Cf. A000108, A346073, A349048.

Programs

  • PARI
    a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=50, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)^2+4*x^4*(1-x))))

Formula

a(n) = 1 - Sum_{k=0..n-4} a(k) * a(n-k-4).
G.f. A(x) satisfies: A(x) = 1/(1-x) - x^4 * A(x)^2.
G.f.: 2 / ( 1-x + sqrt((1-x)^2 + 4*x^4*(1-x)) ).
D-finite with recurrence +(n+4)*a(n) +2*(-n-3)*a(n-1) +(n+2)*a(n-2) +4*(n-2)*a(n-4) +2*(-2*n+5)*a(n-5)=0. - R. J. Mathar, Jan 25 2023

A346076 a(n) = 1 + Sum_{k=1..n-4} a(k) * a(n-k-4).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 11, 17, 25, 36, 54, 84, 131, 201, 307, 475, 745, 1172, 1837, 2878, 4531, 7173, 11381, 18057, 28669, 45624, 72796, 116336, 186066, 297865, 477505, 766621, 1232214, 1982292, 3191693, 5143974, 8298640, 13399691, 21652705, 35014373, 56663700
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2021

Keywords

Links

Crossrefs

Cf. A105633, A217282, A307971, A343305, A346048, A346073, A346075, A346077.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 4], {k, 1, n - 4}]; Table[a[n], {n, 0, 44}]
nmax = 44; A[] = 0; Do[A[x] = 1/(1 - x) + x^4 A[x] (A[x] - 1) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • SageMath
    @CachedFunction
def a(n): # a = A346076
    if (n<5): return 1
    else: return 1 + sum(a(k)*a(n-k-4) for k in range(1,n-3))
[a(n) for n in range(51)] # G. C. Greubel, Nov 27 2022

Formula

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

A364591 G.f. satisfies A(x) = 1/(1-x) + x^4*A(x)^4.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 11, 21, 40, 85, 197, 457, 1028, 2289, 5193, 12069, 28338, 66445, 155563, 365701, 865815, 2061133, 4919431, 11758741, 28165412, 67657225, 162977081, 393445865, 951438682, 2304494349, 5591221729, 13588455861, 33075115578, 80616857525, 196742749155
Offset: 0

Views

Author

Seiichi Manyama, Jul 29 2023

Keywords

Crossrefs

Cf. A346073, A364590.
Cf. A226974, A349289, A364588.

Programs

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

Formula

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

A364590 G.f. satisfies A(x) = 1/(1-x) + x^4*A(x)^3.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 7, 11, 19, 37, 74, 142, 268, 518, 1033, 2077, 4152, 8290, 16687, 33899, 69148, 141160, 288650, 592354, 1220086, 2519226, 5210164, 10794088, 22408556, 46613554, 97125751, 202662419, 423459427, 886048249, 1856448852, 3894362560, 8178530890
Offset: 0

Views

Author

Seiichi Manyama, Jul 29 2023

Keywords

Crossrefs

Cf. A346073, A364591.
Cf. A049130, A199475, A364589.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n-2*k,2*k) * binomial(3*k,k) / (2*k+1).
Showing 1-5 of 5 results.

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