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.

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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

A349047 G.f. A(x) satisfies: A(x) = 1 / (1 - x + x^3 * A(x)).

Original entry on oeis.org

1, 1, 1, 0, -2, -5, -7, -4, 10, 38, 70, 68, -40, -329, -767, -1012, -214, 2842, 8642, 14332, 10136, -21622, -96578, -196412, -213080, 96264, 1037344, 2608788, 3698996, 1121127, -10234567, -33425980, -58537486, -45735382, 83471346, 408899204, 871127040
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 06 2021

Keywords

Links

Crossrefs

Cf. A000108, A007440, A023431, A100223, A214649, A343773, A349048.

Programs

  • Mathematica
    nmax = 36; A[] = 0; Do[A[x] = 1/(1 - x + x^3 A[x]) + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] - Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 36}]
Table[Sum[(-1)^k Binomial[n - k, 2 k] CatalanNumber[k], {k, 0, Floor[n/3]}], {n, 0, 36}]

Formula

G.f.: (-1 + x + sqrt((1 - x)^2 + 4*x^3)) / (2*x^3).
a(0) = 1; a(n) = a(n-1) - Sum_{k=0..n-3} a(k) * a(n-k-3).
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-k,2*k) * Catalan(k).
a(n) = F([(1-n)/3, (2-n)/3, -n/3], [2, -n], -27), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 07 2021
Showing 1-2 of 2 results.

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