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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A367056 G.f. satisfies A(x) = 1 + x*A(x)^2 + x^3*A(x).

Original entry on oeis.org

1, 1, 2, 6, 17, 52, 168, 561, 1922, 6719, 23871, 85938, 312823, 1149421, 4257460, 15880036, 59594517, 224856450, 852491806, 3245959002, 12407332166, 47592364107, 183139542306, 706794663136, 2735053815771, 10609811267757, 41251228784198
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2023

Keywords

Crossrefs

Cf. A049140, A071969, A125305, A216490, A367042.

Programs

  • Maple
    A367056 := proc(n)
    add(binomial(n-2*k+1,k) * binomial(2*n-5*k,n-3*k)/(n-2*k+1),k=0..floor(n/3)) ;
end proc:
seq(A367056(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    a(n) = sum(k=0, n\3, binomial(n-2*k+1, k)*binomial(2*n-5*k, n-3*k)/(n-2*k+1));

Formula

G.f.: A(x) = 2 / (1-x^3+sqrt((1-x^3)^2-4*x)).
a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k+1,k) * binomial(2*n-5*k,n-3*k)/(n-2*k+1).
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-2*n+7)*a(n-3) +(n-8)*a(n-6)=0. - R. J. Mathar, Dec 04 2023

A108074 Triangle in A071944 with rows reversed.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 19, 16, 9, 4, 1, 63, 54, 31, 14, 5, 1, 219, 188, 111, 52, 20, 6, 1, 787, 676, 405, 197, 80, 27, 7, 1, 2897, 2492, 1508, 752, 320, 116, 35, 8, 1, 10869, 9361, 5712, 2900, 1276, 489, 161, 44, 9, 1, 41414, 35702, 21933, 11296, 5095, 2034, 714
Offset: 0

Views

Author

N. J. A. Sloane, Jun 05 2005

Keywords

Comments

A convolution triangle of numbers based on A071969. - Philippe Deléham, Sep 15 2005

Examples

			Triangle begins:
   1;
   1,  1;
   2,  2,  1;
   6,  5,  3,  1;
  19, 16,  9,  4,  1; ...

Crossrefs

Cf. A071944, A071969.

Formula

T(0, 0) = 1; T(n, k) = 0 if k<0 or if k>n; T(n, k) = Sum_{j>=0} T(n-1, k-1+j) + Sum_{j>=0} T(n-1, k+2+j). - Philippe Deléham, Sep 15 2005

Extensions

More terms from Philippe Deléham, Sep 15 2005

A370246 Coefficient of x^n in the expansion of ( 1/(1-x) * (1+x^3) )^n.

Original entry on oeis.org

1, 1, 3, 13, 51, 201, 813, 3333, 13779, 57361, 240153, 1010109, 4264989, 18066777, 76745763, 326796213, 1394494803, 5961639969, 25528971369, 109482236013, 470145451401, 2021360463849, 8700225608583, 37484437325157, 161647475666301, 697673760945201
Offset: 0

Views

Author

Seiichi Manyama, Feb 13 2024

Keywords

Crossrefs

Cf. A071969, A370212.

Programs

  • PARI
    a(n, s=3, t=1, u=1) = sum(k=0, n\s, binomial(t*n, k)*binomial((u+1)*n-s*k-1, n-s*k));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n,k) * binomial(2*n-3*k-1,n-3*k).
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x) / (1+x^3) ). See A071969.
Previous Showing 11-13 of 13 results.

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

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