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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A348593 Triangle read by rows: T(n,m) = Sum_{j=0..min(m,n-m)} C(2j,j)*C(n-2j-1,m-j)*C(n-m,j)/(j+1).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 1, 6, 7, 1, 1, 8, 18, 6, 1, 1, 10, 34, 30, 7, 1, 1, 12, 55, 88, 33, 8, 1, 1, 14, 81, 195, 145, 42, 9, 1, 1, 16, 112, 366, 460, 184, 52, 10, 1, 1, 18, 148, 616, 1146, 763, 248, 63, 11, 1, 1, 20, 189, 960, 2422, 2544, 1060, 324, 75, 12, 1, 1, 22, 235, 1413, 4558, 6916, 4282, 1490, 413, 88, 13, 1
Offset: 0

Views

Author

Vladimir Kruchinin, Jan 25 2022

Keywords

Examples

			Triangle begins
  1;
  1;
  1,  2;
  1,  4,  1;
  1,  6,  7,  1;
  1,  8, 18,  6,  1;
  1, 10, 34, 30,  7,  1;
  1, 12, 55, 88, 33,  8,  1;
		

Crossrefs

Row sums give A173992.

Programs

  • Maxima
    T(n,m):=sum(binomial(2*j,j)*binomial(n-2*j-1,m-j)*binomial(n-m,j)/(j+1), j,0,min(m,n-m));

Formula

G.f.: (1-sqrt(1-4*x^2*y*(1-x*y)/(1-x-x*y)))/(2*x^2*y).
Sum_{m>=0} (-1)^m * T(n,m) = A307374(n). - Alois P. Heinz, Jan 26 2022

A351879 a(0) = a(1) = 1; a(n) = -Sum_{k=0..n-2} binomial(n-2,k) * a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, -1, -2, 0, 10, 10, -60, -220, 400, 4200, 2200, -90200, -290400, 1892000, 15796000, -24024000, -775676000, -1592492000, 36509880000, 240055640000, -1435950560000, -23703057840000, 7376731120000, 2082346354000000, 9478853472000000, -162472029808000000
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 23 2022

Keywords

Crossrefs

Programs

  • Mathematica
    a[0] = a[1] = 1; a[n_] := a[n] = -Sum[Binomial[n - 2, k] a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 26}]

Formula

E.g.f. A(x) satisfies: A(x) = 1 + x - Integral( Integral A(x)^2 dx) dx.
Showing 1-2 of 2 results.