A307374 -id:A307374 - cp's OEIS frontend

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).

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

Vladimir Kruchinin, Jan 25 2022

			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;

Row sums give A173992.

Cf. A000108, A307374.

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));

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

Ilya Gutkovskiy, Feb 23 2022

sign

Cf. A007558, A307374, A346078.

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}]

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

cp's OEIS Frontend

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).

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Maxima

Formula

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

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Formula

cp's OEIS Frontend

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).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maxima

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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).