A110488 -id:A110488 - cp's OEIS frontend

A110489 Row sums of a triangle based on the Catalan numbers.

1, 2, 5, 14, 43, 142, 497, 1828, 7037, 28326, 119361, 527748, 2454929, 12041410, 62354641, 340840118, 1963757863, 11896370734, 75549183725, 501393978466, 3467199478543, 24916100775758, 185646100106929, 1431332539961350

Offset: 0

Paul Barry, Jul 22 2005

Row sums of A110488.

G. C. Greubel, Table of n, a(n) for n = 0..595

Cf. A110488.

T[n_, 0] := CatalanNumber[n]; T[n_, 1] := CatalanNumber[n]; T[n_, n_] := 1; T[n_, k_] := Sum[2*(j + 1)*(k - 1)^j*Binomial[2 (n - k) + 1, n - k - j]/(n - k + j + 2), {j, 0, n - k}]; Table[Sum[T[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 29 2017 *)

a(n) = sum(k=0, n, sum(j=0, (n-k), 2*(j+1)*(k-1)^j*binomial(2*(n-k)+1, n-k-j)/ (n-k+j+2))); \\ Michel Marcus, Aug 29 2017

a(n) = Sum_{k=0..n} Sum_{j=0..(n-k)} 2*(j+1)*(k-1)^j*C(2*(n-k)+1, n-k-j)/ (n-k+j+2).

A110490 Diagonal sums of a triangle based on the Catalan numbers.

Original entry on oeis.org

1, 1, 3, 7, 20, 59, 185, 600, 2003, 6833, 23727, 83606, 298313, 1076155, 3920823, 14416987, 53482012, 200151737, 755894009, 2882782933, 11115015138, 43400057683, 172016505877, 694208585423, 2863726993748, 12130698802645

Offset: 0

Paul Barry, Jul 22 2005

Diagonal sums of A110488.

G. C. Greubel, Table of n, a(n) for n = 0..675

Cf. A110488.

T[n_, 0] := CatalanNumber[n]; T[n_, 1] := CatalanNumber[n]; T[n_, n_] := 1; T[n_, k_] := Sum[2*(j + 1)*(k - 1)^j*Binomial[2 (n - k) + 1, n - k - j]/(n - k + j + 2), {j, 0, n - k}]; Join[{1, 1}, Table[Sum[T[n - k, k], {k, 0, n}], {n,2,50}]] (* G. C. Greubel, Aug 29 2017 *)

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..(n-2*k)} 2*(j+1)*(k-1)^j*C(2*(n-2*k)+1, n-2*k-j)/(n-2*k+j+2).

cp's OEIS Frontend

A110489 Row sums of a triangle based on the Catalan numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula