A156717 Triangle read by rows: T(n,m) = binomial(n + m - 1, 2*m) + binomial(2*n - m - 2, 2*(n - m - 1)).

2, 2, 2, 2, 6, 2, 2, 11, 11, 2, 2, 17, 30, 17, 2, 2, 24, 63, 63, 24, 2, 2, 32, 115, 168, 115, 32, 2, 2, 41, 192, 375, 375, 192, 41, 2, 2, 51, 301, 748, 990, 748, 301, 51, 2, 2, 62, 450, 1379, 2288, 2288, 1379, 450, 62, 2, 2, 74, 648, 2396, 4823, 6006, 4823, 2396, 648, 74, 2

Offset

1,1

Links

Stefano Spezia, First 150 rows of the triangle, flattened.

Formula

T(n,m) = binomial(n + m - 1, 2*m) + binomial(2*n - m - 2, 2*(n - m - 1)).

From Stefano Spezia, Dec 26 2018: (Start)

T(n,m) = A007318(n + m - 1, 2*m) + A007318(2*n - m - 2, 2*(n - m - 1)).

Sum_{m=0..n-1} T(n,m) = A052995(n).

(End)

Example

n\m|  0     1     2     3     4     5     6     7     8
---+---------------------------------------------------
1  |  2
2  |  2     2
3  |  2     6     2
4  |  2    11    11     2
5  |  2    17    30    17     2
6  |  2    24    63    63    24     2
7  |  2    32   115   168   115    32     2
8  |  2    41   192   375   375   192    41     2
9  |  2    51   301   748   990   748   301    51     2

Maple

a := (n, m) -> binomial(n+m-1, 2*m)+binomial(2*n-m-2, 2*(n-m-1)): seq(seq(a(n, m), m = 0 .. n-1), n = 1 .. 10) # Stefano Spezia, Dec 26 2018

Mathematica

Flatten[Table[Table[Binomial[n + m - 1, 2*m] + Binomial[2*n - m - 2, 2*(n - m - 1)], {m, 0, n - 1}], {n, 1, 10}]]

Prog

(GAP) Flat(List([1..10], n->List([0..n-1], m->Binomial(n + m - 1, 2*m) + Binomial(2*n - m - 2, 2*(n - m - 1))))); # Stefano Spezia, Dec 26 2018
(PARI) T(n, m) = binomial(n+m-1, 2*m)+binomial(2*n-m-2, 2*(n-m-1)); \\ Stefano Spezia, Dec 26 2018

Crossrefs

Cf. A007318, A052995 (row sums).

Sequence in context: A073124 A278260 A070877 * A198889 A329814 A130754

Adjacent sequences: A156714 A156715 A156716 * A156718 A156719 A156720

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 14 2009

Extensions

Edited by Stefano Spezia, Dec 26 2018

Status

approved