A167024 Triangle read by rows: T(n, m) = binomial(n, m)* Sum_{k=0..m} binomial(n, k) for 0 <= m <= n.

1, 1, 2, 1, 6, 4, 1, 12, 21, 8, 1, 20, 66, 60, 16, 1, 30, 160, 260, 155, 32, 1, 42, 330, 840, 855, 378, 64, 1, 56, 609, 2240, 3465, 2520, 889, 128, 1, 72, 1036, 5208, 11410, 12264, 6916, 2040, 256, 1, 90, 1656, 10920, 32256, 48132, 39144, 18072, 4599, 512

Offset: 0

Author

Roger L. Bagula, Oct 27 2009

Keywords

• nonn
• tabl

Comments

Row sums are A032443(n).

Examples

			1,
1, 2,
1, 6, 4,
1, 12, 21, 8,
1, 20, 66, 60, 16,
1, 30, 160, 260, 155, 32,
1, 42, 330, 840, 855, 378, 64,
1, 56, 609, 2240, 3465, 2520, 889, 128,
1, 72, 1036, 5208, 11410, 12264, 6916, 2040, 256,
1, 90, 1656, 10920, 32256, 48132, 39144, 18072, 4599, 512,
1, 110, 2520, 21120, 81060, 160776, 178080, 116160, 45585, 10230, 1024
		

Links

• Muniru A Asiru, Rows n=0..100 of triangle, flattened

Crossrefs

Cf. A008949, A032443.

Programs

GAP

t:=Flat(List([0..10],n->List([0..n],m->Binomial(n,m)*Sum([0..m],k->Binomial(n,k)))));; Print(t); # Muniru A Asiru, Dec 28 2018

Maple

T:=(n, m)-> binomial(n, m)*add(binomial(n, k), k=0..m): seq(seq(T(n, m), m=0..n), n=0..9); # Muniru A Asiru, Dec 28 2018

Mathematica

T[m_, n_] = If[m == 0 && n == 0, 1, Sum[Binomial[m, n]*Binomial[m, k], {k, 0, n}]]
Flatten[Table[Table[T[m, n], {n, 0, m}], {m, 0, 10}]]
T[n_,k_] := Binomial[n, k] (2^n - Binomial[n, k + 1] Hypergeometric2F1[1, 1 -n + k, k + 2, -1]); Table[T[n,k], {n,0,8}, {k,0,n}] // Flatten (* Peter Luschny, Dec 28 2018 *)

Formula

T(n, m) = binomial(n,m)*A008949(n,m). [Nov 03 2009]

G.f.: (1/x)*d(arctanh(N(x,y)))/dy, where N(x,y) is g.f. of Narayana numbers (A001263). - Vladimir Kruchinin, Apr 11 2018

T(n, k) = binomial(n, k)*(2^n - binomial(n, 1+k)*hypergeom([1, 1+k-n], [k+2], -1)). - Peter Luschny, Dec 28 2018

Extensions

Introduced OEIS notational standards in the definition - The Assoc. Editors of the OEIS, Nov 05 2009