A167024 - OEIS

%I #29 Dec 28 2018 23:24:59

%S 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,

%T 855,378,64,1,56,609,2240,3465,2520,889,128,1,72,1036,5208,11410,

%U 12264,6916,2040,256,1,90,1656,10920,32256,48132,39144,18072,4599,512

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

%C Row sums are A032443(n).

%H Muniru A Asiru, <a href="/A167024/b167024.txt">Rows n=0..100 of triangle, flattened</a>

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

%F 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

%F 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

%e 1,

%e 1, 2,

%e 1, 6, 4,

%e 1, 12, 21, 8,

%e 1, 20, 66, 60, 16,

%e 1, 30, 160, 260, 155, 32,

%e 1, 42, 330, 840, 855, 378, 64,

%e 1, 56, 609, 2240, 3465, 2520, 889, 128,

%e 1, 72, 1036, 5208, 11410, 12264, 6916, 2040, 256,

%e 1, 90, 1656, 10920, 32256, 48132, 39144, 18072, 4599, 512,

%e 1, 110, 2520, 21120, 81060, 160776, 178080, 116160, 45585, 10230, 1024

%p 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

%t T[m_, n_] = If[m == 0 && n == 0, 1, Sum[Binomial[m, n]*Binomial[m, k], {k, 0, n}]]

%t Flatten[Table[Table[T[m, n], {n, 0, m}], {m, 0, 10}]]

%t 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 *)

%o (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

%Y Cf. A008949, A032443.

%K nonn,tabl

%O 0,3

%A _Roger L. Bagula_, Oct 27 2009

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