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A167024 Triangle read by rows: T(n, m) = binomial(n, m)* Sum_{k=0..m} binomial(n, k) for 0 <= m <= n. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Row sums are A032443(n).
LINKS
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
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
Sequence in context: A208911 A208761 A123519 * A114687 A137594 A112360
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Oct 27 2009
EXTENSIONS
Introduced OEIS notational standards in the definition - The Assoc. Editors of the OEIS, Nov 05 2009
STATUS
approved

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Last modified July 20 11:48 EDT 2024. Contains 374445 sequences. (Running on oeis4.)