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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 Muniru A Asiru, Rows n=0..100 of triangle, flattened 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 Cf. A008949, A032443. Sequence in context: A208911 A208761 A123519 * A114687 A137594 A112360 Adjacent sequences: A167021 A167022 A167023 * A167025 A167026 A167027 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.)