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 A185943 Riordan array ((1/(1-x))^m, x*A000108(x)), m = 2. 3
 1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 16, 12, 5, 1, 6, 39, 34, 18, 6, 1, 7, 104, 98, 59, 25, 7, 1, 8, 301, 294, 190, 92, 33, 8, 1, 9, 927, 919, 618, 324, 134, 42, 9, 1, 10, 2983, 2974, 2047, 1128, 510, 186, 52, 10, 1, 11, 9901, 9891, 6908, 3934, 1887, 759, 249, 63, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened FORMULA R(n,k,m) = k*Sum_{i=0..n-k} binomial(i+m-1, m-1)*binomial(2*(n-i)-k-1, n-i-1)/(n-i), m = 2, k > 0. R(n,0,2) = n + 1. Conjecture: R(n,1,2) = A014140(n-1). R(n,2,2) = A014143(n-2). - R. J. Mathar, Feb 11 2011 EXAMPLE Array begins 1; 2, 1; 3, 3, 1; 4, 7, 4, 1; 5, 16, 12, 5, 1; 6, 39, 34, 18, 6, 1; 7, 104, 98, 59, 25, 7, 1; 8, 301, 294, 190, 92, 33, 8, 1; Production matrix begins: 2, 1; -1, 1, 1; 1, 1, 1, 1; 0, 1, 1, 1, 1; 0, 1, 1, 1, 1, 1; 0, 1, 1, 1, 1, 1, 1; 0, 1, 1, 1, 1, 1, 1, 1; 0, 1, 1, 1, 1, 1, 1, 1, 1; ... Philippe Deléham, Sep 20 2014 MATHEMATICA r[n_, k_, m_] := k*Sum[ Binomial[i + m - 1, m - 1]*Binomial[2*(n - i) - k - 1, n - i - 1]/(n - i), {i, 0, n - k}]; r[n_, 0, 2] := n + 1; Table[r[n, k, 2], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 13 2012, from formula *) PROG (Sage) @CachedFunction def A(n, k): if n==k: return n+1 return add(A(n-1, j) for j in (0..k)) A185943 = lambda n, k: A(n, n-k) for n in (0..7) : print([A185943(n, k) for k in (0..n)]) # Peter Luschny, Nov 14 2012 CROSSREFS Cf. A091491 (m=1), A185944 (m=3), A185945 (m=4). Row sums A014140. Cf. A000108, A014143. Sequence in context: A253273 A055129 A133804 * A352001 A208337 A208335 Adjacent sequences: A185940 A185941 A185942 * A185944 A185945 A185946 KEYWORD nonn,tabl AUTHOR Vladimir Kruchinin, Feb 07 2011 STATUS approved

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Last modified August 6 15:42 EDT 2024. Contains 374974 sequences. (Running on oeis4.)