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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 * A208337 A208335 A208597

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 May 17 05:13 EDT 2021. Contains 343965 sequences. (Running on oeis4.)