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A185943 Riordan array ((1/(1-x))^m, x*A000108(x)), m = 2. 3

%I

%S 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,

%T 301,294,190,92,33,8,1,9,927,919,618,324,134,42,9,1,10,2983,2974,2047,

%U 1128,510,186,52,10,1,11,9901,9891,6908,3934,1887,759,249,63,11,1

%N Riordan array ((1/(1-x))^m, x*A000108(x)), m = 2.

%H G. C. Greubel, <a href="/A185943/b185943.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F 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.

%F R(n,0,2) = n + 1.

%F Conjecture: R(n,1,2) = A014140(n-1). R(n,2,2) = A014143(n-2). - _R. J. Mathar_, Feb 11 2011

%e Array begins

%e 1;

%e 2, 1;

%e 3, 3, 1;

%e 4, 7, 4, 1;

%e 5, 16, 12, 5, 1;

%e 6, 39, 34, 18, 6, 1;

%e 7, 104, 98, 59, 25, 7, 1;

%e 8, 301, 294, 190, 92, 33, 8, 1;

%e Production matrix begins:

%e 2, 1;

%e -1, 1, 1;

%e 1, 1, 1, 1;

%e 0, 1, 1, 1, 1;

%e 0, 1, 1, 1, 1, 1;

%e 0, 1, 1, 1, 1, 1, 1;

%e 0, 1, 1, 1, 1, 1, 1, 1;

%e 0, 1, 1, 1, 1, 1, 1, 1, 1;

%e ... _Philippe Deléham_, Sep 20 2014

%t 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 *)

%o (Sage)

%o @CachedFunction

%o def A(n, k):

%o if n==k: return n+1

%o return add(A(n-1, j) for j in (0..k))

%o A185943 = lambda n,k: A(n, n-k)

%o for n in (0..7) :

%o print([A185943(n, k) for k in (0..n)]) # _Peter Luschny_, Nov 14 2012

%Y Cf. A091491 (m=1), A185944 (m=3), A185945 (m=4).

%Y Row sums A014140. Cf. A000108, A014143.

%K nonn,tabl

%O 0,2

%A _Vladimir Kruchinin_, Feb 07 2011

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Last modified June 22 16:35 EDT 2021. Contains 345388 sequences. (Running on oeis4.)