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A104259
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Triangle T read by rows: matrix product of Pascal and Catalan triangle.
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10
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1, 2, 1, 5, 4, 1, 15, 14, 6, 1, 51, 50, 27, 8, 1, 188, 187, 113, 44, 10, 1, 731, 730, 468, 212, 65, 12, 1, 2950, 2949, 1956, 970, 355, 90, 14, 1, 12235, 12234, 8291, 4356, 1785, 550, 119, 16, 1, 51822, 51821, 35643, 19474, 8612, 3021, 805, 152, 18, 1
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OFFSET
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0,2
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COMMENTS
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Also, Riordan array (G,G), G(t)=(1 - ((1-5*t)/(1-t))^(1/2))/(2*t).
From Emanuele Munarini, May 18 2011: (Start)
Row sums = A002212.
Diagonal sums = A190737.
Central coefficients = A190738. (End)
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LINKS
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Robert Israel, Table of n, a(n) for n = 0..5049
D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139, 2000. [alternative link]
D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000.
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FORMULA
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T(n,k) = sum(binomial(n,i)*binomial(2*i-k,i-k)*(k+1)/(i+1),i=k..n).
T(n+1,k+2) = T(n+1,k+1) + T(n,k+2) - T(n,k+1) - T(n,k). - Emanuele Munarini, May 18 2011
T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + Sum_{i, i>=0} T(n-1,k+1+i). - Philippe Deléham, Feb 23 2012
T(n,k) = C(n,k)*hypergeom([k/2+1/2,k/2+1,k-n],[k+1,k+2],-4). - Peter Luschny, Sep 23 2014
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EXAMPLE
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Triangle begins:
1
2, 1
5, 4, 1
15, 14, 6, 1
51, 50, 27, 8, 1
188, 187, 113, 44, 10, 1
731, 730, 468, 212, 65, 12, 1
2950, 2949, 1956, 970, 355, 90, 14, 1
12235, 12234, 8291, 4356, 1785, 550, 119, 16, 1
Production matrix begins
2, 1
1, 2, 1
1, 1, 2, 1
1, 1, 1, 2, 1
1, 1, 1, 1, 2, 1
1, 1, 1, 1, 1, 2, 1
1, 1, 1, 1, 1, 1, 2, 1
... - Philippe Deléham, Mar 01 2013
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MAPLE
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T := (n, k) -> binomial(n, k)*hypergeom([k/2+1/2, k/2+1, k-n], [k+1, k+2], -4); seq(print(seq(round(evalf(T(n, k), 99)), k=0..n)), n=0..8); # Peter Luschny, Sep 23 2014
# Alternative:
N:= 12: # to get the first N rows
P:= Matrix(N, N, (i, j) -> binomial(i-1, j-1), shape=triangular[lower]):
C:= Matrix(N, N, (i, j) -> binomial(2*i-j-1, i-j)*j/i, shape=triangular[lower]):
T:= P . C:
for i from 1 to N do
seq(T[i, j], j=1..i)
od; # Robert Israel, Sep 23 2014
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MATHEMATICA
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Flatten[Table[Sum[Binomial[n, i]Binomial[2i-k, i-k](k+1)/(i+1), {i, k, n}], {n, 0, 100}, {k, 0, n}]] (* Emanuele Munarini, May 18 2011 *)
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PROG
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(Maxima) create_list(sum(binomial(n, i)*binomial(2*i-k, i-k)*(k+1)/(i+1), i, k, n), n, 0, 12, k, 0, n); /* Emanuele Munarini, May 18 2011 */
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CROSSREFS
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T = A007318 * A033184.
Left-hand columns include A007317, A007317 - 1. Row sums are in A002212.
Sequence in context: A193673 A126181 A154930 * A137650 A171515 A110271
Adjacent sequences: A104256 A104257 A104258 * A104260 A104261 A104262
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KEYWORD
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nonn,tabl
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AUTHOR
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Ralf Stephan, Mar 17 2005
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STATUS
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approved
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