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A104259 Triangle T read by rows: matrix product of Pascal and Catalan triangle. 10
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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)
LINKS
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.
FORMULA
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
EXAMPLE
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
MAPLE
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
MATHEMATICA
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 *)
PROG
(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 */
CROSSREFS
Left-hand columns include A007317, A007317 - 1. Row sums are in A002212.
Sequence in context: A126181 A362924 A154930 * A137650 A363732 A171515
KEYWORD
nonn,tabl
AUTHOR
Ralf Stephan, Mar 17 2005
STATUS
approved

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Last modified July 24 05:18 EDT 2024. Contains 374575 sequences. (Running on oeis4.)