A104259 Triangle T read by rows: matrix product of Pascal and Catalan triangle.

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

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

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.

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

T = A007318 * A033184.

Left-hand columns include A007317, A007317 - 1. Row sums are in A002212.

Sequence in context: A126181 A362924 A154930 * A137650 A363732 A171515

Adjacent sequences: A104256 A104257 A104258 * A104260 A104261 A104262

Keyword

nonn,tabl

Author

Ralf Stephan, Mar 17 2005

Status

approved