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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

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

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: A193673 A126181 A154930 * A137650 A171515 A110271

Adjacent sequences:  A104256 A104257 A104258 * A104260 A104261 A104262

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Mar 17 2005

STATUS

approved

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Last modified October 22 10:44 EDT 2018. Contains 316436 sequences. (Running on oeis4.)