OFFSET
0,5
COMMENTS
Row sums are A014137 (partial sums of Catalan numbers A000108). Columns equal the partial sums of the columns of the Catalan convolution triangle A033184. Columns include A014137, A014138, A001453.
Apart from the first column, any term is the partial sum of terms of the row above, when summing from the right. - Ralf Stephan, Apr 27 2004
Matrix inverse equals triangle A104402.
Riordan array (1/(1-x), x*c(x)) where c(x) is the g.f. of A000108. - Philippe Deléham, Nov 04 2009
LINKS
Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
FORMULA
T(n, k) = Sum_{j=0..n-k} T(n-k, j)*C(k+j-1, k-1).
G.f.: 2/(2-y*(1-sqrt(1-4*x)))/(1-x).
T(n, k) = T(n-1, k-1) + T(n, k+1) for n>0, with T(n, 0)=1.
Recurrence: for k>0, T(n, k) = Sum_{j=k..n} T(n-1, j). - Ralf Stephan, Apr 27 2004
T(n+2,2)= |A099324(n+2)|. - Philippe Deléham, Nov 25 2009
T(n,k) = k * Sum_{i=0..n-k} binomial(2*(n-i)-k-1, n-i-1)/(n-i) for k>0; T(n,0)=1. - Vladimir Kruchinin, Feb 07 2011
From Gary W. Adamson, Jul 26 2011: (Start)
The n-th row of the triangle is the top row of M^n, where M is the following infinite square production matrix in which a column of (1,0,0,0,...) is prepended to an infinite lower triangular matrix of all 1's and the rest zeros:
1, 1, 0, 0, 0, 0, ...
0, 1, 1, 0, 0, 0, ...
0, 1, 1, 1, 0, 0, ...
0, 1, 1, 1, 1, 0, ...
0, 1, 1, 1, 1, 1, ...
(End)
EXAMPLE
T(7,3) = T(4,0)*C(2,2) + T(4,1)*C(3,2) + T(4,2)*C(5,2) + T(4,3)*C(6,2) = (1)*1 + (4)*3 + (3)*6 + (1)*10 = 41.
Rows begin:
1;
1, 1;
1, 2, 1;
1, 4, 3, 1;
1, 9, 8, 4, 1;
1, 23, 22, 13, 5, 1;
1, 65, 64, 41, 19, 6, 1;
1, 197, 196, 131, 67, 26, 7, 1;
1, 626, 625, 428, 232, 101, 34, 8, 1;
1, 2056, 2055, 1429, 804, 376, 144, 43, 9, 1;
1, 6918, 6917, 4861, 2806, 1377, 573, 197, 53, 10, 1;
1, 23714, 23713, 16795, 9878, 5017, 2211, 834, 261, 64, 11, 1;
1, 82500, 82499, 58785, 35072, 18277, 8399, 3382, 1171, 337, 76, 12, 1;
...
As to the production matrix M, top row of M^3 = [1, 4, 3, 1, 0, 0, 0, ...].
MATHEMATICA
nmax = 11; t[n_, k_] := k*(2n-k-1)!*HypergeometricPFQ[{1, k-n, -n}, {k/2-n+1/2, k/2-n+1}, 1/4]/(n!*(n-k)!); t[_, 0] = 1; Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)
PROG
(PARI) T(n, k)=if(k>n || n<0 || k<0, 0, if(k==0 || k==n, 1, sum(j=0, n-k, T(n-k, j)*binomial(k+j-1, k-1)); ); )
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff(2/(2-Y*(1-sqrt(1-4*X)))/(1-X), n, x), k, y)
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k || k==0, 1, T(n-1, k-1)+T(n, k+1)))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(Haskell)
a091491 n k = a091491_tabl !! n !! k
a091491_row n = a091491_tabl !! n
a091491_tabl = iterate (\row -> 1 : scanr1 (+) row) [1]
-- Reinhard Zumkeller, Jul 12 2012
(Magma)
A091491:= func< n, k | k eq 0 select 1 else k*(&+[Binomial(2*(n-j)-k-1, n-j-1)/(n-j): j in [0..n-k]]) >;
[A091491(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 30 2021
(Sage)
def A091491(n, k): return 1 if (k==0) else k*sum(binomial(2*(n-j)-k-1, n-j-1)/(n-j) for j in (0..n-k))
flatten([[A091491(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 30 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jan 14 2004
STATUS
approved