OFFSET
0,2
COMMENTS
Triangle T(n,k), 0 <= k <= n, read by rows given by [2,1,1,1,1,1,1,...] DELTA [1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jun 05 2007
Inverse of Riordan array (1/(1+2*x), x*(1+x)/(1+2*x)^2) (see A123876). - Philippe Deléham, Oct 25 2007
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
Joseph Pappe, Digjoy Paul and Anne Schilling, An area-depth symmetric q,t-Catalan polynomial, arXiv:2109.06300 [math.CO], 2021. See Remark 2.4 p. 4.
FORMULA
Number triangle T(n,k) = (4^n/2^k)*Sum_{j=0..k} C(k,j)*C(n+(j-1)/2,n)*(-1)^(k-j).
Sum_{k=0..n} (-1)^k*T(n,k) = A000108(n), Catalan numbers. - Philippe Deléham, Nov 07 2006
T(n,k) = Sum_{j>=0} A039599(n,j)*binomial(j,k). - Philippe Deléham, Mar 30 2007
Sum_{k=0..n} T(n,k)*x^k = A127053(n), A126985(n), A127016(n), A127017(n), A126987(n), A126986(n), A126982(n), A126984(n), A126983(n), A000007(n), A000108(n), A000984(n), A007854(n), A076035(n), A076036(n), A127628(n), A126694(n), A115970(n) for x = -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Oct 25 2007
EXAMPLE
Triangle begins:
1;
2, 1;
6, 5, 1;
20, 22, 8, 1;
70, 93, 47, 11, 1;
252, 386, 244, 81, 14, 1;
MATHEMATICA
T[n_, k_]:= (4^n/2^k)*Sum[(-1)^(k-j)*Binomial[k, j]*Binomial[n+(j-1)/2, n], {j, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 28 2019 *)
PROG
(PARI) {T(n, k) = (4^n/2^k)*sum(j=0, k, (-1)^(k-j)*binomial(k, j)* binomial(n+(j-1)/2, n))}; \\ G. C. Greubel, May 28 2019
(Magma) [[ Round((4^n/2^k)*(&+[ (-1)^(k-j)*Binomial(k, j)*Gamma(n+(j+1)/2)/(Factorial(n)*Gamma((j+1)/2)) : j in [0..k]])) : k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 28 2019
(Sage) [[(4^n/2^k)*sum( (-1)^(k-j)*binomial(k, j)* binomial(n+(j-1)/2, n) for j in (0..k)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 28 2019
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Feb 12 2006
STATUS
approved