A116395 Riordan array (1/sqrt(1-4*x), (1/sqrt(1-4*x)-1)/2).

1, 2, 1, 6, 5, 1, 20, 22, 8, 1, 70, 93, 47, 11, 1, 252, 386, 244, 81, 14, 1, 924, 1586, 1186, 500, 124, 17, 1, 3432, 6476, 5536, 2794, 888, 176, 20, 1, 12870, 26333, 25147, 14649, 5615, 1435, 237, 23, 1, 48620, 106762, 112028, 73489, 32714, 10135, 2168, 307, 26, 1

Offset

0,2

Comments

Row sums are A007854. Diagonal sums are A116396.

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

Sequence in context: A193723 A260914 A159965 * A159924 A133367 A179456

Adjacent sequences: A116392 A116393 A116394 * A116396 A116397 A116398

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 12 2006

Status

approved