OFFSET
0,4
LINKS
Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
Emanuele Munarini, A generalization of André-Jeannin's symmetric identity, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 98-118.
FORMULA
G.f.: 1/(1-x*(1+x)*(1+y)). - Vladeta Jovovic, Oct 11 2003
Riordan array (1/(1-x-x^2), x(1+x)/(1-x-x^2)). The inverse of the signed version (1/(1+x-x^2),x(1-x)/(1+x-x^2)) is abs(A091698). - Paul Barry, Jun 10 2005
T(n, k) = Sum_{j=0..n} C(j, n-j)C(j, k). - Paul Barry, Nov 09 2005
Diagonal sums are A002478. - Paul Barry, Nov 09 2005
Central coefficients T(2*n,n) are A137644. - Paul Barry, Apr 15 2010
Product of Riordan arrays (1, x(1+x))*(1/(1-x), x/(1-x)), that is, A026729*A007318. - Paul Barry, Mar 14 2011
Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 12 2011
EXAMPLE
T(3,1) = T(2,1) + T(1,1) + T(2,0) + T(1,0) = 3 + 1 + 2 + 1 = 7.
Triangle begins:
1,
1, 1,
2, 3, 1,
3, 7, 5, 1,
5, 15, 16, 7, 1,
8, 30, 43, 29, 9, 1,
13, 58, 104, 95, 46, 11, 1,
21, 109, 235, 271, 179, 67, 13, 1,
34, 201, 506, 705, 591, 303, 92, 15, 1
MATHEMATICA
T[n_, k_] := Sum[Binomial[j, n - j]*Binomial[j, k], {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 11 2017, after Paul Barry *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[1/(1 - x - x^2), 11] // Flatten (* Peter Luschny, Nov 27 2018 *)
PROG
(Haskell)
a063967_tabl = [1] : [1, 1] : f [1] [1, 1] where
f us vs = ws : f vs ws where
ws = zipWith (+) ([0] ++ us ++ [0]) $
zipWith (+) (us ++ [0, 0]) $ zipWith (+) ([0] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Apr 17 2013
CROSSREFS
Row sums are A002605.
Cf. A321620.
Sum_{k=0..n} x^k*T(n,k) is (-1)^n*A057086(n) (x=-11), (-1)^n*A057085(n+1) (x=-10), (-1)^n*A057084(n) (x=-9), (-1)^n*A030240(n) (x=-8), (-1)^n*A030192(n) (x=-7), (-1)^n*A030191(n) (x=-6), (-1)^n*A001787(n+1) (x=-5), A000748(n) (x=-4), A108520(n) (x=-3), A049347(n) (x=-2), A000007(n) (x=-1), A000045(n) (x=0), A002605(n) (x=1), A030195(n+1) (x=2), A057087(n) (x=3), A057088(n) (x=4), A057089(n) (x=5), A057090(n) (x=6), A057091(n) (x=7), A057092(n) (x=8), A057093(n) (x=9). - Philippe Deléham, Nov 03 2006
KEYWORD
AUTHOR
Henry Bottomley, Sep 05 2001
STATUS
approved