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%I #48 Apr 24 2022 09:15:27
%S 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,
%T 21,109,235,271,179,67,13,1,34,201,506,705,591,303,92,15,1,55,365,
%U 1051,1717,1746,1140,475,121,17,1,89,655,2123,3979,4759,3780,2010,703,154,19,1
%N Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
%H Reinhard Zumkeller, <a href="/A063967/b063967.txt">Rows n = 0..120 of triangle, flattened</a>
%H E. Deutsch, L. Ferrari and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.aam.2004.05.002">Production Matrices</a>, Advances in Mathematics, 34 (2005) pp. 101-122.
%H Emanuele Munarini, <a href="https://doi.org/10.1515/puma-2015-0028">A generalization of André-Jeannin's symmetric identity</a>, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 98-118.
%F G.f.: 1/(1-x*(1+x)*(1+y)). - _Vladeta Jovovic_, Oct 11 2003
%F 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
%F T(n, k) = Sum_{j=0..n} C(j, n-j)C(j, k). - _Paul Barry_, Nov 09 2005
%F Diagonal sums are A002478. - _Paul Barry_, Nov 09 2005
%F A026729*A007318 as infinite lower triangular matrices. - _Philippe Deléham_, Dec 11 2008
%F Central coefficients T(2*n,n) are A137644. - _Paul Barry_, Apr 15 2010
%F Product of Riordan arrays (1, x(1+x))*(1/(1-x), x/(1-x)), that is, A026729*A007318. - _Paul Barry_, Mar 14 2011
%F 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
%e T(3,1) = T(2,1) + T(1,1) + T(2,0) + T(1,0) = 3 + 1 + 2 + 1 = 7.
%e Triangle begins:
%e 1,
%e 1, 1,
%e 2, 3, 1,
%e 3, 7, 5, 1,
%e 5, 15, 16, 7, 1,
%e 8, 30, 43, 29, 9, 1,
%e 13, 58, 104, 95, 46, 11, 1,
%e 21, 109, 235, 271, 179, 67, 13, 1,
%e 34, 201, 506, 705, 591, 303, 92, 15, 1
%t 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_ *)
%t (* Function RiordanSquare defined in A321620. *)
%t RiordanSquare[1/(1 - x - x^2), 11] // Flatten (* _Peter Luschny_, Nov 27 2018 *)
%o (Haskell)
%o a063967_tabl = [1] : [1,1] : f [1] [1,1] where
%o f us vs = ws : f vs ws where
%o ws = zipWith (+) ([0] ++ us ++ [0]) $
%o zipWith (+) (us ++ [0,0]) $ zipWith (+) ([0] ++ vs) (vs ++ [0])
%o -- _Reinhard Zumkeller_, Apr 17 2013
%Y Row sums are A002605.
%Y Columns include: A000045(n+1), A023610(n-1).
%Y Main diagonal: A000012, a(n, n-1) = A005408(n-1).
%Y Matrix inverse: A091698, matrix square: A091700.
%Y Cf. A321620.
%Y 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
%K easy,nonn,tabl
%O 0,4
%A _Henry Bottomley_, Sep 05 2001
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