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%I #20 Jul 27 2022 10:30:24
%S 1,1,1,1,2,1,2,3,3,1,3,5,6,4,1,4,8,11,10,5,1,6,12,19,21,15,6,1,9,18,
%T 31,40,36,21,7,1,13,27,49,71,76,57,28,8,1,19,40,76,120,147,133,85,36,
%U 9,1,28,59,116,196,267,280,218,121,45,10,1,41,87,175,312,463,547,498,339,166,55,11,1
%N Riordan array (1/(1-x-x^3), 1/(1-x)).
%C Inverse matrix is A099569.
%C Subtriangle of the triangle in A144903. - _Philippe Deléham_, Dec 29 2013
%H G. C. Greubel, <a href="/A099567/b099567.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Number triangle T(n, k) = Sum_{j=0..floor(n/3)} binomial(n-2*j, k+j).
%F Columns have g.f. (1/(1-x-x^3))*(x/(1-x))^k.
%F Sum_{k=0..n} T(n, k) = A099568(n).
%F T(n,0) = A000930(n), T(n,n) = 1, T(n,k) = T(n-1,k-1) + T(n-1,k) for 0<k<n. - _Philippe Deléham_, Dec 29 2013
%F exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(2 + 3*x + 3*x^2/2! + x^3/3!) = 2 + 5*x + 11*x^2/2! + 21*x^3/3! + 36*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - _Peter Bala_, Dec 21 2014
%F From _G. C. Greubel_, Jul 27 2022: (Start)
%F T(n, n-1) = n, for n >= 1.
%F T(n, n-2) = A000217(n-1), for n >= 2.
%F T(n, n-3) = A050407(n+1), for n >= 3.
%F T(2*n, n) = A144904(n+1), for n >= 1. (End)
%e Rows begin:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 2, 3, 3, 1;
%e 3, 5, 6, 4, 1;
%e 4, 8, 11, 10, 5, 1;
%e 6, 12, 19, 21, 15, 6, 1;
%e 9, 18, 31, 40, 36, 21, 7, 1;
%e 13, 27, 49, 71, 76, 57, 28, 8, 1;
%e 19, 40, 76, 120, 147, 133, 85, 36, 9, 1;
%e 28, 59, 116, 196, 267, 280, 218, 121, 45, 10, 1;
%t T[n_, 0]:=T[n,0]=HypergeometricPFQ[{(1-n)/3,(2-n)/3,-n/3}, {(1-n)/2,-n/2}, -27/4];
%t T[n_, k_]:= T[n,k]= If[k==n, 1, T[n-1,k-1] +T[n-1,k]];
%t Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 28 2017 *)
%o (Magma)
%o T:= func< n,k | (&+[Binomial(n-2*j, k+j): j in [0..Floor(n/3)]]) >;
%o [[T(n,k): k in [0..n]]: n in [0..15]]; // _G. C. Greubel_, Jul 27 2022
%o (SageMath)
%o @CachedFunction
%o def A099567(n, k): return sum( binomial(n-2*j, k+j) for j in (0..(n//3)) )
%o flatten([[A099567(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jul 27 2022
%Y Cf. A000217, A050407, A099568 (row sums), A099569, A144903, A144904.
%Y Columns: A000930, A077868, A050228, A226405, A144898, A144899, A144900, A144901, A144902.
%K easy,nonn,tabl
%O 0,5
%A _Paul Barry_, Oct 22 2004
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