A099567 - OEIS

%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