A173117 - OEIS

%I #8 Apr 28 2021 01:59:15

%S 1,1,1,1,3,1,1,4,4,1,1,5,8,5,1,1,6,14,14,6,1,1,7,20,28,20,7,1,1,8,27,

%T 49,49,27,8,1,1,9,35,76,98,76,35,9,1,1,10,44,111,175,175,111,44,10,1,

%U 1,11,54,155,286,350,286,155,54,11,1

%N Triangle T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 1, read by rows.

%H G. C. Greubel, <a href="/A173117/b173117.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 1.

%F Sum_{k=0..n} T(n, k, q) = [n=0] + q*[n=2] + Sum_{j=0..5} q^j*2^(n-2*j)*[n > 2*j] for q = 1. - _G. C. Greubel_, Apr 27 2021

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1,  3,  1;

%e   1,  4,  4,   1;

%e   1,  5,  8,   5,   1;

%e   1,  6, 14,  14,   6,   1;

%e   1,  7, 20,  28,  20,   7,   1;

%e   1,  8, 27,  49,  49,  27,   8,   1;

%e   1,  9, 35,  76,  98,  76,  35,   9,  1;

%e   1, 10, 44, 111, 175, 175, 111,  44, 10,  1;

%e   1, 11, 54, 155, 286, 350, 286, 155, 54, 11, 1;

%t T[n_, k_, q_]:= If[k==0 || k==n, 1, q*Boole[n==2] + Sum[q^j*Binomial[n-2*j, k-j]*Boole[n>2*j], {j,0,5}]];

%t Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 27 2021 *)

%o (Sage)

%o @CachedFunction

%o def T(n,k,q): return 1 if (k==0 or k==n) else q*bool(n==2) + sum( q^j*binomial(n-2*j, k-j)*bool(n>2*j) for j in (0..5) )

%o flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 27 2021

%Y Cf. A007318 (q=0), A072405 (q= -1), this sequence (q=1), A173118 (q=2), A173119 (q=3), A173120 (q= -4), A173122.

%K nonn,tabl,easy,less

%O 0,5

%A _Roger L. Bagula_, Feb 10 2010

%E Edited by _G. C. Greubel_, Apr 27 2021