A173120 - OEIS

%I #12 Apr 28 2021 02:01:30

%S 1,1,1,1,-2,1,1,-1,-1,1,1,0,-2,0,1,1,1,14,14,1,1,1,2,15,28,15,2,1,1,3,

%T 17,-21,-21,17,3,1,1,4,20,-4,-42,-4,20,4,1,1,5,24,16,210,210,16,24,5,

%U 1,1,6,29,40,226,420,226,40,29,6,1

%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 = -4, read by rows.

%H G. C. Greubel, <a href="/A173120/b173120.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 = -4.

%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 = -4. - _G. C. Greubel_, Apr 27 2021

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1, -2,  1;

%e   1, -1, -1,   1;

%e   1,  0, -2,   0,   1;

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

%e   1,  2, 15,  28,  15,   2,   1;

%e   1,  3, 17, -21, -21,  17,   3,  1;

%e   1,  4, 20,  -4, -42,  -4,  20,  4,  1;

%e   1,  5, 24,  16, 210, 210,  16, 24,  5, 1;

%e   1,  6, 29,  40, 226, 420, 226, 40, 29, 6, 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,-4], {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,-4) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 27 2021

%Y Cf. A007318 (q=0), A072405 (q= -1), A173117 (q=1), A173118 (q=2), A173119 (q=3), this sequence (q= -4), A173122.

%K sign,tabl,easy,less

%O 0,5

%A _Roger L. Bagula_, Feb 10 2010

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