A173046 - OEIS

%I #5 Feb 16 2021 17:49:00

%S 1,1,1,1,5,1,1,10,10,1,1,19,37,19,1,1,36,105,105,36,1,1,69,270,403,

%T 270,69,1,1,134,660,1314,1314,660,134,1,1,263,1563,3895,5189,3895,

%U 1563,263,1,1,520,3619,10835,18045,18045,10835,3619,520,1,1,1033,8236,28791,57553,71931,57553,28791,8236,1033,1

%N Triangle T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 2, read by rows.

%C The triangle sequences having the form T(n,k,q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 have the row sums Sum_{k=0..n} T(n,k,q) = 2^(n-2)*q^n + 2^n - (n-1) - (5/4)*[n=0] -(q/2)*[n=1]. - _G. C. Greubel_, Feb 16 2021

%H G. C. Greubel, <a href="/A173046/b173046.txt">Rows n = 0..100 of the triangle, flattened</a>

%F T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 2.

%F Sum_{k=0..n} T(n, k, 2) = 4^(n-1) + 2^n - (n-1) - (5/4)*[n=0] = A000302(n-1) + A132045(n) - (5/4)*[n=0]. - [n=1]. - _G. C. Greubel_, Feb 16 2021

%e Triangle begins as:

%e   1;

%e   1,    1;

%e   1,    5,    1;

%e   1,   10,   10,     1;

%e   1,   19,   37,    19,     1;

%e   1,   36,  105,   105,    36,     1;

%e   1,   69,  270,   403,   270,    69,     1;

%e   1,  134,  660,  1314,  1314,   660,   134,     1;

%e   1,  263, 1563,  3895,  5189,  3895,  1563,   263,    1;

%e   1,  520, 3619, 10835, 18045, 18045, 10835,  3619,  520,    1;

%e   1, 1033, 8236, 28791, 57553, 71931, 57553, 28791, 8236, 1033, 1;

%t T[n_, m_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] +(q^n)*Binomial[n-2, k-1] -1];

%t Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 16 2021 *)

%o (Sage)

%o def T(n,k,q): return 1 if (k==0 or k==n) else binomial(n,k) + q^n*binomial(n-2,k-1) -1

%o flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 16 2021

%o (Magma)

%o T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) + q^n*Binomial(n-2,k-1) -1 >;

%o [T(n,k,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 16 2021

%Y Cf. A132044 (q=0), A173075 (q=1), this sequence (q=2), A173047 (q=3).

%Y Cf. A000302, A132045.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 08 2010

%E Edited by _G. C. Greubel_, Feb 16 2021