A174718 - OEIS

%I #9 Feb 09 2021 21:37:34

%S 1,1,1,1,-2,1,1,-13,-13,1,1,-44,-74,-44,1,1,-123,-278,-278,-123,1,1,

%T -314,-881,-1196,-881,-314,1,1,-761,-2539,-4317,-4317,-2539,-761,1,1,

%U -1784,-6884,-14024,-17594,-14024,-6884,-1784,1,1,-4087,-17884,-42412,-63874,-63874,-42412,-17884,-4087,1

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

%C The row sums of this class of sequences, for varying q, is given by Sum_{k=0..n} T(n, k, q) = q^n * (n+1) + 2^n * (1 - q^n). - _G. C. Greubel_, Feb 09 2021

%H G. C. Greubel, <a href="/A174718/b174718.txt">Rows n = 0..100 of the triangle, flattened</a>

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

%F Sum_{k=0..n} T(n, k, 2) = 2^n *(n + 2 - 2^n) = A001787(n+1) - A020522(n). - _G. C. Greubel_, Feb 09 2021

%e Triangle begins as:

%e   1;

%e   1,     1;

%e   1,    -2,      1;

%e   1,   -13,    -13,      1;

%e   1,   -44,    -74,    -44,      1;

%e   1,  -123,   -278,   -278,   -123,      1;

%e   1,  -314,   -881,  -1196,   -881,   -314,      1;

%e   1,  -761,  -2539,  -4317,  -4317,  -2539,   -761,      1;

%e   1, -1784,  -6884, -14024, -17594, -14024,  -6884,  -1784,     1;

%e   1, -4087, -17884, -42412, -63874, -63874, -42412, -17884, -4087, 1;

%t T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1);

%t Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten

%o (Sage)

%o def T(n,k,q): return 1 + (1-q^n)*(binomial(n,k) - 1)

%o flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 09 2021

%o (Magma)

%o T:= func< n,k,q | 1 + (1-q^n)*(Binomial(n,k) -1) >;

%o [T(n,k,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 09 2021

%Y Cf. A000012 (q=1), this sequence (q=2), A174719 (q=3), A174720 (q=4).

%Y Cf. A001787, A020522.

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, Mar 28 2010

%E Edited by _G. C. Greubel_, Feb 09 2021