A168557 - OEIS

%I #15 Nov 22 2018 20:20:32

%S 1,-2,2,2,-2,-3,-3,2,4,6,4,-2,-5,-10,-10,-5,2,6,15,20,15,6,-2,-7,-21,

%T -35,-35,-21,-7,2,8,28,56,70,56,28,8,-2,-9,-36,-84,-126,-126,-84,-36,

%U -9,2,10,45,120,210,252,210,120,45,10,-2,-11,-55,-165,-330,-462,-462,-330

%N Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial (-1)^n*((x + 1)^n - x^n + 1), 0 <= k <= max(0, n - 1).

%C A variant of Pascal's triangle, the first column replaced by 2 (if n > 0), the last column dropped, and then odd rows multiplied by (-1)^n.

%C Absolute value row sums are A000079.

%F From _Franck Maminirina Ramaharo_, Nov 22 2018: (Start)

%F T(n,k) = (-1)^n*binomial(n, k) + (-1)^n*delta(0, k) - delta(0, n), where delta is Kronecker's delta-symbol.

%F G.f.: (1 + 2*x*y - (1 - x - x^2)*y^2)/((1 + y)*(1 + x*y)*(1 + y + x*y)).

%F E.g.f.: (1 - exp(y) + exp(x*y))*exp(-(1 + x)*y). (End)

%e Triangle begins:

%e    1;

%e   -2;

%e    2,   2;

%e   -2,  -3,  -3;

%e    2,   4,   6,    4;

%e   -2,  -5, -10,  -10,   -5;

%e    2,   6,  15,   20,   15,    6;

%e   -2,  -7, -21,  -35,  -35,  -21,   -7;

%e    2,   8,  28,   56,   70,   56,   28,    8;

%e   -2,  -9, -36,  -84, -126, -126,  -84,  -36,   -9;

%e    2,  10,  45,  120,  210,  252,  210,  120,   45,  10;

%e   -2, -11, -55, -165, -330, -462, -462, -330, -165, -55, -11;

%e    2,  12,  66,  220,  495,  792,  924,  792,  495,  220, 66, 12;

%e    ...

%t Table[CoefficientList[(-1)^n*(x + 1)^n - (-1)^n*(x^n - 1), x], {n, 0, 12}]

%o (Maxima) create_list((-1)^n*binomial(n, k) + (-1)^n*kron_delta(0, k) - kron_delta(0, n), n, 0, 12, k, 0, max(0, n - 1)); /* _Franck Maminirina Ramaharo_, Nov 21 2018 */

%Y Cf. A074909, A007318, A108086, A117440, A130595.

%K sign,tabf,easy

%O 0,2

%A _Roger L. Bagula_, Nov 29 2009