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A303872 Triangle read by rows: T(0,0) = 1; T(n,k) = -T(n-1,k) + 2 T(n-1,k-1) for k = 0,1,...,n; T(n,k)=0 for n or k < 0.

%I A303872 #49 Aug 08 2025 09:07:39
%S A303872 1,-1,2,1,-4,4,-1,6,-12,8,1,-8,24,-32,16,-1,10,-40,80,-80,32,1,-12,60,
%T A303872 -160,240,-192,64,-1,14,-84,280,-560,672,-448,128,1,-16,112,-448,1120,
%U A303872 -1792,1792,-1024,256,-1,18,-144,672,-2016,4032,-5376,4608,-2304,512
%N A303872 Triangle read by rows: T(0,0) = 1; T(n,k) = -T(n-1,k) + 2 T(n-1,k-1) for k = 0,1,...,n; T(n,k)=0 for n or k < 0.
%C A303872 Row n gives coefficients in expansion of (-1+2x)^n. Row sums=1.
%C A303872 In the center-justified triangle, the numbers in skew diagonals pointing top-Left give the triangle in A133156 (coefficients of Chebyshev polynomials of the second kind), and the numbers in skew diagonals pointing top-right give the triangle in A305098. The coefficients in the expansion of 1/(1-x) are given by the sequence generated by the row sums. The generating function of the central terms is 1/sqrt(1+8x), signed version of A059304.
%D A303872 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 389-391.
%H A303872 Shara Lalo, <a href="/A303872/a303872.pdf">Skew diagonals in center-justified triangle</a>
%H A303872 Paweł Lorek and Piotr Markowski, <a href="https://arxiv.org/abs/1812.00690">Absorption time and absorption probabilities for a family of multidimensional gambler models</a>, arXiv:1812.00690 [math.PR], 2018.
%F A303872 G.f.: 1 / (1 + t - 2t*x).
%F A303872 T(n,k) = (-1)^(n+k)*2^k*binomial(n,k). - _Stefano Spezia_, Aug 08 2025
%e A303872 Triangle begins:
%e A303872    1;
%e A303872   -1,   2;
%e A303872    1,  -4,   4;
%e A303872   -1,   6, -12,    8;
%e A303872    1,  -8,  24,  -32,   16;
%e A303872   -1,  10, -40,   80,  -80,    32;
%e A303872    1, -12,  60, -160,  240,  -192,   64;
%e A303872   -1,  14, -84,  280, -560,   672, -448,   128;
%e A303872    1, -16, 112, -448, 1120, -1792, 1792, -1024, 256;
%t A303872 T[0, 0] = 1; T[n_, k_] := If[n < 0 || k < 0, 0, - T[n - 1, k] + 2 T[n - 1, k - 1]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
%t A303872 For[i = 0, i < 4, i++, Print[CoefficientList[Expand[(-1 +2 x)^i], x]]]
%o A303872 (PARI) T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, -T(n-1, k) + 2*T(n-1, k-1)));
%o A303872 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, May 26 2018
%Y A303872 Row sums give A000012.
%Y A303872 Signed version of A013609 ((1+2*x)^n).
%Y A303872 Cf. A033999 (column 0).
%K A303872 tabl,easy,sign
%O A303872 0,3
%A A303872 _Shara Lalo_, May 25 2018