A185342 Triangle of successive recurrences in columns of A117317(n).
2, 4, -4, 6, -12, 8, 8, -24, 32, -16, 10, -40, 80, -80, 32, 12, -60, 160, -240, 192, -64, 14, -84, 280, -560, 672, -448, 128, 16, -112, 448, -1120, 1792, -1792, 1024, -256, 18, -144, 672, -2016, 4032, -5376, 4608, -2304, 512, 20, -180, 960, -3360, 8064
Offset: 0
Examples
Triangle T(n,k),for 1<=k<=n, begins : 2 (1) 4 -4 (2) 6 -12 8 (3) 8 -24 32 -16 (4) 10 -40 80 -80 32 (5) 12 -60 160 -240 192 -64 (6) 14 -84 280 -560 672 -448 128 (7) 16 -112 448 -1120 1792 -1792 1024 -256 (8) Successive rows can be divided by A171977.
Links
- G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Crossrefs
Programs
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Mathematica
Table[(-1)*Binomial[n, k]*(-2)^k, {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Jun 27 2017 *) -
PARI
for(n=1,20, for(k=1,n, print1((-2)^(k+1)*binomial(n,k)/2, ", "))) \\ G. C. Greubel, Jun 27 2017
Formula
Take A133156(n) without 1's or -1's double triangle (C)=
2
4
8 -4
16 -12
32 -32 6
64 -80 24
128 -192 80 -8
256 -448 240 -40
512 -1024 672 -160 10;
a(n) is increasing odd diagonals and increasing (sign changed) even diagonals. Rows sum of (C) = A201629 (?) Another link between Chebyshev polynomials and cos( ).
T(n,k) = T(n-1,k) - 2*T(n-1,k-1) for k>1, T(n,1) = 2*n = 2*T(n-1,1) - T(n-2,1). - Philippe Deléham, Feb 11 2012
T(n,k) = (-1)* Binomial(n,k)*(-2)^k, 1<=k<=n. - Philippe Deléham, Feb 11 2012
Comments