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Triangle of successive recurrences in columns of A117317(n).
1

%I #33 Oct 18 2024 20:33:16

%S 2,4,-4,6,-12,8,8,-24,32,-16,10,-40,80,-80,32,12,-60,160,-240,192,-64,

%T 14,-84,280,-560,672,-448,128,16,-112,448,-1120,1792,-1792,1024,-256,

%U 18,-144,672,-2016,4032,-5376,4608,-2304,512,20,-180,960,-3360,8064

%N Triangle of successive recurrences in columns of A117317(n).

%C A117317 (A):

%C 1

%C 2 1

%C 4 5 1

%C 8 16 9 1

%C 16 44 41 14 1

%C 32 112 146 85 20 1

%C 64 272 456 377 155 27 1

%C have for their columns successive signatures

%C (2) (4,-4) (6,-12,8) (8,-24, 32, -16) (10,-40,80,-80,32) i.e. a(n).

%C Take based on abs(A133156) (B):

%C 1

%C 2 0

%C 4 1 0

%C 8 4 0 0

%C 16 12 1 0 0

%C 32 32 6 0 0 0

%C 64 80 24 1 0 0 0.

%C The recurrences of successive columns are also a(n). a(n) columns: A005843(n+1), A046092(n+1), A130809, A130810, A130811, A130812, A130813.

%C A053220 + A001787 = A014480.

%H G. C. Greubel, <a href="/A185342/b185342.txt">Table of n, a(n) for the first 100 rows, flattened</a>

%F Take A133156(n) without 1's or -1's double triangle (C)=

%F 2

%F 4

%F 8 -4

%F 16 -12

%F 32 -32 6

%F 64 -80 24

%F 128 -192 80 -8

%F 256 -448 240 -40

%F 512 -1024 672 -160 10;

%F a(n) is increasing odd diagonals and increasing (sign changed) even diagonals. Rows sum of (C) = A201629 (?) Another link between Chebyshev polynomials and cos( ).

%F Absolute values: A013609(n) without 1's. Also 2*A193862 = (2*A002260)*A135278.

%F 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

%F T(n,k) = (-1)* Binomial(n,k)*(-2)^k, 1<=k<=n. - _Philippe Deléham_, Feb 11 2012

%e Triangle T(n,k),for 1<=k<=n, begins :

%e 2 (1)

%e 4 -4 (2)

%e 6 -12 8 (3)

%e 8 -24 32 -16 (4)

%e 10 -40 80 -80 32 (5)

%e 12 -60 160 -240 192 -64 (6)

%e 14 -84 280 -560 672 -448 128 (7)

%e 16 -112 448 -1120 1792 -1792 1024 -256 (8)

%e Successive rows can be divided by A171977.

%t Table[(-1)*Binomial[n, k]*(-2)^k, {n, 1, 20}, {k, 1, n}] // Flatten (* _G. C. Greubel_, Jun 27 2017 *)

%o (PARI) for(n=1,20, for(k=1,n, print1((-2)^(k+1)*binomial(n,k)/2, ", "))) \\ _G. C. Greubel_, Jun 27 2017

%Y Cf. For (A): A053220, A056243. For (B): A000079, A001787, A001788, A001789. For A193862: A115068 (a Coxeter group). For (2): A014480 (also (3),(4),(5),..); also A053220 and A001787.

%Y Cf. A007318.

%K sign,tabl

%O 0,1

%A _Paul Curtz_, Jan 26 2012