This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A185342 #33 Oct 18 2024 20:33:16
%S A185342 2,4,-4,6,-12,8,8,-24,32,-16,10,-40,80,-80,32,12,-60,160,-240,192,-64,
%T A185342 14,-84,280,-560,672,-448,128,16,-112,448,-1120,1792,-1792,1024,-256,
%U A185342 18,-144,672,-2016,4032,-5376,4608,-2304,512,20,-180,960,-3360,8064
%N A185342 Triangle of successive recurrences in columns of A117317(n).
%C A185342 A117317 (A):
%C A185342 1
%C A185342 2 1
%C A185342 4 5 1
%C A185342 8 16 9 1
%C A185342 16 44 41 14 1
%C A185342 32 112 146 85 20 1
%C A185342 64 272 456 377 155 27 1
%C A185342 have for their columns successive signatures
%C A185342 (2) (4,-4) (6,-12,8) (8,-24, 32, -16) (10,-40,80,-80,32) i.e. a(n).
%C A185342 Take based on abs(A133156) (B):
%C A185342 1
%C A185342 2 0
%C A185342 4 1 0
%C A185342 8 4 0 0
%C A185342 16 12 1 0 0
%C A185342 32 32 6 0 0 0
%C A185342 64 80 24 1 0 0 0.
%C A185342 The recurrences of successive columns are also a(n). a(n) columns: A005843(n+1), A046092(n+1), A130809, A130810, A130811, A130812, A130813.
%C A185342 A053220 + A001787 = A014480.
%H A185342 G. C. Greubel, <a href="/A185342/b185342.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F A185342 Take A133156(n) without 1's or -1's double triangle (C)=
%F A185342 2
%F A185342 4
%F A185342 8 -4
%F A185342 16 -12
%F A185342 32 -32 6
%F A185342 64 -80 24
%F A185342 128 -192 80 -8
%F A185342 256 -448 240 -40
%F A185342 512 -1024 672 -160 10;
%F A185342 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 A185342 Absolute values: A013609(n) without 1's. Also 2*A193862 = (2*A002260)*A135278.
%F A185342 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 A185342 T(n,k) = (-1)* Binomial(n,k)*(-2)^k, 1<=k<=n. - _Philippe Deléham_, Feb 11 2012
%e A185342 Triangle T(n,k),for 1<=k<=n, begins :
%e A185342 2 (1)
%e A185342 4 -4 (2)
%e A185342 6 -12 8 (3)
%e A185342 8 -24 32 -16 (4)
%e A185342 10 -40 80 -80 32 (5)
%e A185342 12 -60 160 -240 192 -64 (6)
%e A185342 14 -84 280 -560 672 -448 128 (7)
%e A185342 16 -112 448 -1120 1792 -1792 1024 -256 (8)
%e A185342 Successive rows can be divided by A171977.
%t A185342 Table[(-1)*Binomial[n, k]*(-2)^k, {n, 1, 20}, {k, 1, n}] // Flatten (* _G. C. Greubel_, Jun 27 2017 *)
%o A185342 (PARI) for(n=1,20, for(k=1,n, print1((-2)^(k+1)*binomial(n,k)/2, ", "))) \\ _G. C. Greubel_, Jun 27 2017
%Y A185342 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 A185342 Cf. A007318.
%K A185342 sign,tabl
%O A185342 0,1
%A A185342 _Paul Curtz_, Jan 26 2012