|
%I #24 Apr 10 2023 14:05:52
%S 2,2,0,3,-3,2,4,-6,4,0,5,-10,10,-5,2,6,-15,20,-15,6,0,7,-21,35,-35,21,
%T -7,2,8,-28,56,-70,56,-28,8,0,9,-36,84,-126,126,-84,36,-9,2,10,-45,
%U 120,-210,252,-210,120,-45,10,0,11,-55,165,-330,462,-462,330,-165,55,-11,2
%N Triangle T(n,k) read by rows: coefficients in the recurrence of sequences which equal their n-th differences.
%C Sequences which equal their p-th differences obey recurrences a(n) = Sum_{s=1..p} T(p,s)*a(n-s).
%C This defines T(p,s) as essentially a signed version of a chopped Pascal triangle A014410, see A130785.
%C For cases like p=2, 4, 6, 8, 10, 12, 14, the denominator of the rational generating function of a(n) contains a factor 1-x; depending on the first terms in the sequences a(n), additional, simpler recurrences may exist if this cancels with a factor in the numerator. - _R. J. Mathar_, Jun 10 2008
%H Alois P. Heinz, <a href="/A135356/b135356.txt">Rows n = 1..141, flattened</a>
%F T(n,k) = (-1)^(k+1)*A007318(n, k). T(n,n) = 1 - (-1)^n.
%F Sum_{k=1..n} T(n, k) = 2.
%F From _G. C. Greubel_, Apr 09 2023: (Start)
%F Sum_{k=1..n} (-1)^(k-1)*T(n, k) = 2*A051049(n-1).
%F Sum_{k=1..n-1} T(n, k) = (1 + (-1)^n).
%F Sum_{k=1..n-1} (-1)^(k-1)*T(n, k) = A000225(n-1).
%F T(2*n, n) = (-1)^(n-1)*A000984(n), n >= 1. (End)
%e Triangle begins with row n=1:
%e 2;
%e 2, 0;
%e 3, -3, 2;
%e 4, -6, 4, 0;
%e 5, -10, 10, -5, 2;
%e 6, -15, 20, -15, 6, 0;
%e 7, -21, 35, -35, 21, -7, 2;
%e 8, -28, 56, -70, 56, -28, 8, 0;
%e 9, -36, 84, -126, 126, -84, 36, -9, 2;
%p T:= (p, s)-> `if`(p=s, 2*irem(p, 2), (-1)^(s+1) *binomial(p, s)):
%p seq(seq(T(p, s), s=1..p), p=1..11); # _Alois P. Heinz_, Aug 26 2011
%t T[p_, s_]:= If[p==s, 2*Mod[s, 2], (-1)^(s+1)*Binomial[p, s]];
%t Table[T[p, s], {p, 12}, {s, p}]//Flatten (* _Jean-François Alcover_, Feb 19 2015, after _Alois P. Heinz_ *)
%o (Magma)
%o A135356:= func< n,k | k eq n select 1-(-1)^n else (-1)^(k+1)*Binomial(n,k) >;
%o [A135356(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 09 2023
%o (SageMath)
%o def A135356(n,k):
%o if (k==n): return 2*(n%2)
%o else: return (-1)^(k+1)*binomial(n,k)
%o flatten([[A135356(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Apr 09 2023
%Y Related sequences: A000079 (n=1), A131577 (n=2), (A131708 , A130785, A131562, A057079) (n=3), (A000749, A038503, A009545, A038505) (n=4), A133476 (n=5), A140343 (n=6), A140342 (n=7).
%Y Cf. A000225, A000984, A051049, A130785.
%K sign,tabl
%O 1,1
%A _Paul Curtz_, Dec 08 2007, Mar 25 2008, Apr 28 2008
%E Edited by _R. J. Mathar_, Jun 10 2008
|
