A172385 a(n) = 1 if n=0, otherwise Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k) *(-1)^k*a(n-1-2k).
1, 1, 1, 0, -2, -4, -1, 14, 34, 2, -189, -439, 263, 3796, 6997, -14437, -96643, -106774, 671097, 2800836, 57519, -31088662, -82674287, 155322877, 1455331563, 1936970102, -14267868745, -66446614533, 19215003803, 1037638182571, 2654391633166, -8675836955120, -67833031653088
Offset: 0
Examples
Eigensequence for the number triangle 1; 1, 0; 0, 1, 0; -1, 0, 1, 0; 0, -2, 0, 1, 0; 1, 0, -3, 0, 1, 0; 0, 3, 0, -4, 0, 1, 0; -1, 0, 6, 0, -5, 0, 1, 0; 0, -4, 0, 10, 0, -6, 0, 1, 0; 1, 0, -10, 0, 15, 0, -7, 0, 1, 0; 0, 5, 0, -20, 0, 21, 0, -8, 0, 1, 0; (augmented version of Riordan array (1/(1+x^2), x/(1+x^2)).
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..949
Programs
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Mathematica
a[n_] := If[n == 0, 1, Sum[Binomial[n - k - 1, k]*(-1)^k*a[n - 2*k - 1], {k, 0, Floor[(n - 1)/2]}]]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Oct 07 2018 *)
Formula
G.f.: A(x) = 1 + (x/(1+x^2))*A(x/(1+x^2)).
Extensions
Name corrected by G. C. Greubel, Oct 07 2018
Terms a(27) and beyond from Seiichi Manyama, Feb 25 2023