A332754 a(n) = Sum_{k=1..n-1} ((-1)^(k+n+1)*binomial(k,floor(k/2))).
0, 1, 1, 2, 4, 6, 14, 21, 49, 77, 175, 287, 637, 1079, 2353, 4082, 8788, 15522, 33098, 59280, 125476, 227240, 478192, 873886, 1830270, 3370030, 7030570, 13027730, 27088870, 50469890, 104647630, 195892565, 405187825, 761615285, 1571990935, 2965576715
Offset: 1
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 1..3326
Programs
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Maple
f:= gfun:-rectoproc({(4*n + 4)*a(n) + (6 + 4*n)*a(n + 1) + (-n - 1)*a(n + 2) + (-n - 3)*a(n + 3), a(0) = 0, a(1) = 0, a(2) = 1, a(3) = 1},a(n),remember): map(f, [$1..100]); # Robert Israel, Oct 08 2020 -
Mathematica
Sum[(-1)^(k + n + 1) Binomial[k, Floor[k/2]], {k, 1, -1 + n}] -
PARI
a(n) = sum(k=1, n-1, (-1)^(k+n+1)*binomial(k, k\2)); \\ Michel Marcus, Feb 22 2020
Formula
a(n) = Sum_{k=1..n-1} ((-1)^(k+n+1)*binomial(k,floor(k/2))).
From Robert Israel, Oct 08 2020: (Start)
D-finite with recurrence: (4*n - 8)*a(n - 3) + (-6 + 4*n)*a(n - 2) + (-n + 2)*a(n - 1) - n*a(n) = 0.
G.f. (sqrt((1+2*x)/(1-2*x))-1-2*x)/(2+2*x). (End)
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