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

1,4

Comments

a(n) is the sum of the alternating series of central binomial coefficients (including all rows, defined as binomial(m,floor(m/2)) or equivalently binomial(m,ceiling(m/2)) for all m odd, A001405).

Links

Robert Israel, Table of n, a(n) for n = 1..3326

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)

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}]

Prog

(PARI) a(n) = sum(k=1, n-1, (-1)^(k+n+1)*binomial(k, k\2)); \\ Michel Marcus, Feb 22 2020

Crossrefs

Sequence in context: A337499 A284886 A249339 * A307676 A084685 A219042

Adjacent sequences: A332751 A332752 A332753 * A332755 A332756 A332757

Keyword

nonn

Author

Brian P Hawkins, Feb 22 2020

Status

approved