A397489 Number of ways to write n as a sum n = 0 + 1 + 2 +- 3 +- 4 ... +- n with nonnegative partial sums.

1, 1, 0, 0, 1, 1, 0, 0, 1, 3, 0, 0, 12, 12, 0, 0, 65, 115, 0, 0, 507, 972, 0, 0, 4931, 8586, 0, 0, 49845, 86086, 0, 0, 517481, 931253, 0, 0, 5643431, 10439967, 0, 0, 64489578, 120255030, 0, 0, 762585871, 1422072284, 0, 0, 9238616655, 17252409833, 0, 0, 114139783947, 214024316649

Offset

0,10

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1001

Formula

a(n) = A397488(n,n).

a(n) = 0 <=> n mod 4 >= 2.

Example

a(4) = 1: 0+1+2-3+4.
a(9) = 3: 0+1+2-3+4+5-6+7+8-9, 0+1+2-3+4+5+6-7-8+9, 0+1+2+3-4+5-6+7-8+9.
a(12) = 12: 0+1+2-3+4+5+6-7+8+9+10-11-12, 0+1+2-3+4+5+6+7-8+9-10+11-12, 0+1+2-3+4+5+6+7+8-9-10-11+12, 0+1+2+3-4+5-6+7+8+9+10-11-12, 0+1+2+3-4+5+6-7+8+9-10+11-12, 0+1+2+3-4+5+6+7-8-9+10+11-12, 0+1+2+3-4+5+6+7-8+9-10-11+12, 0+1+2+3+4-5+6-7+8-9+10+11-12, 0+1+2+3+4-5+6-7+8+9-10-11+12, 0+1+2+3+4-5+6+7-8-9+10-11+12, 0+1+2+3+4+5-6-7+8-9+10-11+12, 0+1+2+3+4+5+6+7+8+9-10-11-12.

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(i=0,
     `if`(n=0, 1, 0), b(n+i, i-1)+ `if`(i>n, 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..53);

Crossrefs

Main diagonal of A397488.

Sequence in context: A370064 A363033 A321711 * A277788 A208848 A277945

Adjacent sequences: A397485 A397486 A397488 * A397490 A397491 A397492

Keyword

nonn

Author

Alois P. Heinz, Jun 27 2026

Status

approved