A307392 Number of partitions of n with at most one part in the interval [i*(i+1)/2, i+(i*(i+1)/2)] for all nonnegative integers i.

1, 1, 1, 1, 2, 3, 3, 3, 3, 4, 6, 9, 11, 12, 12, 12, 13, 15, 18, 22, 27, 34, 42, 50, 56, 60, 63, 66, 70, 76, 84, 94, 106, 120, 136, 154, 177, 206, 241, 279, 317, 352, 381, 404, 423, 442, 464, 492, 528, 574, 630, 694, 764, 839, 920, 1008, 1104, 1213, 1341, 1494, 1674, 1878

Offset

0,5

Comments

The intervals are: [1,2], [3,5], [6,9], [10,14], [15,20], [21,27], [28,35], [36,44], [45,54], [55,65], ... .

Links

Table of n, a(n) for n=0..61.

Formula

G.f.: Product_{n>=0} (1 + Sum_{k=(n*(n+1)/2)..(n*(n+3)/2)} x^k).

Example

a(0)=1 by definition of the empty partition.
a(10)=6 because 10=9+1=8+2=7+3=6+4=6+3+1 (for example, you cannot take 5+5 or 7+2+1 because of the definition of a(n)).

Maple

f:= n-> 1+add(x^j, j=n*(n+1)/2..n*(n+3)/2):
a:= n-> coeff(mul(f(k), k=1..ceil((sqrt(9+8*n)-3)/2)), x, n):
seq(a(n), n=0..61);

Prog

(PARI) f(n, x) = (1+sum(j=n*(n+1)/2, n*(n+3)/2, x^j));
a(n) = polcoef(prod(k=1, ceil((sqrt(9+8*n)-3)/2), f(k, x)), n, x); \\ version 2.11.0 or newer; Michel Marcus, Apr 08 2019
(PARI) first(n) = v = Vecrev(Vec(a(n))); vector(n, i, v[i]) \\ using a(n) from above \\ David A. Corneth, Apr 08 2019

Crossrefs

Cf. A000217, A057944.

Sequence in context: A029089 A358468 A173924 * A046886 A257246 A056206

Adjacent sequences: A307389 A307390 A307391 * A307393 A307394 A307395

Keyword

nonn

Author

Igor Haladjian, Apr 06 2019

Status

approved