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

Author

Igor Haladjian, Apr 06 2019

Keywords

• nonn

Comments

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

Examples

			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)).
		

Crossrefs

Cf. A000217, A057944.

Programs

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);

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

Formula

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