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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. 0
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 (list; graph; refs; listen; history; text; internal format)
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: A029090 A029089 A173924 * A046886 A257246 A056206

Adjacent sequences:  A307389 A307390 A307391 * A307393 A307394 A307395

KEYWORD

nonn

AUTHOR

Igor Haladjian, Apr 06 2019

STATUS

approved

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Last modified October 27 02:00 EDT 2021. Contains 348270 sequences. (Running on oeis4.)