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A238866 Number of partitions of n where the difference between consecutive parts is at most 6. 10
1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 52, 71, 91, 121, 155, 202, 255, 328, 410, 520, 647, 810, 1000, 1244, 1525, 1879, 2293, 2804, 3401, 4135, 4988, 6028, 7241, 8701, 10404, 12447, 14818, 17645, 20931, 24822, 29334, 34658, 40817, 48052, 56416, 66190, 77471, 90621, 105756, 123338, 143555, 166956, 193815, 224828, 260352 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most 6 times (by taking conjugates).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^(7*i))/(1-q^i) ) ).

a(n) = Sum_{k=0..6} A238353(n,k). - Alois P. Heinz, Mar 09 2014

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(b(n-i*j, i-1), j=0..min(6, n/i))))

    end:

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(b(n-i*j, i-1), j=1..n/i)))

    end:

a:= n-> add(g(n, k), k=0..n):

seq(a(n), n=0..60);  # Alois P. Heinz, Mar 09 2014

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 0, Min[6, n/i]}]]]; g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 1, n/i}]]]; a[n_] := Sum[g[n, k], {k, 0, n}]; Table[a[n], {n, 0, 60}] (* Jean-Fran├žois Alcover, Feb 18 2015, after Alois P. Heinz *)

PROG

(PARI) N=66;  q = 'q + O('q^N);

Vec( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1, k-1, (1-q^(7*i))/(1-q^i) ) ) )

CROSSREFS

Sequences "number of partitions with max diff d": A000005 (d=0, for n>=1),  A034296 (d=1), A224956 (d=2), A238863 (d=3), A238864 (d=4), A238865 (d=5), this sequence, A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).

Sequence in context: A026814 A008637 A008631 * A035978 A319475 A319454

Adjacent sequences:  A238863 A238864 A238865 * A238867 A238868 A238869

KEYWORD

nonn

AUTHOR

Joerg Arndt, Mar 08 2014

STATUS

approved

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Last modified November 17 11:02 EST 2019. Contains 329226 sequences. (Running on oeis4.)