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A238867 Number of partitions of n where the difference between consecutive parts is at most 7. 10
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 73, 95, 125, 162, 210, 268, 344, 434, 549, 688, 861, 1069, 1328, 1637, 2016, 2472, 3023, 3682, 4479, 5424, 6558, 7905, 9508, 11404, 13657, 16307, 19440, 23123, 27454, 32526, 38479, 45424, 53545, 63006, 74024, 86824, 101701, 118931, 138899, 161983, 188656, 219419, 254895, 295709 (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 seven 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^(8*i))/(1-q^i) ) ).

a(n) = Sum_{k=0..7} 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(7, 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[7, 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^(8*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), A238866 (d=6), this sequence, A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).

Sequence in context: A341913 A008638 A008632 * A035988 A088669 A091580

Adjacent sequences:  A238864 A238865 A238866 * A238868 A238869 A238870

KEYWORD

nonn

AUTHOR

Joerg Arndt, Mar 08 2014

STATUS

approved

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Last modified June 15 15:16 EDT 2021. Contains 345049 sequences. (Running on oeis4.)