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A238865 Number of partitions of n where the difference between consecutive parts is at most 5. 10
1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 50, 67, 87, 114, 146, 188, 238, 302, 379, 476, 593, 737, 911, 1124, 1379, 1688, 2058, 2504, 3034, 3669, 4422, 5319, 6378, 7634, 9114, 10859, 12908, 15316, 18134, 21434, 25283, 29775, 35001, 41080, 48133, 56312, 65778, 76727, 89366, 103947, 120739, 140065, 162271, 187769, 217006, 250504 (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 five 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^(6*i))/(1-q^i) ) ).

a(n) = Sum_{k=0..5} 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(5, 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[5, 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 *)

Table[Count[IntegerPartitions[n], _?(Max[Abs[Differences[#]]]<6&)], {n, 0, 60}] (* Harvey P. Dale, Feb 04 2017 *)

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^(6*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), this sequence, A238866 (d=6), A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).

Sequence in context: A026813 A008636 A008630 * A326978 A035969 A332745

Adjacent sequences:  A238862 A238863 A238864 * A238866 A238867 A238868

KEYWORD

nonn

AUTHOR

Joerg Arndt, Mar 08 2014

STATUS

approved

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Last modified August 3 18:36 EDT 2021. Contains 346440 sequences. (Running on oeis4.)