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A238864 Number of partitions of n where the difference between consecutive parts is at most 4. 10
1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 36, 46, 63, 79, 104, 131, 169, 210, 269, 332, 418, 515, 640, 782, 967, 1173, 1435, 1736, 2108, 2534, 3062, 3663, 4398, 5243, 6259, 7429, 8834, 10441, 12356, 14559, 17159, 20144, 23661, 27686, 32403, 37807, 44102, 51306, 59680, 69235, 80297, 92924, 107482, 124070, 143157, 164862, 189763, 218057 (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 four 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^(5*i))/(1-q^i) ) ).

a(n) = Sum_{k=0..4} 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(4, 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[4, 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^(5*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), this sequence, A238865 (d=5), A238866 (d=6), A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).

Sequence in context: A026812 A001402 A008629 * A070289 A035961 A051056

Adjacent sequences:  A238861 A238862 A238863 * A238865 A238866 A238867

KEYWORD

nonn

AUTHOR

Joerg Arndt, Mar 08 2014

STATUS

approved

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Last modified July 28 15:08 EDT 2021. Contains 346335 sequences. (Running on oeis4.)