A238869 Number of partitions of n where the difference between consecutive parts is at most 9.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 99, 131, 170, 221, 283, 364, 461, 586, 737, 926, 1154, 1439, 1779, 2199, 2703, 3317, 4051, 4942, 6001, 7278, 8796, 10610, 12760, 15323, 18344, 21928, 26148, 31127, 36971, 43848, 51890, 61321, 72327, 85183, 100149, 117588, 137827, 161343, 188583, 220139, 256607, 298761, 347360

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 nine times (by taking conjugates).

In general, for d > 0, "number of partitions with max diff d" is asymptotic to d^(1/4) * exp(Pi*sqrt(2*d*n/(3*(d+1)))) / (2^(5/4) * 3^(1/4) * (d+1)^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jan 26 2022

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Formula

G.f.: 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^(10*i))/(1-q^i) ) ).

a(n) = Sum_{k=0..9} A238353(n,k). - Alois P. Heinz, Mar 09 2014

a(n) ~ 3^(1/4) * exp(Pi*sqrt(3*n/5)) / (4 * 5^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jan 26 2022

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(9, 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[9, 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-François Alcover, Feb 18 2015, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n], _?(Max[Abs[Differences[#]]]<10&)], {n, 0, 60}] (* Harvey P. Dale, Nov 24 2024 *)

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^(10*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), A238867 (d=7), A238868 (d=8), this sequence, A000041 (d --> infinity).

Sequence in context: A008640 A008634 A347577 * A326333 A036011 A325856

Adjacent sequences: A238866 A238867 A238868 * A238870 A238871 A238872

Keyword

nonn

Author

Joerg Arndt, Mar 08 2014

Status

approved