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

%I A238869 #21 Nov 24 2024 15:08:00
%S A238869 1,1,2,3,5,7,11,15,22,30,42,56,76,99,131,170,221,283,364,461,586,737,
%T A238869 926,1154,1439,1779,2199,2703,3317,4051,4942,6001,7278,8796,10610,
%U A238869 12760,15323,18344,21928,26148,31127,36971,43848,51890,61321,72327,85183,100149,117588,137827,161343,188583,220139,256607,298761,347360
%N A238869 Number of partitions of n where the difference between consecutive parts is at most 9.
%C A238869 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).
%C A238869 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
%H A238869 Vaclav Kotesovec, <a href="/A238869/b238869.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A238869 G.f.: 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^(10*i))/(1-q^i) ) ).
%F A238869 a(n) = Sum_{k=0..9} A238353(n,k). - _Alois P. Heinz_, Mar 09 2014
%F A238869 a(n) ~ 3^(1/4) * exp(Pi*sqrt(3*n/5)) / (4 * 5^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Jan 26 2022
%p A238869 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A238869       add(b(n-i*j, i-1), j=0..min(9, n/i))))
%p A238869     end:
%p A238869 g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A238869       add(b(n-i*j, i-1), j=1..n/i)))
%p A238869     end:
%p A238869 a:= n-> add(g(n, k), k=0..n):
%p A238869 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 09 2014
%t A238869 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_ *)
%t A238869 Table[Count[IntegerPartitions[n],_?(Max[Abs[Differences[#]]]<10&)],{n,0,60}] (* _Harvey P. Dale_, Nov 24 2024 *)
%o A238869 (PARI) N=66;  q = 'q + O('q^N);
%o A238869 Vec( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1,k-1, (1-q^(10*i))/(1-q^i) ) ) )
%Y A238869 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).
%K A238869 nonn
%O A238869 0,3
%A A238869 _Joerg Arndt_, Mar 08 2014