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A241131 Number of partitions p of n such that (maximal multiplicity over the parts of p) = number of 1s in p.

%I A241131 #23 Jun 21 2025 15:14:57
%S A241131 0,1,1,2,3,4,7,9,13,18,26,32,47,60,79,104,137,173,227,285,365,461,583,
%T A241131 724,912,1129,1403,1729,2137,2611,3211,3906,4765,5777,7010,8450,10213,
%U A241131 12263,14738,17637,21113,25158,30008,35638,42333,50130,59346,70035,82663
%N A241131 Number of partitions p of n such that (maximal multiplicity over the parts of p) = number of 1s in p.
%H A241131 Vaclav Kotesovec, <a href="/A241131/b241131.txt">Table of n, a(n) for n = 0..9500</a> (terms 0..300 from John Tyler Rascoe, terms 301..2000 from Alois P. Heinz)
%F A241131 G.f.: Sum_{i>0} x^i * Product_{j>1} ((1 - x^(j*(i+1)))/(1 - x^j)). - _John Tyler Rascoe_, Mar 12 2024
%F A241131 a(n) ~ c * exp(Pi*sqrt(2*n/3)) / n, where c = 0.07449179... - _Vaclav Kotesovec_, Jun 21 2025
%e A241131 a(6) counts these 7 partitions:  51, 411, 321, 3111, 2211, 21111, 111111.
%p A241131 b:= proc(n,i,m) option remember; `if`(i=1, `if`(n>=m, 1, 0),
%p A241131       add(b(n-i*j, i-1, max(j, m)), j=0..n/i))
%p A241131     end:
%p A241131 a:= n-> `if`(n=0, 0, b(n$2, 0)):
%p A241131 seq(a(n), n=0..48);  # _Alois P. Heinz_, Mar 15 2024
%t A241131 z = 30; m[p_] := Max[Map[Length, Split[p]]]; Table[Count[IntegerPartitions[n], p_ /; m[p] == Count[p, 1]], {n, 0, z}]
%o A241131 (PARI)
%o A241131 A_x(N)={my(x='x+O('x^N),g=sum(i=1, N, x^i*prod(j=2, N, (1-x^(j*(i+1)))/(1-x^j))));
%o A241131 concat([0],Vec(g))}
%o A241131 A_x(50) \\ _John Tyler Rascoe_, Mar 12 2024
%Y A241131 Cf. A241090, A241132.
%K A241131 nonn,easy
%O A241131 0,4
%A A241131 _Clark Kimberling_, Apr 24 2014