cp's OEIS Frontend

A237756 Number of partitions of n such that 3*(greatest part) = (number of parts).

%I A237756 #28 Oct 18 2024 03:54:59
%S A237756 0,0,1,0,0,0,1,1,1,1,2,2,2,2,4,4,6,7,10,10,13,14,19,21,27,31,40,45,55,
%T A237756 64,79,91,111,127,154,177,211,243,290,333,394,455,538,618,726,834,977,
%U A237756 1121,1304,1495,1738,1989,2302,2633,3041,3473,3999,4562,5241
%N A237756 Number of partitions of n such that 3*(greatest part) = (number of parts).
%C A237756 Also, the number of partitions of n such that (greatest part) = 3*(number of parts).
%H A237756 Vaclav Kotesovec, <a href="/A237756/b237756.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Seiichi Manyama)
%F A237756 G.f.: Sum_{k>=1} x^(4*k-1) * Product_{j=1..k-1} (1-x^(3*k+j-1))/(1-x^j). - _Seiichi Manyama_, Jan 24 2022
%F A237756 a(n) ~ Pi^3 * exp(Pi*sqrt(2*n/3)) / (3*2^(5/2)*n^(5/2)). - _Vaclav Kotesovec_, Oct 17 2024
%e A237756 a(15) = 4 counts these partitions: [12,1,1,1], [9,5,1], [9,4,2], [9,3,3].
%t A237756 z = 50; Table[Count[IntegerPartitions[n], p_ /; Max[p] = = 3 Length[p]], {n, z}]
%t A237756 (* or *)
%t A237756 nmax = 100; Rest[CoefficientList[Series[Sum[x^(4*k-1) * Product[(1 - x^(3*k+j-1)) / (1 - x^j), {j, 1, k-1}], {k, 1, nmax/4 + 1}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Oct 15 2024 *)
%t A237756 nmax = 100; p = x^2; s = x^2; Do[p = Normal[Series[p*x^4*(1 - x^(4*k - 1))*(1 - x^(4*k))*(1 - x^(4*k + 1))*(1 - x^(4*k + 2))/((1 - x^(3*k + 2))*(1 - x^(3*k + 1))*(1 - x^(3*k))*(1 - x^k)), {x, 0, nmax}]]; s += p;, {k, 1, nmax/4 + 1}]; Take[CoefficientList[s, x], nmax] (* _Vaclav Kotesovec_, Oct 16 2024 *)
%o A237756 (PARI) my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(4*k-1)*prod(j=1, k-1, (1-x^(3*k+j-1))/(1-x^j))))) \\ _Seiichi Manyama_, Jan 24 2022
%Y A237756 Column 3 of A350879.
%Y A237756 Cf. A064173, A237753, A350892, A350894.
%K A237756 nonn,easy
%O A237756 1,11
%A A237756 _Clark Kimberling_, Feb 13 2014