A015742 - OEIS

%I #21 Feb 03 2019 16:40:45

%S 0,0,0,0,0,0,1,1,1,2,2,3,4,4,5,7,8,10,12,14,18,22,25,30,36,42,50,58,

%T 67,79,92,106,123,142,164,189,217,248,284,325,370,421,479,543,616,698,

%U 788,890,1005,1131,1273,1432,1606,1802

%N Number of 7's in all the partitions of n into distinct parts.

%H Vaclav Kotesovec, <a href="/A015742/b015742.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: x^7*(Product_{j>=1} (1+x^j))/(1+x^7). - _Emeric Deutsch_, Apr 17 2006

%F a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Oct 30 2015

%e a(9)=1 because in the 8 (=A000009(9)) partitions of 9 into distinct parts, namely [9], [8,1], [7,2], [6,3], [6,2,1], [5,4], [5,3,1] and [4,3,2] we have altogether one part equal to 7.

%p g:=x^7*product(1+x^j,j=1..60)/(1+x^7): gser:=series(g,x=0,57): seq(coeff(gser,x,n),n=1..54); # _Emeric Deutsch_, Apr 17 2006

%t n7[n_]:=Count[Flatten[Select[IntegerPartitions[n],Max[Transpose[ Tally[#]][[2]]]==1&]],7]; Table[n7[n],{n,60}] (* _Harvey P. Dale_, Aug 30 2013 *)

%t nmax = 100; Rest[CoefficientList[Series[x^7/(1+x^7) * Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Oct 30 2015 *)

%K nonn

%O 1,10

%A _Clark Kimberling_

%E More terms from _Emeric Deutsch_, Apr 17 2006