A015742 Number of 7's in all the partitions of n into distinct parts.

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, 67, 79, 92, 106, 123, 142, 164, 189, 217, 248, 284, 325, 370, 421, 479, 543, 616, 698, 788, 890, 1005, 1131, 1273, 1432, 1606, 1802

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Examples

			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.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Programs

Maple

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

Mathematica

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

Formula

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

a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015

Extensions

More terms from Emeric Deutsch, Apr 17 2006