A098716 Number of partitions of the n-th partition number into integers not greater than the (n-1)-th partition number.
1, 1, 2, 5, 13, 49, 169, 972, 5559, 52979, 526450, 10617149, 214475363, 9035782113, 476715641982, 51820049305123, 7479565064189887, 2645418340373829359, 1318520401609595443835, 1774758704783778068230273
Offset: 1
Keywords
Examples
n=7: A000041(7)=15 has A000041(15)=176 partitions, seven of them with integers greater than A000041(7-1)=11: 12+3, 12+2+1, 12+1+1, 13+2, 13+1+1, 14+1 and 15, therefore a(7)=176-7=169.
Crossrefs
Cf. A058699.
Programs
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Maple
with(combinat): a:=proc(n) local G, Gser: G:=1/product(1-x^j,j=1..numbpart(n-1)): Gser:=series(G,x=0,20+numbpart(n)): coeff(Gser,x^numbpart(n)) end: seq(a(n),n=1..22); # Emeric Deutsch, Apr 23 2006
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Mathematica
a[n_] := SeriesCoefficient[1/Product[1 - x^j, {j, 1, PartitionsP[n - 1]}], {x, 0, PartitionsP[n]}]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, May 28 2024, after Emeric Deutsch *)
Extensions
More terms from Emeric Deutsch, Apr 23 2006