OFFSET
1,16
COMMENTS
Row sums yield A015723. T(n,1)=A025147(n-1); T(n,2)=A015744(n-2); T(n,3)=A015745(n-3); T(n,4)=A015746(n-4); T(n,5)=A015750(n-5). - Emeric Deutsch, Mar 29 2006
Number of parts of size k in all partitions of n into distinct parts. Number of partitions of n-k into distinct parts not including a part of size k. - Franklin T. Adams-Watters, Jan 24 2012
LINKS
Mircea Merca, Table of n, a(n) for n = 1..7260
FORMULA
G.f.: G(t,x) = Product_{j>=1} (1+x^j) * Sum_{i>=1} t^i*x^i/(1+x^i). - Emeric Deutsch, Mar 29 2006
From Mircea Merca, Feb 28 2014: (Start)
T(n,k) = Sum_{j=1..floor(n/k)} (-1)^(j-1)*A000009(n-j*k).
G.f.: for column k: q^k/(1+q^k)*(-q;q)_{inf}. (End)
EXAMPLE
T(8,3)=2 because we have [5,3] and [4,3,1].
Triangle begins:
n/k 1 2 3 4 5 6 7 8 9 10
01: 1
02: 0 1
03: 1 1 1
04: 1 0 1 1
05: 1 1 1 1 1
06: 2 2 1 1 1 1
07: 2 2 1 2 1 1 1
08: 3 2 2 1 2 1 1 1
09: 3 3 3 2 2 2 1 1 1
10: 5 4 4 3 2 2 2 1 1 1
...
The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with multiset union {1,1,2,2,3,4,5,6}, with multiplicities (2,2,1,1,1,1), which is row n = 6. - Gus Wiseman, May 07 2019
MAPLE
g:=product(1+x^j, j=1..50)*sum(t^i*x^i/(1+x^i), i=1..50): gser:=simplify(series(g, x=0, 18)): for n from 1 to 14 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 14 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Mar 29 2006
seq(seq(coeff(x^k*(product(1+x^j, j=1..n))/(1+x^k), x, n), k=1..n), n=1..13); # Mircea Merca, Feb 28 2014
MATHEMATICA
z = 15; d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]; p[n_, k_] := p[n, k] = d[n][[k]]; s[n_] := s[n] = Flatten[Table[p[n, k], {k, 1, PartitionsQ[n]}]]; t[n_, k_] := Count[s[n], k]; u = Table[t[n, k], {n, 1, z}, {k, 1, n}]; TableForm[u] (* A015716 as a triangle *)
v = Flatten[u] (* A015716 as a sequence *)
(* Clark Kimberling, Mar 14 2014 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved