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A015716 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, one of which is k (1<=k<=n). 5
1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 1, 1, 3, 3, 3, 2, 2, 2, 1, 1, 1, 5, 4, 4, 3, 2, 2, 2, 1, 1, 1, 5, 5, 4, 3, 3, 3, 2, 2, 1, 1, 1, 7, 6, 5, 5, 4, 3, 3, 2, 2, 1, 1, 1, 8, 7, 6, 6, 4, 4, 4, 3, 2, 2, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
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)

a(n) = A238450(n) + A238451(n).

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

Cf. A015723, A015744, A015745, A015746, A015750, A025147, A027293, A066633.

Cf. A006128, A022629, A066189, A246867, A325504, A325506, A325513.

Sequence in context: A204909 A214211 A099384 * A308884 A101598 A342461

Adjacent sequences:  A015713 A015714 A015715 * A015717 A015718 A015719

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified April 18 20:36 EDT 2021. Contains 343089 sequences. (Running on oeis4.)