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A116598 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 1 (n>=0, 0<=k<=n). 4
1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 2, 2, 1, 1, 0, 1, 4, 4, 2, 2, 1, 1, 0, 1, 7, 4, 4, 2, 2, 1, 1, 0, 1, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 21, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 24, 21, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

Row sums yield the partition numbers (A000041).

Reversed rows converge to A002865. - Joerg Arndt, Jul 07 2014

T(n,k) is the number of partitions of n for which the difference between the two largest, not necessarily distinct, parts is k (in partitions having only 1 part, we assume that 0 is also a part). This follows easily from the definition by taking the conjugate partitions. Example: T(6,2) = 2 becasue we have [3,1,1,1] and [4,2]. - Emeric Deutsch, Dec 05 2015

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

G.f.: G(t,x) = 1/( (1-t*x)*prod(j>=2, 1-x^j ) ).

T(n,k) = p(n-k)-p(n-k-1) for k<n, where p(n) are the partition numbers (A000041).

T(n,0) = A002865(n).

Sum(k*T(n,k),k=0..n) = A000070(n-1) for n>=1.

Column k has g.f. x^k/prod(j>=2, 1-x^j ) (k>=0).

EXAMPLE

T(6,2) = 2 because we have [4,1,1] and [2,2,1,1].

Triangle starts:

00:   1,

01:   0,  1,

02:   1,  0,  1,

03:   1,  1,  0,  1,

04:   2,  1,  1,  0,  1,

05:   2,  2,  1,  1,  0,  1,

06:   4,  2,  2,  1,  1,  0,  1,

07:   4,  4,  2,  2,  1,  1,  0,  1,

08:   7,  4,  4,  2,  2,  1,  1,  0,  1,

09:   8,  7,  4,  4,  2,  2,  1,  1,  0,  1,

10:  12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1,

11:  14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1,

12:  21, 14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1,

13:  24, 21, 14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1,

14:  34, 24, 21, 14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1,

15:  41, 34, 24, 21, 14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1,

...

MAPLE

with(combinat): T:=proc(n, k) if k<n then numbpart(n-k)-numbpart(n-k-1) elif k=n then 1 else 0 fi end: for n from 0 to 14 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

MATHEMATICA

nn = 20; p = Product[1/(1 - x^i), {i, 2, nn}]; Prepend[CoefficientList[Table[Coefficient[Series[p /(1 - x y), {x, 0, nn}], x^n], {n, 1, nn}], y], 1] // Flatten  (* Geoffrey Critzer, Jan 22 2012 *)

CROSSREFS

Cf. A000041, A002865, A000070.

Sequence in context: A069713 A072233 A264391 * A244925 A068914 A090824

Adjacent sequences:  A116595 A116596 A116597 * A116599 A116600 A116601

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Feb 18 2006

STATUS

approved

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Last modified August 16 13:42 EDT 2017. Contains 290623 sequences.