A116599 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 2 (n>=0, 0<=k<=floor(n/2)).

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 6, 3, 1, 1, 8, 4, 2, 1, 11, 6, 3, 1, 1, 15, 8, 4, 2, 1, 20, 11, 6, 3, 1, 1, 26, 15, 8, 4, 2, 1, 35, 20, 11, 6, 3, 1, 1, 45, 26, 15, 8, 4, 2, 1, 58, 35, 20, 11, 6, 3, 1, 1, 75, 45, 26, 15, 8, 4, 2, 1, 96, 58, 35, 20, 11, 6, 3, 1, 1, 121, 75, 45, 26, 15, 8, 4, 2, 1

Offset

0,5

Comments

Row n has 1 + floor(n/2) terms.

Row sums are the partition numbers (A000041).

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Formula

T(n,0) = A027336(n), Sum_{k=0..floor(n/2)} k*T(n,k) = A024786(n).

Column k has g.f.: x^(2*k)/[(1-x)*Product_{j>=0} ((1-x^j))] (k=0,1,2,...).

G.f.: 1/[(1-x)*(1-t*x^2)*Product_{j>=3}( (1-x^j) )].

T(n,k) = p(n-2*k) - p(n-2*k-2) for k<=(n-2)/2;

T(n, floor(n/2))=1 (follows at once from the g.f.).

Example

T(6,1)=3 because we have [4,2], [3,2,1] and [2,1,1,1,1].
Triangle starts:
1;
1;
1,1;
2,1;
3,1,1;
4,2,1;
6,3,1,1;
8,4,2,1;

Maple

with(combinat): T:=proc(n, k) if k=floor(n/2) then 1 elif k<=(n-2)/2 then numbpart(n-2*k)-numbpart(n-2*k-2) fi end: for n from 0 to 18 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

Mathematica

nn = 20; p = Product[1/(1 - x^i), {i, 3, nn}]; f[list_] := Select[list, # > 0 &]; Map[f, CoefficientList[Series[p /(1 - x)/(1 - y x^2), {x, 0, nn}], {x, y}]] // Flatten  (* Geoffrey Critzer, Jan 22 2012 *)

Crossrefs

Cf. A000041, A027336, A024786.

Sequence in context: A089178 A187489 A355145 * A138121 A138151 A207378

Adjacent sequences: A116596 A116597 A116598 * A116600 A116601 A116602

Keyword

nonn,tabf

Author

Emeric Deutsch, Feb 18 2006

Extensions

Keyword tabl changed to tabf by Michel Marcus, Apr 09 2013

Status

approved