login
Triangle read by rows: row n lists number of unordered partitions of n into k parts which are partition numbers (members of A000041).
4

%I #11 Mar 23 2018 20:52:58

%S 1,2,1,3,2,1,5,7,2,1,7,11,7,2,1,11,26,19,7,2,1,15,40,38,19,7,2,1,22,

%T 83,78,54,19,7,2,1,30,120,168,102,54,19,7,2,1,42,223,301,244,134,54,

%U 19,7,2,1,56,320,557,471,292,134,54,19,7,2,1,77,566,1035,1000,623,356,134,54

%N Triangle read by rows: row n lists number of unordered partitions of n into k parts which are partition numbers (members of A000041).

%C A060642 describes the ordered case.

%C Number of twice-partitions of n of length k. A twice-partition of n is a choice of a partition of each part in a partition of n. - _Gus Wiseman_, Mar 23 2018

%F G.f.: 1/Product_{k>0} (1-y*A000041(k)*x^k). - _Vladeta Jovovic_, May 21 2006

%e Triangle begins:

%e 1

%e 2 1

%e 3 2 1

%e 5 7 2 1

%e 7 11 7 2 1

%e 11 26 19 7 2 1

%e 15 40 38 19 7 2 1

%e 22 83 78 54 19 7 2 1

%e 30 120 168 102 54 19 7 2 1

%e 42 223 301 244 134 54 19 7 2 1

%e 56 320 557 471 292 134 54 19 7 2 1

%e The T(5,3) = 7 twice-partitions: (3)(1)(1), (21)(1)(1), (111)(1)(1), (2)(2)(1), (2)(11)(1), (11)(2)(1), (11)(11)(1). - _Gus Wiseman_, Mar 23 2018

%t nn=12;

%t ser=Product[1/(1-PartitionsP[n]x^n y),{n,nn}];

%t Table[SeriesCoefficient[ser,{x,0,n},{y,0,k}],{n,nn},{k,n}] (* _Gus Wiseman_, Mar 23 2018 *)

%Y Cf. A000041, A001970, A008284, A036036, A048996, A055887, A061260, A063834, A273873, A281145, A289501, A299200, A299201.

%K nonn,tabl

%O 0,2

%A _Alford Arnold_, May 19 2006

%E More terms and better definition from _Vladeta Jovovic_, May 21 2006