0,13

Conjecture: there exists a unique lower triangular matrix T such that the row sums of T^n yields the n-dimensional partitions for all n>0. Specifically, row sums of T form A000041 (linear partitions); row sums of T^2 form A000219 (planar partitions); row sums of T^3 form A000293 (solid partitions); row sums of T^4 form A000334(4-D); row sums of T^5 form A000390(5-D); row sums of T^6 form A000416(6-D); row sums of T^7 form A000427(7-D). Rows indexed 9-13 were calculated by Wouter Meeussen (wouter.meeussen(AT)pandora.be).

For n>=0: T(0, 0)=1, T(0, n+1)=0, T(1, n+1)=1. For n>=1: T(n, n)=1, T(n+1, n)=1, T(n+2, n)=n, T(n+3, n)=1, T(n+4, n)=n*(5+n^2)/6, T(n+5, n)=(-48+90*n-7*n^2-6*n^3-5*n^4)/24, T(n+6, n)=(400-382*n-55*n^2+30*n^3+35*n^4+12*n^5)/40 (Wouter Meeussen).

G.f.: A(x, y) = Product_{n>=1} 1/(1-x^n)^[P_n(y)/n], where P_n(y) is the n-th row polynomial of triangle A096800.

Triangle T begins:

{1},

{0,1},

{0,1,1},

{0,1,1,1},

{0,1,2,1,1},

{0,1,1,3,1,1},

{0,1,3,1,4,1,1},

{0,1,-1,7,1,5,1,1},

{0,1,15,-17,14,1,6,1,1},

{0,1,-78,133,-61,25,1,7,1,1},

{0,1,632,-1020,529,-152,41,1,8,1,1},

{0,1,-6049,9826,-4989,1506,-314,63,1,9,1,1},

{0,1,68036,-110514,56161,-16668,3532,-576,92,1,10,1,1},

{0,1,-878337,1427046,-724881,214528,-44703,7276,-972,129,1,11,1,1},...

with row sums: {1,1,2,3,5,7,11,15,22,...} (A000041).

T^2 begins:

{1},

{0,1},

{0,2,1},

{0,3,2,1},

{0,5,5,2,1},

{0,7,7,7,2,1},

{0,11,16,9,9,2,1},

{0,15,15,31,11,11,2,1},

{0,22,59,-4,54,13,13,2,1},...

with row sums: {1,1,3,6,13,24,48,86,...} (A000219).

Cf. A000041, A000219, A000293, A000334, A000390, A000416, A000427, A096652(T^2), A096653(T^3), A096642-A096645(columns).

Cf. A096800, A096751.

Sequence in context: A156749 A039803 A147809 this_sequence A114640 A056890 A048825

Adjacent sequences: A096648 A096649 A096650 this_sequence A096652 A096653 A096654

nice,sign,tabl

Paul D. Hanna (pauldhanna(AT)juno.com) and Wouter Meeussen (wouter.meeussen(AT)pandora.be), Jul 02 2004