OFFSET
0,13
COMMENTS
Hanna's Triangle: There exists a unique lower triangular matrix T, with ones on its diagonal, 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.
Existence and integrality of Hanna's triangle has been proved in arXiv:1203.4419. (Suresh Govindarajan)
LINKS
S. Govindarajan Notes on higher-dimensional partitions, arXiv:1203.4419
Wouter Meeussen, Rows 14-17 added
FORMULA
For n>=0: T(0, 0)=1, T(n+1,0)=0, T(n+1,1)=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). Corrected entry for the zeroth and first columns of the matrix T -- entry had columns and rows interchanged (Corrected by Suresh Govindarajan)
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.
EXAMPLE
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).
CROSSREFS
KEYWORD
AUTHOR
Paul D. Hanna and Wouter Meeussen, Jul 02 2004
EXTENSIONS
Rows 14-17 calculated (using extra terms in A096642-A096645 provided by Sean A. Irvine) by Wouter Meeussen, Jan 08 2011
STATUS
approved