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A096651
Lower triangular matrix T, read by rows, such that the row sums of T^n form the n-dimensional partitions.
18
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, 0, 1, 12817659, -20827070, 10576885, -3123249, 647092, -103476, 13644, -1541, 175, 1, 12, 1, 1
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
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
nice,sign,tabl
AUTHOR
EXTENSIONS
Rows 14-17 calculated (using extra terms in A096642-A096645 provided by Sean A. Irvine) by Wouter Meeussen, Jan 08 2011
STATUS
approved