

A096651


Lower triangular matrix T, read by rows, such that the row sums of T^n form the ndimensional 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
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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 ndimensional 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(4D); row sums of T^5 form A000390(5D); row sums of T^6 form A000416(6D); row sums of T^7 form A000427(7D). Rows indexed 913 were calculated by Wouter Meeussen.
Existence and integrality of Hanna's triangle has been proved in arXiv:1203.4419. (Suresh Govindarajan)


LINKS

Table of n, a(n) for n=0..119.
S. Govindarajan Notes on higherdimensional partitions, arXiv:1203.4419
Wouter Meeussen, Rows 1417 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*n7*n^26*n^35*n^4)/24, T(n+6, n)=(400382*n55*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/(1x^n)^[P_n(y)/n], where P_n(y) is the nth 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

Cf. A000041, A000219, A000293, A000334, A000390, A000416, A000427, A096652(T^2), A096653(T^3), A096642A096645(columns).
Cf. A096800, A096751.
Sequence in context: A039803 A147809 A217605 * A209354 A114640 A056890
Adjacent sequences: A096648 A096649 A096650 * A096652 A096653 A096654


KEYWORD

nice,sign,tabl


AUTHOR

Paul D. Hanna and Wouter Meeussen, Jul 02 2004


EXTENSIONS

Rows 1417 calculated (using extra terms in A096642A096645 provided by Sean A. Irvine) by Wouter Meeussen, Jan 08 2011


STATUS

approved



