login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..119.

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

Cf. A000041, A000219, A000293, A000334, A000390, A000416, A000427, A096652(T^2), A096653(T^3), A096642-A096645(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 14-17 calculated (using extra terms in A096642-A096645 provided by Sean A. Irvine) by Wouter Meeussen, Jan 08 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 18 14:28 EST 2017. Contains 294894 sequences.