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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 (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 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

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