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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.

%I A096651 #23 Jan 20 2025 03:52:59
%S A096651 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,
%T A096651 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,
%U A096651 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
%N A096651 Lower triangular matrix T, read by rows, such that the row sums of T^n form the n-dimensional partitions.
%C A096651 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_.
%C A096651 Existence and integrality of Hanna's triangle has been proved in arXiv:1203.4419. (Suresh Govindarajan)
%H A096651 S. Govindarajan <a href="http://arxiv.org/abs/1203.4419">Notes on higher-dimensional partitions</a>, arXiv:1203.4419 [math.CO], 2012.
%H A096651 Wouter Meeussen, <a href="/A096651/a096651.txt">Rows 14-17 added</a>
%F A096651 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)
%F A096651 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.
%e A096651 Triangle T begins:
%e A096651   {1},
%e A096651   {0,1},
%e A096651   {0,1,1},
%e A096651   {0,1,1,1},
%e A096651   {0,1,2,1,1},
%e A096651   {0,1,1,3,1,1},
%e A096651   {0,1,3,1,4,1,1},
%e A096651   {0,1,-1,7,1,5,1,1},
%e A096651   {0,1,15,-17,14,1,6,1,1},
%e A096651   {0,1,-78,133,-61,25,1,7,1,1},
%e A096651   {0,1,632,-1020,529,-152,41,1,8,1,1},
%e A096651   {0,1,-6049,9826,-4989,1506,-314,63,1,9,1,1},
%e A096651   {0,1,68036,-110514,56161,-16668,3532,-576,92,1,10,1,1},
%e A096651   {0,1,-878337,1427046,-724881,214528,-44703,7276,-972,129,1,11,1,1},
%e A096651   ...
%e A096651   with row sums: {1,1,2,3,5,7,11,15,22,...} (A000041).
%e A096651 T^2 begins:
%e A096651   {1},
%e A096651   {0,1},
%e A096651   {0,2,1},
%e A096651   {0,3,2,1},
%e A096651   {0,5,5,2,1},
%e A096651   {0,7,7,7,2,1},
%e A096651   {0,11,16,9,9,2,1},
%e A096651   {0,15,15,31,11,11,2,1},
%e A096651   {0,22,59,-4,54,13,13,2,1},
%e A096651   ...
%e A096651   with row sums: {1,1,3,6,13,24,48,86,...} (A000219).
%Y A096651 Cf. A000041, A000219, A000293, A000334, A000390, A000416, A000427, A096652(T^2), A096653(T^3), A096642-A096645(columns).
%Y A096651 Cf. A096800, A096751.
%K A096651 nice,sign,tabl
%O A096651 0,13
%A A096651 _Paul D. Hanna_ and _Wouter Meeussen_, Jul 02 2004
%E A096651 Rows 14-17 calculated (using extra terms in A096642-A096645 provided by _Sean A. Irvine_) by _Wouter Meeussen_, Jan 08 2011