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A096651
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Lower triangular matrix T, read by rows, such that the row sums of T^n form the n-dimensional partitions.
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18
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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
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0,13
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COMMENTS
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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)
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LINKS
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FORMULA
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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.
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EXAMPLE
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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).
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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