A220265 Triangle where the g.f. of row n is: Sum_{k=0..n^2-n+1} T(n,k)*y^k = (2*(1+y)^n - 1) * ((1+y)^n - 1)^(n-1) / y^(n-1), as read by rows.
1, 2, 2, 9, 8, 2, 9, 72, 177, 222, 163, 72, 18, 2, 64, 800, 3696, 9800, 17408, 22284, 21340, 15554, 8652, 3633, 1120, 240, 32, 2, 625, 11250, 82500, 365000, 1131750, 2654250, 4922750, 7425000, 9274150, 9704600, 8566200, 6398000, 4042345, 2152890, 959690, 354020
Offset: 1
Examples
Triangle begins: 1, 2; 2, 9, 8, 2; 9, 72, 177, 222, 163, 72, 18, 2; 64, 800, 3696, 9800, 17408, 22284, 21340, 15554, 8652, 3633, 1120, 240, 32, 2; 625, 11250, 82500, 365000, 1131750, 2654250, 4922750, 7425000, 9274150, 9704600, 8566200, 6398000, 4042345, 2152890, 959690, 354020, 106251, 25300, 4600, 600, 50, 2; 7776, 190512, 2015280, 13222440, 62141310, 225598527, 662159412, 1618976925, 3366367410, 6041884575, 9462175520, 13034476980, 15886286910, 17202209995, 16595155500, 14285514705, 10978477070, 7528219125, 4599186000, 2496823900, 1200043026, 508072257, 188241900, 60515895, 16695030, 3895573, 753984, 117810, 14280, 1260, 72, 2; ... where the alternating antidiagonal sums equal zero (after the initial '1'): 0 = 2 - 2; 0 = 9 - 9; 0 = 64 - 72 + 8; 0 = 625 - 800 + 177 - 2; 0 = 7776 - 11250 + 3696 - 222; 0 = 117649 - 190512 + 82500 - 9800 + 163; ... Column 0 forms A000169(n) = n^(n-1) and column 1 equals n^(n-2)*n*(n+1)^2/2. The largest term in row n, found at position ceiling(n^2/2) - (n-1), begins: [2, 9, 222, 22284, 9704600, 17202209995, 123106610062800, 3600033286934164416, 421003580776636784633028, 200645860378226792820279591852, ...].
Links
- Paul D. Hanna, Triangle of Rows 1..20, flattened.
Programs
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PARI
{T(n,k)=polcoeff((2*(1+x)^n-1)*((1+x)^n-1)^(n-1)/x^(n-1),k)} for(n=1,6,for(k=0,n^2-n+1,print1(T(n,k),", "));print(("")))
Comments