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

Author

Paul D. Hanna, Dec 09 2012

Keywords

• nonn
• tabf

Comments

Based on the identity:

1 = Sum_{n>=1} (2*G(x)^n - 1) * (1 - G(x)^n)^(n-1) for all G(x) such that G(0)=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.

Crossrefs

Cf. A220266, A220267, A000169.

Programs

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(("")))

Formula

0 = Sum_{k=0..n-1} (-1)^k * T(n-k,k) for n>1.

Antidiagonal sums equal A220266.

Main diagonal equals A220267.

Row sums equal (2^(n+1) - 1)*(2^n - 1)^(n-1).

Position of largest term in row n is: A099392(n) = ceiling(n^2/2) - (n-1).