A220265 - OEIS

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

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OFFSET

1,2

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.

LINKS

Paul D. Hanna, Triangle of Rows 1..20, flattened.

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).

EXAMPLE

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, ...].

PROG

(PARI) {T(n, k)=polcoeff((2*(1+x)^n-1)*((1+x)^n-1)^(n-1)/x^(n-1), k)}