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

1,2

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)}
for(n=1, 6, for(k=0, n^2-n+1, print1(T(n, k), ", ")); print(("")))

Crossrefs

Cf. A220266, A220267, A000169.

Sequence in context: A256591 A011149 A212990 * A243597 A021439 A198423

Adjacent sequences: A220262 A220263 A220264 * A220266 A220267 A220268

Keyword

nonn,tabf

Author

Paul D. Hanna, Dec 09 2012

Status

approved