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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 7 22:50 EST 2021. Contains 349590 sequences. (Running on oeis4.)