A220265 - OEIS

%I #19 Dec 09 2012 02:52:21

%S 1,2,2,9,8,2,9,72,177,222,163,72,18,2,64,800,3696,9800,17408,22284,

%T 21340,15554,8652,3633,1120,240,32,2,625,11250,82500,365000,1131750,

%U 2654250,4922750,7425000,9274150,9704600,8566200,6398000,4042345,2152890,959690,354020

%N 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.

%C Based on the identity:

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

%H Paul D. Hanna, <a href="/A220265/b220265.txt">Triangle of Rows 1..20, flattened.</a>

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

%F Antidiagonal sums equal A220266.

%F Main diagonal equals A220267.

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

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

%e Triangle begins:

%e 1, 2;

%e 2, 9, 8, 2;

%e 9, 72, 177, 222, 163, 72, 18, 2;

%e 64, 800, 3696, 9800, 17408, 22284, 21340, 15554, 8652, 3633, 1120, 240, 32, 2;

%e 625, 11250, 82500, 365000, 1131750, 2654250, 4922750, 7425000, 9274150, 9704600, 8566200, 6398000, 4042345, 2152890, 959690, 354020, 106251, 25300, 4600, 600, 50, 2;

%e 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; ...

%e where the alternating antidiagonal sums equal zero (after the initial '1'):

%e 0 = 2 - 2;

%e 0 = 9 - 9;

%e 0 = 64 - 72 + 8;

%e 0 = 625 - 800 + 177 - 2;

%e 0 = 7776 - 11250 + 3696 - 222;

%e 0 = 117649 - 190512 + 82500 - 9800 + 163; ...

%e Column 0 forms A000169(n) = n^(n-1) and column 1 equals n^(n-2)*n*(n+1)^2/2.

%e The largest term in row n, found at position ceiling(n^2/2) - (n-1), begins:

%e [2, 9, 222, 22284, 9704600, 17202209995, 123106610062800, 3600033286934164416, 421003580776636784633028, 200645860378226792820279591852, ...].

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

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

%Y Cf. A220266, A220267, A000169.

%K nonn,tabf

%O 1,2

%A _Paul D. Hanna_, Dec 09 2012