This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A220265 #19 Dec 09 2012 02:52:21
%S A220265 1,2,2,9,8,2,9,72,177,222,163,72,18,2,64,800,3696,9800,17408,22284,
%T A220265 21340,15554,8652,3633,1120,240,32,2,625,11250,82500,365000,1131750,
%U A220265 2654250,4922750,7425000,9274150,9704600,8566200,6398000,4042345,2152890,959690,354020
%N 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.
%C A220265 Based on the identity:
%C A220265 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 A220265 Paul D. Hanna, <a href="/A220265/b220265.txt">Triangle of Rows 1..20, flattened.</a>
%F A220265 0 = Sum_{k=0..n-1} (-1)^k * T(n-k,k) for n>1.
%F A220265 Antidiagonal sums equal A220266.
%F A220265 Main diagonal equals A220267.
%F A220265 Row sums equal (2^(n+1) - 1)*(2^n - 1)^(n-1).
%F A220265 Position of largest term in row n is: A099392(n) = ceiling(n^2/2) - (n-1).
%e A220265 Triangle begins:
%e A220265 1, 2;
%e A220265 2, 9, 8, 2;
%e A220265 9, 72, 177, 222, 163, 72, 18, 2;
%e A220265 64, 800, 3696, 9800, 17408, 22284, 21340, 15554, 8652, 3633, 1120, 240, 32, 2;
%e A220265 625, 11250, 82500, 365000, 1131750, 2654250, 4922750, 7425000, 9274150, 9704600, 8566200, 6398000, 4042345, 2152890, 959690, 354020, 106251, 25300, 4600, 600, 50, 2;
%e A220265 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 A220265 where the alternating antidiagonal sums equal zero (after the initial '1'):
%e A220265 0 = 2 - 2;
%e A220265 0 = 9 - 9;
%e A220265 0 = 64 - 72 + 8;
%e A220265 0 = 625 - 800 + 177 - 2;
%e A220265 0 = 7776 - 11250 + 3696 - 222;
%e A220265 0 = 117649 - 190512 + 82500 - 9800 + 163; ...
%e A220265 Column 0 forms A000169(n) = n^(n-1) and column 1 equals n^(n-2)*n*(n+1)^2/2.
%e A220265 The largest term in row n, found at position ceiling(n^2/2) - (n-1), begins:
%e A220265 [2, 9, 222, 22284, 9704600, 17202209995, 123106610062800, 3600033286934164416, 421003580776636784633028, 200645860378226792820279591852, ...].
%o A220265 (PARI) {T(n,k)=polcoeff((2*(1+x)^n-1)*((1+x)^n-1)^(n-1)/x^(n-1),k)}
%o A220265 for(n=1,6,for(k=0,n^2-n+1,print1(T(n,k),", "));print(("")))
%Y A220265 Cf. A220266, A220267, A000169.
%K A220265 nonn,tabf
%O A220265 1,2
%A A220265 _Paul D. Hanna_, Dec 09 2012