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%I #3 Mar 30 2012 18:37:04
%S 1,1,1,3,2,1,7,7,3,1,25,25,12,4,1,75,75,50,18,5,1,275,275,200,83,25,6,
%T 1,927,927,652,377,125,33,7,1,3438,3438,2511,1584,617,177,42,8,1,
%U 12306,12306,8868,5430,2919,932,240,52,9,1,46131,46131,33825,21519,12651,4759
%N Square array, read by antidiagonals, where row n+1 equals the partial sums of the sequence resulting from removing the terms in the first column and main diagonal from row n, for n>=0, with row 0 consisting of all 1's.
%C G.f. of n-th upper diagonal: D(x,n) = D(x,0)*G(x)^n where G(x) = 1 + x*G(x)^4 ; i.e., the g.f. of the n-th upper diagonal, D(x,n), equals the product of the g.f. of the main diagonal, D(x,0) and the n-th power of G(x) where G(x) is the g.f. of A002293.
%e Square array begins:
%e (1), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
%e (1), (2), 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...;
%e (3), 7, (12), 18, 25, 33, 42, 52, 63, 75, 88, 102, 117, 133, 150, ...;
%e (7), 25, 50, (83), 125, 177, 240, 315, 403, 505, 622, 755, 905, ...;
%e (25), 75, 200, 377, (617), 932, 1335, 1840, 2462, 3217, 4122, 5195, ...;
%e (75), 275, 652, 1584, 2919, (4759), 7221, 10438, 14560, 19755, 26210,..;
%e (275), 927, 2511, 5430, 12651, 23089, (37649), 57404, 83614, 117746,...;
%e (927), 3438, 8868, 21519, 44608, 102012, 185626, (303372), 464867, ...;
%e (3438), 12306, 33825, 78433, 180445, 366071, 830938, 1512611, (2480181), ...; ...
%e For each row, removing terms enclosed in parenthesis and then taking partial sums yields the next row.
%o (PARI) {T(n,k)=if(k<0,0,if(n==0,1,T(n,k-1) + if(n-1>k+1,T(n-1,k+1),T(n-1,k+2))))}
%Y Cf. A130463 (first column), A130464 (main diagonal); A002293.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, May 26 2007
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