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A128888 Table with g.f. [1-x*n-sqrt(x^2*n^2-2*n*x+1+4*x^2-4*x)]/(2*x).

%I A128888 #4 Mar 30 2012 17:39:19
%S A128888 1,1,0,1,1,0,1,2,3,0,1,3,8,10,0,1,4,15,36,36,0,1,5,24,84,176,137,0,1,
%T A128888 6,35,160,510,912,543,0,1,7,48,270,1152,3279,4928,2219,0,1,8,63,420,
%U A128888 2240,8768,21975,27472,9285,0,1,9,80,616,3936,19605,69504,151905,156864
%N A128888 Table with g.f. [1-x*n-sqrt(x^2*n^2-2*n*x+1+4*x^2-4*x)]/(2*x).
%C A128888 Column m=2 is essentially the same as A005563 or A067998 or A106230. Row n=1 is essentially the same as A025238 and A002212. The table is read along diagonals and provides the Taylor coefficient of x^m in column m. It also is the slice t=1 through the trivariate g.f. defined in A129170, which provides an implicit proof that all values are nonnegative.
%e A128888 Table with rows n>=0 and columns m>=0 starts
%e A128888 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A128888 1, 1, 3, 10, 36, 137, 543, 2219, 9285, 39587, 171369, ...
%e A128888 1, 2, 8, 36, 176, 912, 4928, 27472, 156864, 912832, 5394176, ...
%e A128888 1, 3, 15, 84, 510, 3279, 21975, 151905, 1075425, 7758777, 56839965, ...
%e A128888 1, 4, 24, 160, 1152, 8768, 69504, 568064, 4753920, 40537088, 350963712, ...
%e A128888 1, 5, 35, 270, 2240, 19605, 178535, 1675495, 16095765, 157527055, 1565170985, ...
%e A128888 1, 6, 48, 420, 3936, 38832, 398208, 4205904, 45459840, 500488512, 5593373184, ...
%e A128888 1, 7, 63, 616, 6426, 70427, 801423, 9387917, 112501809, 1372985957, 17007257421,...
%p A128888 H := proc(n,x) (-x*n+1-(x^2*n^2-2*n*x+1+4*x^2-4*x)^(1/2))/(2*x) ; end: T := proc(n,m) coeftayl( H(n,x),x=0,m) ; end: for diag from 0 to 20 do for m from 0 to diag do n := diag-m ; printf("%d, ",T(n,m)) ; od ; od;
%Y A128888 Cf. A005563, A002212, A129170.
%K A128888 easy,nonn,tabl
%O A128888 0,8
%A A128888 _R. J. Mathar_, Apr 19 2007