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

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, 6, 35, 160, 510, 912, 543, 0, 1, 7, 48, 270, 1152, 3279, 4928, 2219, 0, 1, 8, 63, 420, 2240, 8768, 21975, 27472, 9285, 0, 1, 9, 80, 616, 3936, 19605, 69504, 151905, 156864

Offset

0,8

Comments

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.

Links

Table of n, a(n) for n=0..63.

Example

Table with rows n>=0 and columns m>=0 starts
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 3, 10, 36, 137, 543, 2219, 9285, 39587, 171369, ...
1, 2, 8, 36, 176, 912, 4928, 27472, 156864, 912832, 5394176, ...
1, 3, 15, 84, 510, 3279, 21975, 151905, 1075425, 7758777, 56839965, ...
1, 4, 24, 160, 1152, 8768, 69504, 568064, 4753920, 40537088, 350963712, ...
1, 5, 35, 270, 2240, 19605, 178535, 1675495, 16095765, 157527055, 1565170985, ...
1, 6, 48, 420, 3936, 38832, 398208, 4205904, 45459840, 500488512, 5593373184, ...
1, 7, 63, 616, 6426, 70427, 801423, 9387917, 112501809, 1372985957, 17007257421,...

Maple

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;

Crossrefs

Cf. A005563, A002212, A129170.

Sequence in context: A055137 A143325 A307910 * A305401 A306100 A294046

Adjacent sequences: A128885 A128886 A128887 * A128889 A128890 A128891

Keyword

easy,nonn,tabl

Author

R. J. Mathar, Apr 19 2007

Status

approved