proposed

approved

editing

proposed

Cf. A322268.

allocated for Ilya GutkovskiySquare array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. (sec(x) + tan(x))^k.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 2, 0, 1, 4, 9, 10, 5, 0, 1, 5, 16, 30, 32, 16, 0, 1, 6, 25, 68, 117, 122, 61, 0, 1, 7, 36, 130, 320, 528, 544, 272, 0, 1, 8, 49, 222, 725, 1684, 2709, 2770, 1385, 0, 1, 9, 64, 350, 1440, 4400, 9856, 15600, 15872, 7936, 0, 1, 10, 81, 520, 2597, 9966, 29125, 63668, 99657, 101042, 50521, 0

0,8

<a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>

E.g.f. of column k: (sec(x) + tan(x))^k.

E.g.f. of column k: A_k(x) = 1 + k*x/1! + k^2*x^2/2! + k*(k^2 + 1)*x^3/3! + k^2*(k^2 + 4)*x^4/4! + ...

Square array begins:

1, 1, 1, 1, 1, 1, ...

0, 1, 2, 3, 4, 5, ...

0, 1, 4, 9, 16, 25, ...

0, 2, 10, 30, 68, 130, ...

0, 5, 32, 117, 320, 725, ...

0, 16, 122, 528, 1684, 4400, ...

Table[Function[k, n! SeriesCoefficient[(Sec[x] + Tan[x])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Columns k=0..3 give A000007, A000111, A001250, A292758.

Main diagonal gives A298244.

allocated

nonn,tabl

Ilya Gutkovskiy, Dec 01 2018

approved

editing

allocated for Ilya Gutkovskiy

allocated

approved