A322268 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x)*(sec(x) + tan(x))^k.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 9, 1, 1, 5, 16, 29, 24, 1, 1, 6, 25, 67, 105, 77, 1, 1, 7, 36, 129, 304, 433, 294, 1, 1, 8, 49, 221, 705, 1519, 2029, 1309, 1, 1, 9, 64, 349, 1416, 4145, 8386, 10709, 6664, 1, 1, 10, 81, 519, 2569, 9601, 26385, 51007, 63025, 38177, 1, 1, 11, 100, 737, 4320, 19777, 69406, 181969, 340024, 409713, 243034, 1

Offset

0,5

Links

Table of n, a(n) for n=0..77.

Index entries for sequences related to boustrophedon transform

Formula

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

Example

E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k + 1)^2*x^2/2! + (k^3 + 3*k^2 + 4*k + 1)*x^3/3! + (k^4 + 4*k^3 + 10*k^2 + 8*k + 1)*x^4/4! + ...
Square array begins:
  1,   1,    1,     1,     1,     1,  ...
  1,   2,    3,     4,     5,     6,  ...
  1,   4,    9,    16,    25,    36,  ...
  1,   9,   29,    67,   129,   221,  ...
  1,  24,  105,   304,   705,  1416,  ...
  1,  77,  433,  1519,  4145,  9601,  ...

Mathematica

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

Crossrefs

Columns k=0..3 give A000012, A000667, A292756, A292759.

Main diagonal gives A296793.

Cf. A322267.

Sequence in context: A071922 A138028 A009999 * A144823 A098446 A098447

Adjacent sequences: A322265 A322266 A322267 * A322269 A322270 A322271

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Dec 01 2018

Status

approved