A322268 -id:A322268 - cp's OEIS frontend

A296793 a(n) = n! * [x^n] exp(x)*(sec(x) + tan(x))^n.

1, 2, 9, 67, 705, 9601, 160429, 3175579, 72638209, 1884974185, 54709142101, 1755923320559, 61748847320545, 2360991253910069, 97518218630249005, 4327060674324941491, 205272207854416078849, 10367500700785078039473, 555414837143457708584101, 31458118283019682610004279

Offset: 0

Ilya Gutkovskiy, Dec 20 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps)
N. J. A. Sloane, Transforms.
Index entries for sequences related to boustrophedon transform

Cf. A000111, A000667, A292756, A292759, A292976.

Main diagonal of A322268.

Table[n! SeriesCoefficient[Exp[x] (Sec[x] + Tan[x])^n, {x, 0, n}], {n, 0, 19}]

a(n) = Vec(serlaplace(exp(x)*(1/cos(x) + tan(x))^n))[n+1] \\ Iain Fox, Dec 20 2017

a(n) ~ c * d^n * n^n, where d = 1.12712316036287986633533456353714856005183790513784733... and c = 1.61865092826915643845148401952113086265743345... - Vaclav Kotesovec, Dec 21 2017

A322267 Square 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.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 01 2018

			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,  ...

Index entries for sequences related to boustrophedon transform

Columns k=0..3 give A000007, A000111, A001250, A292758.
Main diagonal gives A298244.
Cf. A322268.

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

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

cp's OEIS Frontend

A296793 a(n) = n! * [x^n] exp(x)*(sec(x) + tan(x))^n.

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