A292756 -id:A292756 - 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

A292759 Expansion of e.g.f. exp(x)*(tan x + sec x)^3.

Original entry on oeis.org

1, 4, 16, 67, 304, 1519, 8386, 51007, 340024, 2469859, 19438606, 164899447, 1500636844, 14587478299, 150891959026, 1655133221887, 19192311085264, 234597922978339, 3015167371458646, 40651421300224327, 573707768015267284, 8458761578948943979, 130059537979390701466

Offset: 0

N. J. A. Sloane, Sep 26 2017

nonn

C. K. Cook, M. R. Bacon, and R. A. Hillman, Higher-order Boustrophedon transforms for certain well-known sequences, Fib. Q., 55(3) (2017), 201-208.

Cf. A000667, A292756.

nmax = 20; CoefficientList[Series[Exp[x]*(Tan[x]+Sec[x])^3, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 02 2019 *)

a(n) ~ 2^(n + 11/2) * n^(n + 5/2) / (Pi^(n + 5/2) * exp(n - Pi/2)). - Vaclav Kotesovec, Jun 02 2019

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.

Original entry on oeis.org

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

Ilya Gutkovskiy, Dec 01 2018

			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,  ...

Index entries for sequences related to boustrophedon transform

Columns k=0..3 give A000012, A000667, A292756, A292759.
Main diagonal gives A296793.
Cf. A322267.

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

E.g.f. of column k: exp(x)*(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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A292759 Expansion of e.g.f. exp(x)*(tan x + sec x)^3.

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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.

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