A292975 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*(sec(x) + tan(x)).

1, 1, 1, 1, 2, 1, 1, 3, 4, 2, 1, 4, 9, 9, 5, 1, 5, 16, 28, 24, 16, 1, 6, 25, 65, 93, 77, 61, 1, 7, 36, 126, 272, 338, 294, 272, 1, 8, 49, 217, 645, 1189, 1369, 1309, 1385, 1, 9, 64, 344, 1320, 3380, 5506, 6238, 6664, 7936, 1, 10, 81, 513, 2429, 8141, 18285, 27365, 31993, 38177, 50521

Offset

0,5

Comments

A(n,k) is the k-th binomial transform of A000111 evaluated at n.

Also column k is the boustrophedon transform of powers of k.

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps)

N. J. A. Sloane, Transforms.

Index entries for sequences related to boustrophedon transform

Formula

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

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 + 3*k + 2)*x^3/3! + (k^4 + 4*k^3 + 6*k^2 + 8*k + 5)*x^4/4! + ...
Square array begins:
   1,   1,    1,     1,     1,     1,  ...
   1,   2,    3,     4,     5,     6,  ...
   1,   4,    9,    16,    25,    36,  ...
   2,   9,   28,    65,   126,   217,  ...
   5,  24,   93,   272,   645,  1320,  ...
  16,  77,  338,  1189,  3380,  8141,  ...

Maple

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
A:= proc(n, k) option remember; `if`(k=0, b(n, 0),
      add(binomial(n, j)*A(j, k-1), j=0..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 27 2017

Mathematica

Table[Function[k, n! SeriesCoefficient[Exp[k x] (Sec[x] + Tan[x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Crossrefs

Columns k=0..2 give A000111, A000667, A000752.

Main diagonal gives A292976.

Sequence in context: A355754 A319844 A193736 * A056863 A120019 A145034

Adjacent sequences: A292972 A292973 A292974 * A292976 A292977 A292978

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Sep 27 2017

Status

approved