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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)). 6
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 (list; table; graph; refs; listen; history; text; internal format)
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 on transform, J. Combin. Theory, 17A 44-54 1996 (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: A238350 A319844 A193736 * A056863 A120019 A145034

Adjacent sequences:  A292972 A292973 A292974 * A292976 A292977 A292978

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, Sep 27 2017

STATUS

approved

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Last modified September 20 09:34 EDT 2020. Contains 337264 sequences. (Running on oeis4.)