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
Examples
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, ...
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
Programs
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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
Formula
E.g.f. of column k: exp(k*x)*(sec(x) + tan(x)).
Comments