|
%I #16 Feb 17 2021 07:28:04
%S 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,
%T 36,126,272,338,294,272,1,8,49,217,645,1189,1369,1309,1385,1,9,64,344,
%U 1320,3380,5506,6238,6664,7936,1,10,81,513,2429,8141,18285,27365,31993,38177,50521
%N 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)).
%C A(n,k) is the k-th binomial transform of A000111 evaluated at n.
%C Also column k is the boustrophedon transform of powers of k.
%H Alois P. Heinz, <a href="/A292975/b292975.txt">Antidiagonals n = 0..140, flattened</a>
%H 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 (<a href="http://neilsloane.com/doc/bous.txt">Abstract</a>, <a href="http://neilsloane.com/doc/bous.pdf">pdf</a>, <a href="http://neilsloane.com/doc/bous.ps">ps</a>)
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F E.g.f. of column k: exp(k*x)*(sec(x) + tan(x)).
%e 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! + ...
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, 6, ...
%e 1, 4, 9, 16, 25, 36, ...
%e 2, 9, 28, 65, 126, 217, ...
%e 5, 24, 93, 272, 645, 1320, ...
%e 16, 77, 338, 1189, 3380, 8141, ...
%p b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p add(b(o-1+j, u-j), j=1..u))
%p end:
%p A:= proc(n, k) option remember; `if`(k=0, b(n, 0),
%p add(binomial(n, j)*A(j, k-1), j=0..n))
%p end:
%p seq(seq(A(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Sep 27 2017
%t Table[Function[k, n! SeriesCoefficient[Exp[k x] (Sec[x] + Tan[x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%Y Columns k=0..2 give A000111, A000667, A000752.
%Y Main diagonal gives A292976.
%K nonn,tabl
%O 0,5
%A _Ilya Gutkovskiy_, Sep 27 2017
|
