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A144871 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where sequence a_k of column k is shadow transform of C(n+k-1,k).

%I A144871 #28 Sep 16 2019 21:25:33
%S A144871 1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,2,1,1,1,2,1,3,2,1,1,1,1,1,4,1,
%T A144871 2,1,1,1,1,2,1,1,3,1,1,1,1,2,3,1,3,4,2,2,1,1,1,1,1,2,3,5,1,1,2,1,1,1,
%U A144871 1,1,3,4,6,2,2,4,2,1,1,1,2,1,4,1,1,1,4,4,3,2,1,1,1,1,2,1,1,1,4,1,1,4,3,2,1
%N A144871 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where sequence a_k of column k is shadow transform of C(n+k-1,k).
%C A144871 Row sequences have periods 1, 1, 3, 8, ... given in A144872.
%H A144871 Alois P. Heinz, <a href="/A144871/b144871.txt">Antidiagonals n = 1..141, flattened</a>
%H A144871 Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150. (<a href="http://math.berkeley.edu/~halbeis/publications/psf/seq.ps">ps</a>, <a href="http://math.berkeley.edu/~halbeis/publications/pdf/seq.pdf">pdf</a>); see Definition 7 for the shadow transform.
%H A144871 OEIS Wiki, <a href="https://oeis.org/wiki/Shadow_transform">Shadow transform</a>.
%H A144871 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%e A144871 Square array begins:
%e A144871   1, 1, 1, 1, 1, 1, ...
%e A144871   1, 1, 1, 1, 1, 1, ...
%e A144871   1, 2, 1, 1, 2, 1, ...
%e A144871   1, 1, 2, 1, 1, 2, ...
%e A144871   1, 2, 3, 4, 1, 1, ...
%e A144871   1, 2, 1, 1, 3, 3, ...
%p A144871 shadow:= proc(p) proc(n) add(`if`(modp(p(j), n)=0, 1, 0), j=0..n-1)
%p A144871          end end:
%p A144871 f:= proc(k) proc(n) binomial(n+k-1, k) end end:
%p A144871 A:= (n, k)-> shadow(f(k))(n):
%p A144871 seq(seq(A(n, 1+d-n), n=1..d), d=1..20);
%t A144871 shadow[p_] := Function[n, Sum[If[Mod[p[j], n] == 0, 1, 0], {j, 0, n-1}]]; f[k_] := Function[n, Binomial[n+k-1, k]]; a[n_, k_] := shadow[f[k]][n]; Table[Table[a[n, 1+d-n], {n, 1, d}], {d, 1, 20}] // Flatten (* _Jean-François Alcover_, Dec 18 2013, translated from Maple *)
%Y A144871 Columns 1-9 give: A000012, A068068, A072457, A144865, A144866, A144867, A144868, A144869, A144870.
%Y A144871 Rows 1+2, 3 give: A000012, A101825.
%Y A144871 Periods of the row sequences: A144872.
%Y A144871 Cf. A007318.
%K A144871 nonn,tabl,look
%O A144871 1,9
%A A144871 _Alois P. Heinz_, Sep 23 2008