%I #25 Nov 16 2017 07:20:41
%S 1,1,1,1,0,1,1,2,0,1,1,0,0,0,1,1,3,3,0,0,1,1,0,0,0,0,0,1,1,4,0,4,0,0,
%T 0,1,1,0,6,0,0,0,0,0,1,1,5,0,0,5,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,1,6,
%U 10,10,0,6,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,7,0,0,0,0,7,0,0,0,0,0,0
%N Triangle T(n,k) read by rows. Column of Pascal's triangle interleaved with k-1 zeros.
%C The rows of the Pascal triangle are here found as square root parabolas like in the plots at www.divisorplot.com. Central binomial coefficients are found at the square root boundary.
%C A156348 * A000010 = A156834: (1, 2, 3, 5, 5, 12, 7, 17, 19, 30, 11, ...). - _Gary W. Adamson_, Feb 16 2009
%C Row sums give A157019.
%H Reinhard Zumkeller, <a href="/A156348/b156348.txt">Rows n = 1..125 of triangle, flattened</a>
%H el Houcein el Abdalaoui, Mohamed Dahmoune and Djelloul Ziadi, <a href="http://arxiv.org/abs/1301.3751">On the transition reduction problem for finite automata</a>, arXiv preprint arXiv:1301.3751 [cs.FL], 2013. - From _N. J. A. Sloane_, Feb 12 2013
%H Jeff Ventrella, <a href="http://www.divisorplot.com">Divisor Plot</a>
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%e Table begins:
%e 1
%e 1 1
%e 1 0 1
%e 1 2 0 1
%e 1 0 0 0 1
%e 1 3 3 0 0 1
%e 1 0 0 0 0 0 1
%e 1 4 0 4 0 0 0 1
%e 1 0 6 0 0 0 0 0 1
%e 1 5 0 0 5 0 0 0 0 1
%e 1 0 0 0 0 0 0 0 0 0 1
%e 1 6 10 10 0 6 0 0 0 0 0 1
%e 1 0 0 0 0 0 0 0 0 0 0 0 1
%e 1 7 0 0 0 0 7 0 0 0 0 0 0 1
%e 1 0 15 0 15 0 0 0 0 0 0 0 0 0 1
%e 1 8 0 20 0 0 0 8 0 0 0 0 0 0 0 1
%e 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
%e 1 9 21 0 0 21 0 0 9 0 0 0 0 0 0 0 0 1
%e 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
%e 1 10 0 35 35 0 0 0 0 10 0 0 0 0 0 0 0 0 0 1
%p A156348 := proc(n,k)
%p if k < 1 or k > n then
%p return 0 ;
%p elif n mod k = 0 then
%p binomial(n/k-2+k,k-1) ;
%p else
%p 0 ;
%p end if;
%p end proc: # _R. J. Mathar_, Mar 03 2013
%t T[n_, k_] := Which[k < 1 || k > n, 0, Mod[n, k] == 0, Binomial[n/k - 2 + k, k - 1], True, 0];
%t Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 16 2017 *)
%o (Haskell) Following Mathar's Maple program.
%o a156348 n k = a156348_tabl !! (n-1) !! (k-1)
%o a156348_tabl = map a156348_row [1..]
%o a156348_row n = map (f n) [1..n] where
%o f n k = if r == 0 then a007318 (n' - 2 + k) (k - 1) else 0
%o where (n', r) = divMod n k
%o -- _Reinhard Zumkeller_, Jan 31 2014
%Y Cf. A007318, A051731,A156834.
%K nonn,tabl,easy,look
%O 1,8
%A _Mats Granvik_, Feb 08 2009