A156348 - OEIS

A156348

Triangle T(n,k) read by rows. Column of Pascal's triangle interleaved with k-1 zeros.

%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