A156348 Triangle T(n,k) read by rows. Column of Pascal's triangle interleaved with k-1 zeros.

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, 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, 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

Offset: 1

Author

Mats Granvik, Feb 08 2009

Keywords

• nonn
• tabl
• easy
• look

Comments

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.

A156348 * A000010 = A156834: (1, 2, 3, 5, 5, 12, 7, 17, 19, 30, 11, ...). - Gary W. Adamson, Feb 16 2009

Row sums give A157019.

Examples

			Table begins:
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  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 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  1
1  0 15  0 15  0  0  0  0  0  0  0  0  0  1
1  8  0 20  0  0  0  8  0  0  0  0  0  0  0  1
1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
1  9 21  0  0 21  0  0  9  0  0  0  0  0  0  0  0  1
1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
1 10  0 35 35  0  0  0  0 10  0  0  0  0  0  0  0  0  0  1
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
• el Houcein el Abdalaoui, Mohamed Dahmoune and Djelloul Ziadi, On the transition reduction problem for finite automata, arXiv preprint arXiv:1301.3751 [cs.FL], 2013. - From _N. J. A. Sloane_, Feb 12 2013
• Jeff Ventrella, Divisor Plot
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A007318, A051731,A156834.

Programs

Haskell

Following Mathar's Maple program.
a156348 n k = a156348_tabl !! (n-1) !! (k-1)
a156348_tabl = map a156348_row [1..]
a156348_row n = map (f n) [1..n] where
   f n k = if r == 0 then a007318 (n' - 2 + k) (k - 1) else 0
           where (n', r) = divMod n k
-- Reinhard Zumkeller, Jan 31 2014

Maple

A156348 := proc(n,k)
    if k < 1 or k > n then
        return 0 ;
    elif n mod k = 0 then
        binomial(n/k-2+k,k-1) ;
    else
        0 ;
    end if;
end proc: # R. J. Mathar, Mar 03 2013

Mathematica

T[n_, k_] := Which[k < 1 || k > n, 0, Mod[n, k] == 0, Binomial[n/k - 2 + k, k - 1], True, 0];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 16 2017 *)