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

1,8

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.

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A007318, A051731,A156834.

Sequence in context: A216282 A147861 A167271 * A306437 A343746 A227990

Adjacent sequences: A156345 A156346 A156347 * A156349 A156350 A156351

Keyword

nonn,tabl,easy,look

Author

Mats Granvik, Feb 08 2009

Status

approved