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A273730 Square array read by antidiagonals: A(n,k) = number of permutations of n elements divided by the number of k-ary heaps on n+1 elements, n>=0, k>=1. 10
1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 24, 1, 1, 1, 1, 3, 120, 1, 1, 1, 1, 2, 6, 720, 1, 1, 1, 1, 1, 3, 9, 5040, 1, 1, 1, 1, 1, 2, 4, 24, 40320, 1, 1, 1, 1, 1, 1, 3, 8, 45, 362880, 1, 1, 1, 1, 1, 1, 2, 4, 12, 108, 3628800, 1, 1, 1, 1, 1, 1, 1, 3, 5, 16, 189, 39916800 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Wikipedia, D-ary heap

FORMULA

A(n,k) = A000142(n)/A273693(n+1,k).

EXAMPLE

Square array A(n,k) begins:

:     1,  1,  1, 1, 1, 1, 1, 1, ...

:     1,  1,  1, 1, 1, 1, 1, 1, ...

:     2,  1,  1, 1, 1, 1, 1, 1, ...

:     6,  2,  1, 1, 1, 1, 1, 1, ...

:    24,  3,  2, 1, 1, 1, 1, 1, ...

:   120,  6,  3, 2, 1, 1, 1, 1, ...

:   720,  9,  4, 3, 2, 1, 1, 1, ...

:  5040, 24,  8, 4, 3, 2, 1, 1, ...

: 40320, 45, 12, 5, 4, 3, 2, 1, ...

MAPLE

with(combinat):

b:= proc(n, k) option remember; local h, i, x, y, z;

      if n<2 then 1 elif k<2 then k

    else h:= ilog[k]((k-1)*n+1);

         if k^h=(k-1)*n+1 then b((n-1)/k, k)^k*

            multinomial(n-1, ((n-1)/k)$k)

       else x, y:=(k^h-1)/(k-1), (k^(h-1)-1)/(k-1);

            for i from 0 do z:= (n-1)-(k-1-i)*y-i*x;

              if y<=z and z<=x then b(y, k)^(k-1-i)*

                 multinomial(n-1, y$(k-1-i), x$i, z)*

                 b(x, k)^i*b(z, k); break fi

            od

      fi fi

    end:

A:= (n, k)-> n!/b(n+1, k):

seq(seq(A(n, 1+d-n), n=0..d), d=0..14);

MATHEMATICA

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, k_] := b[n, k] = Module[{h, i, x, y, z}, Which[n<2, 1, k<2, k, True, h = Floor @ Log[k, (k - 1)*n + 1]; If [k^h == (k-1)*n+1, b[(n-1)/k, k]^k*multinomial[n-1, Array[(n-1)/k&, k]], {x, y} = {(k^h-1)/(k-1), (k^(h-1)-1)/(k-1)}; For[i = 0, True, i++, z = (n-1) - (k-1-i)*y - i*x; If[y <= z && z <= x, b[y, k]^(k-1-i)*multinomial[n-1, Join[Array[y&, k-1-i], Array[x&, i], {z}]] * b[x, k]^i*b[z, k] // Return]]]]]; A[n_, k_] :=  n!/b[n+1, k]; Table[A[n, 1+d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-Fran├žois Alcover, Mar 13 2017, translated from Maple *)

CROSSREFS

Columns k=1-10 give: A000142, A133385, A273731, A273732, A273733, A273734, A273735, A273736, A273737, A273738.

Cf. A273693.

Sequence in context: A199958 A112734 A260685 * A326047 A320637 A280491

Adjacent sequences:  A273727 A273728 A273729 * A273731 A273732 A273733

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, May 28 2016

STATUS

approved

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Last modified December 15 00:30 EST 2019. Contains 329988 sequences. (Running on oeis4.)