cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A292712 Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 4, 3, 0, 1, 1, 4, 8, 5, 0, 1, 1, 4, 14, 25, 7, 0, 1, 1, 4, 14, 43, 53, 11, 0, 1, 1, 4, 14, 67, 139, 148, 15, 0, 1, 1, 4, 14, 67, 223, 495, 328, 22, 0, 1, 1, 4, 14, 67, 343, 951, 1544, 858, 30, 0, 1, 1, 4, 14, 67, 343, 1431, 3680, 5111, 1938, 42, 0
Offset: 0

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Author

Alois P. Heinz, Sep 21 2017

Keywords

Examples

			A(2,3) = 4: {aa}, {ab}, {ba}, {a,a}.
A(3,2) = 8: {aaa}, {aab}, {aba}, {baa}, {aa,a}, {ab,a}, {ba,a}, {a,a,a}.
A(3,3) = 14: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}, {a,a,a}.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,     1,     1,     1,      1, ...
  0,  1,   1,    1,     1,     1,     1,     1,      1, ...
  0,  2,   4,    4,     4,     4,     4,     4,      4, ...
  0,  3,   8,   14,    14,    14,    14,    14,     14, ...
  0,  5,  25,   43,    67,    67,    67,    67,     67, ...
  0,  7,  53,  139,   223,   343,   343,   343,    343, ...
  0, 11, 148,  495,   951,  1431,  2151,  2151,   2151, ...
  0, 15, 328, 1544,  3680,  6620,  9860, 14900,  14900, ...
  0, 22, 858, 5111, 16239, 31539, 53739, 78939, 119259, ...
		

Crossrefs

Rows n=0-1 give: A000012, A057427.
Main diagonal gives A292713.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
          add(b(n-j, j, t-1)/j!, j=i..n/t))
        end:
    g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
    A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
          g(d, k), d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)
        end:
    seq(seq(A(n, d-n), n=0..d), d=0..14);
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
    g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
    A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*g[d, k], {d, Divisors[j]}]* A[n - j, k], {j, 1, n}]/n];
    Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 07 2018, from Maple *)

Formula

G.f. of column k: Product_{j>=1} 1/(1-x^j)^A226873(j,k).
A(n,n) = A(n,k) for all k >= n.
A(n,k) = Sum_{j=0..n} A319495(n,j).
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