The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A144074

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A144074

Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.
(history; published version)

#43 by Susanna Cuyler at Wed Dec 30 08:22:15 EST 2020

STATUS

proposed

approved

#42 by Jean-François Alcover at Wed Dec 30 07:42:47 EST 2020

STATUS

editing

proposed

#41 by Jean-François Alcover at Wed Dec 30 07:42:24 EST 2020

MATHEMATICA

a[n_, k_] := SeriesCoefficient[ Product[1/(1-x^j)^(k^j), {j, 1, n}], {x, 0, n}]; a[0, _] = 1; a[_?Positive, 0] = 0; Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jan 15 2014 *)

Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 15 2014 *)

etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d p[d], {d, Divisors[j]}] b[n-j], {j, 1, n}]/n]; b];

A[n_, k_] := etr[k^#&][n];

Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 30 2020, after Alois P. Heinz *)

STATUS

approved

editing

#40 by Alois P. Heinz at Fri Sep 21 16:04:38 EDT 2018

STATUS

editing

approved

#39 by Alois P. Heinz at Fri Sep 21 16:01:11 EDT 2018

CROSSREFS

Cf. A004248, A257740, A256142, A292804.

#38 by Alois P. Heinz at Fri Sep 21 15:54:25 EDT 2018

EXAMPLE

A(4,1) = 5: {aaaa}, {aaa,a}, {aa,aa}, {aa,a,a}, {a,a,a,a}.

#37 by Alois P. Heinz at Fri Sep 21 15:48:26 EDT 2018

EXAMPLE

A(3,2) = 20: {aaa}, {a,aa}, {a,a,a}, {bbb}, {b,bb}, {b,b,b}, {aab}, {aba}, {baa}, {a,ab}, {a,ba}, {aa,b}, {a,a,b}, {bba}, {bab}, {abb}, {b,ba}, {b,ab}, {bb,a}, {b,b,a}.

#36 by Alois P. Heinz at Fri Sep 21 15:39:51 EDT 2018

NAME

Square Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of the powers of k.

EXAMPLE

A(2,2) = 7: {aa}, {a,a}, {bb}, {b,b}, {ab}, {ba}, {a,b}.

A(2,3) = 15: {aa}, {a,a}, {bb}, {b,b}, {cc}, {c,c}, {ab}, {ba}, {a,b}, {ac}, {ca}, {a,c}, {bc}, {cb}, {b,c}.

#35 by Alois P. Heinz at Fri Sep 21 15:37:45 EDT 2018

FORMULA

Column k is Euler transform of the powers of k.

EXTENSIONS

Name changed by Alois P. Heinz, Sep 21 2018

STATUS

approved

editing

#34 by Alois P. Heinz at Thu Sep 20 15:26:19 EDT 2018

STATUS

editing

approved