login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 15
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 20, 5, 0, 1, 5, 26, 64, 59, 7, 0, 1, 6, 40, 148, 276, 162, 11, 0, 1, 7, 57, 285, 843, 1137, 449, 15, 0, 1, 8, 77, 488, 2020, 4632, 4648, 1200, 22, 0, 1, 9, 100, 770, 4140, 13876, 25124, 18585, 3194, 30, 0, 1, 10, 126 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Column k > 1 is asymptotic to k^n * exp(2*sqrt(n) - 1/2 + c(k)) / (2 * sqrt(Pi) * n^(3/4)), where c(k) = Sum_{m>=2} 1/(m*(k^(m-1)-1)). - Vaclav Kotesovec, Mar 14 2015

LINKS

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

N. J. A. Sloane, Transforms

FORMULA

G.f. of column k: Product_{j>=1} 1/(1-x^j)^(k^j).

Column k is Euler transform of the powers of k.

T(n,k) = Sum_{i=0..k} C(k,i) * A257740(n,k-i). - Alois P. Heinz, May 08 2015

EXAMPLE

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

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}.

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}.

Square array begins:

  1, 1,   1,    1,    1,     1, ...

  0, 1,   2,    3,    4,     5, ...

  0, 2,   7,   15,   26,    40, ...

  0, 3,  20,   64,  148,   285, ...

  0, 5,  59,  276,  843,  2020, ...

  0, 7, 162, 1137, 4632, 13876, ...

MAPLE

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n, k)-> etr(j->k^j)(n); seq(seq(A(n, d-n), n=0..d), d=0..14);

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-Fran├žois Alcover, Jan 15 2014 *)

CROSSREFS

Columns k=0-10 give: A000007, A000041, A034899, A144067, A144068, A144069, A144070, A144071, A144072, A144073, A292837.

Rows n=0-2 give: A000012, A001477, A005449.

Main diagonal gives A252654.

Cf. A004248, A257740, A256142, A292804.

Sequence in context: A292860 A265609 A261718 * A261780 A124540 A124550

Adjacent sequences:  A144071 A144072 A144073 * A144075 A144076 A144077

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 09 2008

EXTENSIONS

Name changed by Alois P. Heinz, Sep 21 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 31 22:45 EDT 2020. Contains 334756 sequences. (Running on oeis4.)