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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. 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-François 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

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