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%I #43 Dec 30 2020 08:22:15
%S 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,
%T 148,276,162,11,0,1,7,57,285,843,1137,449,15,0,1,8,77,488,2020,4632,
%U 4648,1200,22,0,1,9,100,770,4140,13876,25124,18585,3194,30,0,1,10,126
%N 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.
%C 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
%H Alois P. Heinz, <a href="/A144074/b144074.txt">Antidiagonals n = 0..140, flattened</a>
%H Vaclav Kotesovec, <a href="https://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f. of column k: Product_{j>=1} 1/(1-x^j)^(k^j).
%F Column k is Euler transform of the powers of k.
%F T(n,k) = Sum_{i=0..k} C(k,i) * A257740(n,k-i). - _Alois P. Heinz_, May 08 2015
%e A(4,1) = 5: {aaaa}, {aaa,a}, {aa,aa}, {aa,a,a}, {a,a,a,a}.
%e A(2,2) = 7: {aa}, {a,a}, {bb}, {b,b}, {ab}, {ba}, {a,b}.
%e 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}.
%e 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}.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 0, 2, 7, 15, 26, 40, ...
%e 0, 3, 20, 64, 148, 285, ...
%e 0, 5, 59, 276, 843, 2020, ...
%e 0, 7, 162, 1137, 4632, 13876, ...
%p 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);
%t a[n_, k_] := SeriesCoefficient[ Product[1/(1-x^j)^(k^j), {j, 1, n}], {x, 0, n}]; a[0, _] = 1; a[_?Positive, 0] = 0;
%t Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jan 15 2014 *)
%t 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];
%t A[n_, k_] := etr[k^#&][n];
%t Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 30 2020, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000007, A000041, A034899, A144067, A144068, A144069, A144070, A144071, A144072, A144073, A292837.
%Y Rows n=0-2 give: A000012, A001477, A005449.
%Y Main diagonal gives A252654.
%Y Cf. A004248, A257740, A256142, A292804.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Sep 09 2008
%E Name changed by _Alois P. Heinz_, Sep 21 2018
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