A285229 - OEIS

%I #37 Feb 22 2023 10:02:40

%S 1,0,1,0,1,1,0,2,1,1,0,2,3,1,1,0,3,4,3,1,1,0,4,8,5,3,1,1,0,5,11,10,5,

%T 3,1,1,0,6,18,16,11,5,3,1,1,0,8,25,29,18,11,5,3,1,1,0,10,38,44,34,19,

%U 11,5,3,1,1,0,12,52,72,55,36,19,11,5,3,1,1

%N Expansion of g.f. Product_{j>=1} 1/(1-y*x^j)^A000009(j), triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%H Alois P. Heinz, <a href="/A285229/b285229.txt">Rows n = 0..200, flattened</a>

%H <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>

%F G.f.: Product_{j>=1} 1/(1-y*x^j)^A000009(j).

%e T(n,k) is the number of multisets of exactly k partitions of positive integers into distinct parts with total sum of parts equal to n.

%e T(4,1) = 2: {4}, {31}.

%e T(4,2) = 3: {3,1}, {21,1}, {2,2}.

%e T(4,3) = 1: {2,1,1}.

%e T(4,4) = 1: {1,1,1,1}.

%e Triangle T(n,k) begins:

%e   1;

%e   0,  1;

%e   0,  1,  1;

%e   0,  2,  1,   1;

%e   0,  2,  3,   1,  1;

%e   0,  3,  4,   3,  1,  1;

%e   0,  4,  8,   5,  3,  1,  1;

%e   0,  5, 11,  10,  5,  3,  1,  1;

%e   0,  6, 18,  16, 11,  5,  3,  1,  1;

%e   0,  8, 25,  29, 18, 11,  5,  3,  1, 1;

%e   0, 10, 38,  44, 34, 19, 11,  5,  3, 1, 1;

%e   0, 12, 52,  72, 55, 36, 19, 11,  5, 3, 1, 1;

%e   0, 15, 75, 110, 96, 60, 37, 19, 11, 5, 3, 1, 1;

%e   ...

%p with(numtheory):

%p g:= proc(n) option remember; `if`(n=0, 1, add(add(

%p       `if`(d::odd, d, 0), d=divisors(j))*g(n-j), j=1..n)/n)

%p     end:

%p b:= proc(n, i) option remember; expand(

%p       `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*

%p        x^j*binomial(g(i)+j-1, j), j=0..n/i))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):

%p seq(T(n), n=0..16);

%t L[n_] := QPochhammer[x^2]/QPochhammer[x] + O[x]^n;

%t A[n_] := Module[{c = L[n]}, CoefficientList[#, y]& /@ CoefficientList[ 1/Product[(1 - x^k*y + O[x]^n)^SeriesCoefficient[c, {x, 0, k}], {k, 1, n}], x]];

%t A[12] // Flatten (* _Jean-François Alcover_, Jan 19 2020, after _Andrew Howroyd_ *)

%t g[n_] := g[n] = If[n==0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n];

%t b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[b[n - i*j, i - 1]*x^j* Binomial[g[i] + j - 1, j], {j, 0, n/i}]]];

%t T[n_] := CoefficientList[b[n, n] + O[x]^(n+1), x];

%t T /@ Range[0, 16] // Flatten (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)

%o (PARI)

%o L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}

%o A(n)={my(c=L(n), v=Vec(1/prod(k=1, n, (1 - x^k*y + O(x*x^n))^polcoef(c,k)))); vector(#v, n, Vecrev(v[n],n))}

%o {my(T=A(12)); for(n=1, #T, print(T[n]))} \\ _Andrew Howroyd_, Dec 29 2019

%Y Columns k=0..10 give: A000007, A000009 (for n>0), A320787, A320788, A320789, A320790, A320791, A320792, A320793, A320794, A320795.

%Y Row sums give A089259.

%Y T(2n,n) give A285230.

%Y Cf. A061260, A360763.

%K nonn,tabl

%O 0,8

%A _Alois P. Heinz_, Apr 14 2017