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A285229 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. 14
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, 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, 11, 5, 3, 1, 1, 0, 12, 52, 72, 55, 36, 19, 11, 5, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

Index entries for triangles generated by the Multiset Transformation

FORMULA

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

EXAMPLE

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.

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

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

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

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

Triangle T(n,k) begins:

  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,  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, 11,  5,  3, 1, 1;

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

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

  ...

MAPLE

with(numtheory):

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

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

    end:

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

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

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

    end:

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

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

MATHEMATICA

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

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]];

A[12] // Flatten (* Jean-François Alcover, Jan 19 2020, after Andrew Howroyd *)

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];

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

T /@ Range[0, 16] // Flatten (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

PROG

(PARI)

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

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))}

{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

CROSSREFS

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

Row sums give A089259.

T(2n,n) give A285230.

Cf. A061260.

Sequence in context: A114638 A123340 A267486 * A227425 A333213 A301636

Adjacent sequences:  A285226 A285227 A285228 * A285230 A285231 A285232

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Apr 14 2017

STATUS

approved

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Last modified August 18 09:18 EDT 2022. Contains 356204 sequences. (Running on oeis4.)