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A320789
Number of multisets of exactly four partitions of positive integers into distinct parts with total sum of parts equal to n.
2
1, 1, 3, 5, 11, 18, 34, 55, 96, 152, 248, 386, 607, 921, 1405, 2092, 3112, 4551, 6635, 9545, 13683, 19401, 27393, 38346, 53441, 73928, 101840, 139398, 190020, 257601, 347836, 467381, 625686, 833917, 1107547, 1465136, 1931754, 2537747, 3323490, 4338012, 5645645
OFFSET
4,3
LINKS
FORMULA
a(n) = [x^n y^4] Product_{j>=1} 1/(1-y*x^j)^A000009(j).
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, add(add(`if`(d::odd,
d, 0), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
b:= proc(n, i) option remember; series(`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))), x, 5)
end:
a:= n-> coeff(b(n$2), x, 4):
seq(a(n), n=4..60);
MATHEMATICA
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] = Series[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}]]], {x, 0, 5}];
a[n_] := SeriesCoefficient[b[n, n], {x, 0, 4}];
a /@ Range[4, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)
CROSSREFS
Column k=4 of A285229.
Cf. A000009.
Sequence in context: A244473 A023597 A281357 * A269628 A162891 A320351
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 21 2018
STATUS
approved