A264405 - OEIS

A264405

Triangle read by rows: T(n,k) is the number of integer partitions of n having k repeated parts (each occurrence is counted).

1, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 2, 0, 1, 3, 0, 2, 1, 0, 1, 4, 0, 2, 2, 2, 0, 1, 5, 0, 4, 2, 1, 2, 0, 1, 6, 0, 6, 2, 3, 2, 2, 0, 1, 8, 0, 7, 4, 4, 2, 2, 2, 0, 1, 10, 0, 8, 6, 6, 4, 3, 2, 2, 0, 1, 12, 0, 13, 6, 6, 8, 3, 3, 2, 2, 0, 1, 15, 0, 15, 9, 11, 6, 9, 4, 3, 2, 2, 0, 1, 18, 0, 21, 10, 13, 12, 7, 8, 4, 3, 2, 2, 0, 1

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OFFSET

0,7

COMMENTS

Compare with A264052 where only one occurrence of a repeated part is counted.

Sum of entries in row n = number of partitions of n = A000041(n).

Sum_{k>=0} k*T(n,k) = A194452(n).

LINKS

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

FORMULA

G.f.: G(t,x) = Product_{j>=1}(1 + x^j + t^2*x^{2j}/(1 - tx^j)).

EXAMPLE

T(4,2) = 2 because each of the partitions [2,2] and [2,1,1] have 2 repeated parts, while [4], [3,1], [1,1,1,1] have 0 or 4 repeated parts.

Triangle starts:

1, 0;

1, 0, 1;

2, 0, 0, 1;

2, 0, 2, 0, 1;

3, 0, 2, 1, 0, 1;

MAPLE

g := product(1+x^j+t^2*x^(2*j)/(1-t*x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n from 0 to 20 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 20 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form

# second Maple program:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

add(expand(b(n-i*j, i-1)*`if`(j>1, x^j, 1)), j=0..n/i)))

end:

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

seq(T(n), n=0..14); # Alois P. Heinz, Dec 07 2015

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Expand[b[n - i*j, i - 1]*If[j > 1, x^j, 1]], {j, 0, n/i}]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, n}]][b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 23 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A000009, A000041, A194452, A264052.

Sequence in context: A087773 A025867 A078646 * A304650 A360670 A347992

Adjacent sequences: A264402 A264403 A264404 * A264406 A264407 A264408

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Dec 07 2015

STATUS

approved