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A276422 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its odd singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once. 4
1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 4, 0, 0, 0, 1, 2, 2, 0, 2, 0, 1, 8, 0, 0, 1, 1, 0, 1, 4, 4, 0, 4, 0, 2, 0, 1, 14, 0, 0, 2, 2, 1, 1, 0, 2, 9, 6, 0, 7, 0, 4, 0, 2, 0, 2, 24, 1, 0, 4, 3, 2, 2, 1, 3, 0, 2, 16, 10, 0, 12, 0, 8, 0, 4, 1, 3, 0, 2, 41, 1, 0, 7, 5, 4, 4, 2, 6, 1, 3, 0, 3, 28, 16, 0, 20, 0, 14, 0, 8, 2, 6, 1, 3, 0, 3 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

T(n,0) = A265256(n).

T(n,n) = A000700(n).

Sum(k*T(n,k), k>=0) = A276423(n).

Sum(T(n,k), k>=0) = A000041(n).

LINKS

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

FORMULA

G.f.:  G(t,x) = Product(((1-x^{2j-1})(1+t^{2j-1}x^{2j-1}) + x^{4j-2})/(1-x^j), j=1..infinity).

EXAMPLE

Row 4 is 4, 0, 0, 0, 1 because in the partitions [1,1,1,1], [1,1,2], [2,2], [1,3], [4] the sums of the odd singletons are 0, 0, 0, 4, 0, respectively.

Row 5 is 2, 2, 0, 2, 0, 1 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5] the sums of the odd singletons are 0, 0, 1, 3, 3, 1, 5, respectively.

Triangle starts:

1;

0,1;

2,0,0;

1,1,0,1;

4,0,0,0,1;

2,2,0,2,0,1.

MAPLE

g := Product(((1-x^(2*j-1))*(1+t^(2*j-1)*x^(2*j-1))+x^(4*j-2))/(1-x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 23)): 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, i), i = 0 .. n) end do; # yields sequence in triangular form

# second Maple program:

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

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

      `if`(j=1 and i::odd, x^i, 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, Sep 14 2016

MATHEMATICA

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*If[j == 1 && OddQ[i], x^i, 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-Fran├žois Alcover, Oct 04 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A000041, A000700, A265256, A276423, A276424, A276425.

Sequence in context: A025922 A161369 A151843 * A323069 A325334 A280287

Adjacent sequences:  A276419 A276420 A276421 * A276423 A276424 A276425

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Sep 14 2016

STATUS

approved

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Last modified September 27 12:30 EDT 2020. Contains 337380 sequences. (Running on oeis4.)