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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
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-François Alcover, Oct 04 2016, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A342322 A161369 A151843 * A323069 A325334 A280287
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Sep 14 2016
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)