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A308654 The overpartition triangle: T(n,k) is the number of overpartitions of n with exactly k positive integer parts, 0 <= k <= n. 0
1, 0, 2, 0, 2, 2, 0, 2, 4, 2, 0, 2, 6, 4, 2, 0, 2, 8, 8, 4, 2, 0, 2, 10, 14, 8, 4, 2, 0, 2, 12, 20, 16, 8, 4, 2, 0, 2, 14, 28, 26, 16, 8, 4, 2, 0, 2, 16, 38, 40, 28, 16, 8, 4, 2, 0, 2, 18, 48, 60, 46, 28, 16, 8, 4, 2, 0, 2, 20, 60, 84, 72, 48, 28, 16, 8, 4, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
T(n,0) = A000007(n).
T(n,1) = A040000(n) for n > 0.
T(n,2) = A005843(n-1).
T(n,3) = 2*A007980(n-3).
T(n,4) = 2*A061866(n-1).
T(n,5) = 2*A091773(n-5).
Conjecture: T(n,k) = 2*(the associated Poincaré series). If T(n,1) were 1 for n>0, then T(n, k>1) would be a Poincaré series.
LINKS
FORMULA
Sum_{k=0..n} T(n,k) = A015128(n), the number of overpartitions of n.
If k > n, T(n,k) = 0.
If n >= k > n/2, T(n,k) = 2*A015128(n-k).
Conjecture: T(n,k) = T(n-k, k) + 2*(T(n-k, k-1) + ... + T(n-k, 1)).
Conjecture: T(n,k) = T(n-1, k-1) + T(n-k, k-1) + T(n-k-1, k-1) + T(n-2k, k-1) + T(n-2k-1) + ...
Conjecture: T(n,1) + T(n-1,2) + ... + T(n-floor(n/2),floor(n/2)) = A300415(n+1).
T(n,2) = 2n - 2.
Conjecture: g.f. T(n,k) = 2*(1+x)(1+x^2)...(1+x^(k-1))/((1-x)...(1-x^k)).
Sum_{k=1..n} k * T(n,k) = A235792(n). - Alois P. Heinz, Jun 15 2019
EXAMPLE
T(5,3) = 8 and counts the overpartitions 3,1,1; 3',1,1; 3,1',1; 3',1',1; 2,2,1; 2',2,1; 2,2,1' and 2',2,1'.
T(16,5) = 404 = T(11,5) + 2*( T(11,4) + T(11,3) + T(11,2) + T(11,1)) = 72 + 2*(84 + 60 + 20 + 2) = 404.
T(16, 5) = T(15,4) + T(11,4) + T(10,4) + T(6,4) + T(5,4) = 248 + 84 + 60 + 8 + 4 = 404.
T(9,1) + T(8,2) + T(7,3) + T(6,4) + T(6,5)= 2 + 14 + 20 + 8 + 2 = 46 =A300415(10).
Triangle: T(n,k) begins:
1;
0, 2;
0, 2, 2;
0, 2, 4, 2;
0, 2, 6, 4, 2;
0, 2, 8, 8, 4, 2;
0, 2, 10, 14, 8, 4, 2;
0, 2, 12, 20, 16, 8, 4, 2;
0, 2, 14, 28, 26, 16, 8, 4, 2;
0, 2, 16, 38, 40, 28, 16, 8, 4, 2;
0, 2, 18, 48, 60, 46, 28, 16, 8, 4, 2;
...
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
expand(`if`(j>0, 2*x^j, 1)*b(n-i*j, i-1)), j=0..n/i)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14); # Alois P. Heinz, Jun 15 2019
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Expand[If[j > 0, 2*x^j, 1]*b[n - i*j, i - 1]], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Dec 10 2019, after Alois P. Heinz *)
CROSSREFS
Row sums give A015128.
Main diagonal T(n,n) gives A040000.
Sequence in context: A029317 A127800 A035692 * A329860 A334209 A143613
KEYWORD
nonn,tabl
AUTHOR
Gregory L. Simay, Jun 14 2019
STATUS
approved

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Last modified March 29 06:34 EDT 2024. Contains 371265 sequences. (Running on oeis4.)