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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 Poincare series). If T(n,1) were 1 for n>0, then T(n, k>1) would be a Poincare 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 CROSSREFS Row sums give A015128. Main diagonal T(n,n) gives A040000. Cf. A005843, A007980, A061866, A091773, A235792, A300415. Sequence in context: A029317 A127800 A035692 * A143613 A208955 A121363 Adjacent sequences:  A308648 A308649 A308650 * A308655 A308656 A308660 KEYWORD nonn,tabl AUTHOR Gregory L. Simay, Jun 14 2019 STATUS approved

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Last modified July 18 15:44 EDT 2019. Contains 325144 sequences. (Running on oeis4.)