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A364892
Row sums of A364891.
1
0, 0, 1, 1, 2, 2, 4, 5, 8, 10, 15, 19, 27, 34, 47, 59, 79, 99, 130, 162, 210, 260, 332, 410, 517, 635, 794, 970, 1202, 1463, 1799, 2180, 2664, 3214, 3904, 4693, 5669, 6789, 8163, 9740, 11658, 13865, 16527, 19592, 23267, 27496, 32538, 38343, 45223, 53142, 62488
OFFSET
1,5
FORMULA
a(n) = Sum_{k=1..n} ((-1)^(k-1)*Sum_{j=0..k-1} (-1)^j*(p(n - j*(2*j + 1)) - p(n - (j + 1)*(2*j + 1)))), where p(n) = A000041(n) is the number of partitions of n.
Conjecture: lim_{n->oo} a(n)/A000041(n) = 1/4.
MATHEMATICA
A364891[n_, k_]:=(-1)^(k-1)*Sum[(-1)^j*(PartitionsP[n-j(2j+1)]-PartitionsP[n-(j+1)(2j+1)]), {j, 0, k-1}]; Table[Sum[A364891[n, k], {k, 1, n}], {n, 1, 51}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Stefano Spezia, Aug 12 2023
STATUS
approved