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A309255
a(n) = n + 1 - Sum_{k=0..n} (Stirling1(n,k) mod 2).
2
0, 1, 1, 2, 3, 4, 3, 4, 7, 8, 7, 8, 9, 10, 7, 8, 15, 16, 15, 16, 17, 18, 15, 16, 21, 22, 19, 20, 21, 22, 15, 16, 31, 32, 31, 32, 33, 34, 31, 32, 37, 38, 35, 36, 37, 38, 31, 32, 45, 46, 43, 44, 45, 46, 39, 40, 49, 50, 43, 44, 45, 46, 31, 32, 63, 64, 63, 64, 65, 66, 63, 64, 69, 70, 67, 68, 69
OFFSET
0,4
COMMENTS
Number of even entries in n-th row of triangle of Stirling numbers of the first kind (A048994).
LINKS
FORMULA
G.f.: 1/(1 - x)^2 - (1 + x) * Product_{k>=0} (1 + 2*x^(2^(k+1))).
a(n) = n + 1 - 2^A000120(floor(n/2)).
MATHEMATICA
Table[n + 1 - Sum[Mod[StirlingS1[n, k], 2], {k, 0, n}], {n, 0, 76}]
nmax = 76; CoefficientList[Series[1/(1 - x)^2 - (1 + x) Product[(1 + 2 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
PROG
(PARI) a(n) = n + 1 - sum(k=0, n, stirling(n, k, 1) % 2); \\ Michel Marcus, Jul 19 2019
(PARI) a(n) = n + 1 - 2^hammingweight(n\2); \\ Amiram Eldar, Jul 25 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jul 19 2019
STATUS
approved