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A363434
Total number of blocks containing only elements of the same parity in all partitions of [n].
7
0, 1, 2, 7, 24, 97, 412, 1969, 9898, 54461, 313944, 1947613, 12603100, 86760255, 620559230, 4682462777, 36586620348, 299664171115, 2534306825064, 22355119509231, 203115201624030, 1917124624702475, 18598998656476220, 186822424157036439, 1925326063016510832
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k=0..n} k * A124424(n,k).
a(n) = A363452(n) + A363453(n).
a(n) mod 2 = A000035(n).
EXAMPLE
a(3) = 7 = 0 + 1 + 2 + 1 + 3 : 123, 12|3, 13|2, 1|23, 1|2|3.
MAPLE
b:= proc(n, e, o, m) option remember; `if`(n=0, e+o,
(e+m)*b(n-1, o, e, m)+b(n-1, o, e+1, m)+
`if`(o=0, 0, o*b(n-1, o-1, e, m+1)))
end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..24);
MATHEMATICA
b[n_, e_, o_, m_] := b[n, e, o, m] = If[n == 0, e + o, (e + m)*b[n-1, o, e, m] + b[n - 1, o, e + 1, m] + If[o == 0, 0, o*b[n - 1, o - 1, e, m + 1]]];
a[n_] := b[n, 0, 0, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Sep 10 2023, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 01 2023
STATUS
approved