login
A206434
Total number of even parts in the last section of the set of partitions of n.
4
0, 1, 0, 3, 1, 6, 4, 13, 10, 24, 23, 46, 46, 81, 88, 143, 159, 242, 278, 404, 470, 657, 776, 1057, 1251, 1663, 1984, 2587, 3089, 3967, 4742, 6012, 7184, 9001, 10753, 13351, 15917, 19594, 23335, 28514, 33883, 41140, 48787, 58894, 69691, 83680, 98809, 118101
OFFSET
1,4
COMMENTS
From Omar E. Pol, Apr 07 2023: (Start)
Convolution of A002865 and A183063.
a(n) is also the total number of even divisors of the terms in the n-th row of the triangle A336811.
a(n) is also the number of even terms in the n-th row of the triangle A207378.
a(n) is also the number of even terms in the n-th row of the triangle A336812. (End)
LINKS
FORMULA
G.f.: (Sum_{i>0} (x^(2*i)-x^(2*i+1))/(1-x^(2*i)))/Product_{i>0} (1-x^i). - Alois P. Heinz, Mar 23 2012
MAPLE
b:= proc(n, i) option remember; local f, g;
if n=0 or i=1 then [1, 0]
else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
[f[1]+g[1], f[2]+g[2]+ ((i+1) mod 2)*g[1]]
fi
end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
seq (a(n), n=1..50); # Alois P. Heinz, Mar 22 2012
MATHEMATICA
b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, 0}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + Mod[i+1, 2]*g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[ a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)
KEYWORD
nonn
AUTHOR
Omar E. Pol, Feb 12 2012
EXTENSIONS
More terms from Alois P. Heinz, Mar 22 2012
STATUS
approved