A206434 Total number of even parts in the last section of the set of partitions of n.

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

Author

Omar E. Pol, Feb 12 2012

Keywords

• nonn

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

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Partial sums give A066898.

Cf. A002865, A006128, A135010, A138121, A138137, A182703, A183063, A206433, A206435, A206436, A207378, A336811, A336812.

Programs

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 *)

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

Extensions

More terms from Alois P. Heinz, Mar 22 2012