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

1,4

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

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

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 *)

Crossrefs

Partial sums give A066898.

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

Sequence in context: A328508 A210841 A007383 * A340179 A120394 A371667

Adjacent sequences: A206431 A206432 A206433 * A206435 A206436 A206437

Keyword

nonn

Author

Omar E. Pol, Feb 12 2012

Extensions

More terms from Alois P. Heinz, Mar 22 2012

Status

approved