A066967 -id:A066967 - cp's OEIS frontend

A102713 Total sum of odd parts in all compositions of n.

1, 2, 8, 18, 48, 110, 260, 586, 1320, 2918, 6412, 13954, 30192, 64926, 138964, 296122, 628664, 1330134, 2805916, 5903090, 12388736, 25942542, 54215268, 113090858, 235502408, 489646150, 1016575020, 2107715426, 4364561680, 9027384958, 18651293172, 38495632794

Offset: 1

Vladeta Jovovic, Feb 06 2005

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).

Cf. A066967, A073371.

a(n)={((15*n+4)*2^(n-1) - 2*(3*n+1)*(-1)^n)/27} \\ Andrew Howroyd, Jan 08 2020

Vec(x*(1 + x^2) / ((1 + x)^2*(1 - 2*x)^2) + O(x^35)) \\ Colin Barker, Jan 08 2020

a(n) = ((15*n+4)*2^(n-1)-2*(3*n+1)*(-1)^n)/27.
From Colin Barker, Jan 08 2020: (Start)
G.f.: x*(1 + x^2) / ((1 + x)^2*(1 - 2*x)^2).
a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3) - 4*a(n-4) for n>4.
(End)

Terms a(26) and beyond from Andrew Howroyd, Jan 08 2020

A208477 Difference between the sum of odd parts and the sum of even parts in all the partitions of n.

Original entry on oeis.org

0, 1, 0, 5, 0, 11, 6, 25, 12, 50, 40, 96, 80, 173, 170, 320, 316, 545, 590, 930, 1020, 1552, 1760, 2537, 2900, 4066, 4736, 6450, 7540, 10045, 11856, 15482, 18280, 23555, 27920, 35461, 42032, 52805, 62662, 77955, 92380, 113963, 135040, 165295, 195540, 237866

Offset: 0

Omar E. Pol, Mar 10 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A066186, A209423.

b:= proc(n,i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    else g:= b(n, i-1);
         h:= `if`(i>n, [0, 0], b(n-i, i));
         [g[1]+h[1], g[2]+h[2] +h[1]*i*(2*(i mod 2)-1)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..60); # Alois P. Heinz, Mar 10 2012

Map[Total[Select[#, OddQ]] - Total[Select[#, EvenQ]] &[Flatten[IntegerPartitions[#]]] &, -1 + Range[30]] (* Peter J. C. Moses, Mar 14 2014 *)
max = 60; s = Sum[x^(2i) (x^(2i) - 2i (x-1) - 1)/(x + x^(4i) - (x+1) x^(2i) ), {i, 1, Floor[max/2]}]/QPochhammer[x] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

a(n) = A066967(n) - A066966(n).

G.f.: (Sum_{i>0} (2*i-1)*x^(2*i-1)/(1-x^(2*i-1))-2*i*x^(2*i)/(1-x^(2*i))) / Product_{j>0} (1-x^j). - Alois P. Heinz, Mar 10 2012

More terms from Alois P. Heinz, Mar 10 2012

cp's OEIS Frontend

A102713 Total sum of odd parts in all compositions of n.

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