A272212 Sum of the odd numbers among the larger parts of the partitions of n into two parts.
0, 0, 1, 0, 3, 3, 8, 5, 12, 12, 21, 16, 27, 27, 40, 33, 48, 48, 65, 56, 75, 75, 96, 85, 108, 108, 133, 120, 147, 147, 176, 161, 192, 192, 225, 208, 243, 243, 280, 261, 300, 300, 341, 320, 363, 363, 408, 385, 432, 432, 481, 456, 507, 507, 560, 533, 588, 588
Offset: 0
Examples
a(5) = 3; the partitions of 5 into two parts are (4,1),(3,2) and the sum of the odd numbers among the larger parts is 3. a(6) = 8; the partitions of 6 into two parts are (5,1),(4,2),(3,3) and the sum of the odd numbers among the larger parts is 5+3 = 8.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for sequences related to partitions
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).
Programs
-
Magma
[(6*n^2-6*n+1+(10*n-5)*(-1)^n-(4*n-2-2*(-1)^n)*(-1)^((2*n+1-(-1)^n) div 4))/32: n in [0..100]];
-
Maple
A272212:=n->(6*n^2-6*n+1+(10*n-5)*(-1)^n-(4*n-2-2*(-1)^n)*(-1)^((2*n+1-(-1)^n)/4))/32: seq(A272212(n), n=0..100);
-
Mathematica
Table[(6n^2-6n+1+(10n-5)(-1)^n-(4n-2-2(-1)^n)(-1)^((2n+1-(-1)^n)/4))/32, {n,0,100}] Table[Total@ Flatten[First /@ IntegerPartitions[n, {2}] /. k_ /; EvenQ@ k -> Nothing], {n, 0, 60}] (* Michael De Vlieger, Apr 26 2016, Version 10.2 *) f[n_] := Sum[(n - i) Mod[n - i, 2], {i, Floor[n/2]}]; Array[f, 58, 0] (* Robert G. Wilson v, Dec 11 2017 *) CoefficientList[ Series[x^2 (1 +x +x^2) (1 -2x +4x^2 -2x^3 +x^4)/((1 -x)^3 (1 +x)^2 (1 +x^2)^2), {x, 0, 57}], x] (* Robert G. Wilson v, Dec 13 2017 *) Table[Total[Select[IntegerPartitions[n,{2}][[All,1]],OddQ]],{n,0,60}] (* Harvey P. Dale, Jun 29 2018 *) -
PARI
concat(vector(2), Vec(x^2*(1+x+x^2)*(1-2*x+4*x^2-2*x^3+x^4)/((1-x)^3*(1+x)^2*(1+x^2)^2) + O(x^50))) \\ Colin Barker, Apr 23 2016
-
PARI
a(n)=([0,1,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0; 0,0,0,0,1,0,0,0,0; 0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,1; 1,-1,0,0,-2,2,0,0,1]^n*[0;0;1;0;3;3;8;5;12])[1,1] \\ Charles R Greathouse IV, Apr 29 2016
Formula
a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n > 8.
a(n) = (6*n^2 - 6*n + 1 + (10*n-5)*(-1)^n - (4*n - 2 - 2*(-1)^n)*(-1)^((2*n+1 - (-1)^n)/4))/32.
G.f.: x^2*(1 + x + x^2)*(1 - 2*x + 4*x^2 - 2*x^3 + x^4) / ((1-x)^3*(1+x)^2*(1+x^2)^2). - Colin Barker, Apr 22 2016
E.g.f.: ((-5*(1 + 2*x))*exp(-x) + (1 + 6*x^2)*exp(x) + 4*(1 + x)*cos(x) + 4*x*sin(x))/32. - Ilya Gutkovskiy, Apr 27 2016
a(n) = Sum_{i=1..floor(n/2)} (n-i) * ((n-i) mod 2). - Wesley Ivan Hurt, Dec 06 2017
Comments