0,3

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

a(n) = A006128(n) - A024786(n+1).

a(n) = Sum_{k=2..n} k*A264405(n,k). - Alois P. Heinz, Dec 07 2015

G.f.: g = Sum_{j>0} (x^{2*j}*(2 - x^j)/(1-x^j))/Product_{k>0}(1 - x^k) (obtained by logarithmic differentiation of the bivariate g.f. given in A264405). - Emeric Deutsch, Feb 02 2016

For n = 6 we have:

--------------------------------------

. Number of

Partitions repeated parts

--------------------------------------

6 .......................... 0

3 + 3 ...................... 2

4 + 2 ...................... 0

2 + 2 + 2 .................. 3

5 + 1 ...................... 0

3 + 2 + 1 .................. 0

4 + 1 + 1 .................. 2

2 + 2 + 1 + 1 .............. 4

3 + 1 + 1 + 1 .............. 3

2 + 1 + 1 + 1 + 1 .......... 4

1 + 1 + 1 + 1 + 1 + 1 ...... 6

------------------------------------

Total ..................... 24

So a(6) = 24.

b:= proc(n, i) option remember; local h, j, t;

if n<0 then [0, 0]

elif n=0 then [1, 0]

elif i<1 then [0, 0]

else h:= [0, 0];

for j from 0 to iquo(n, i) do

t:= b(n-i*j, i-1);

h:= [h[1]+t[1], h[2]+t[2]+`if`(j<2, 0, t[1]*j)]

od; h

fi

end:

a:= n-> b(n, n)[2]:

seq(a(n), n=0..50); # Alois P. Heinz, Nov 20 2011

g := add(x^(2*j)*(2-x^j)/(1-x^j), j = 1 .. 80)/mul(1-x^j, j = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 45); # Emeric Deutsch, Feb 02 2016

myCount[p_List] := Module[{t}, If[p == {}, 0, t = Transpose[Tally[p]][[2]]; Sum[If[t[[i]] == 1, 0, t[[i]]], {i, Length[t]}]]]; Table[Total[Table[myCount[p], {p, IntegerPartitions[i]}]], {i, 0, 20}] (* T. D. Noe, Nov 19 2011 *)

b[n_, i_] := b[n, i] = Module[{h, j, t}, Which[n<0, {0, 0}, n==0, {1, 0}, i < 1, {0, 0}, True, h={0, 0}; For[j=0, j <= Quotient[n, i], j++, t = b[n - i*j, i-1]; h = {h[[1]]+t[[1]], h[[2]]+t[[2]] + If[j<2, 0, t[[1]]*j]}]; h] ]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 25 2015, after Alois P. Heinz *)

Table[Length[Flatten[Select[Flatten[Split[#]&/@IntegerPartitions[n], 1], Length[#]>1&]]], {n, 0, 60}] (* Harvey P. Dale, Jun 12 2024 *)

Cf. A006128, A024786, A047967, A135010, A138121, A194544, A264405.

Sequence in context: A303851 A218542 A379646 * A242516 A282281 A321175

Adjacent sequences: A194449 A194450 A194451 * A194453 A194454 A194455

Omar E. Pol, Nov 19 2011

approved