A276423
Sum of the odd singletons in all partitions of n (n>=0). A singleton in a partition is a part that occurs exactly once.
Original entry on oeis.org
0, 1, 0, 4, 4, 13, 13, 33, 41, 79, 98, 171, 223, 354, 458, 692, 905, 1306, 1694, 2375, 3077, 4202, 5401, 7238, 9260, 12200, 15495, 20145, 25446, 32686, 41020, 52170, 65117, 82071, 101852, 127374, 157277, 195289, 239915, 296023, 362000, 444063, 540595, 659662
Offset: 0
a(4) = 4 because in the partitions [1,1,1,1], [1,1,2], [2,2], [1,3], [4] the sums of the odd singletons are 0,0,0,4,0, respectively; their sum is 4.
a(5) = 13 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5] the sums of the odd singletons are 0,0,1,3,3,1,5, respectively; their sum is 13.
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g := x*(1-x+3*x^2+3*x^4-x^5+x^6)/((1-x^4)^2*(product(1-x^i, i = 1..120))): gser := series(g, x = 0, 60); seq(coeff(gser, x, n), n = 0..50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<1, 0, add((p-> p+`if`(i::odd and j=1,
[0, i*p[1]], 0))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50); # Alois P. Heinz, Sep 14 2016
-
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, 0, Sum[Function[p, p + If[OddQ[i] && j == 1, {0, If[p === 0, 0, i*p[[1]]]}, 0]][b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 04 2016 after Alois P. Heinz *)
Table[Total[Select[Flatten[Tally/@IntegerPartitions[n],1],#[[2]]==1 && OddQ[ #[[1]]]&][[All,1]]],{n,0,50}] (* Harvey P. Dale, May 25 2018 *)
A276424
Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its even singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once.
Original entry on oeis.org
1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 0, 1, 4, 0, 2, 0, 1, 0, 6, 0, 2, 0, 1, 0, 2, 8, 0, 3, 0, 2, 0, 2, 0, 11, 0, 4, 0, 3, 0, 2, 0, 2, 15, 0, 5, 0, 4, 0, 4, 0, 2, 0, 19, 0, 7, 0, 6, 0, 5, 0, 2, 0, 3, 25, 0, 9, 0, 8, 0, 7, 0, 4, 0, 3, 0, 34, 0, 11, 0, 10, 0, 10, 0, 5, 0, 3, 0, 4
Offset: 0
Row 4 is 3, 0, 1, 0, 1 because in the partitions [1,1,1,1], [1,1,2], [2,2], [1,3], [4] the sums of the even singletons are 0, 2, 0, 0, 4, respectively.
Row 5 is 4, 0, 2, 0, 1, 0 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5] the sums of the even singletons are 0, 2, 0, 0, 2, 4, 0, respectively.
Triangle starts:
1;
1,0;
1,0,1;
2,0,1,0;
3,0,1,0,1;
4,0,2,0,1,0;
6,0,2,0,1,0,2.
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g := Product(((1-x^(2*j))*(1+t^(2*j)*x^(2*j))+x^(4*j))/(1-x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 23)): for n from 0 to 20 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 20 do seq(coeff(P[n], t, i), i = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
`if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
`if`(j=1 and i::even, x^i, 1), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..14); # Alois P. Heinz, Sep 14 2016
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b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*If[j == 1 && EvenQ[i], x^i, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 11 2016, after Alois P. Heinz *)
A276425
Sum of the even singletons in all partitions of n (n>=0). A singleton in a partition is a part that occurs exactly once.
Original entry on oeis.org
0, 0, 2, 2, 6, 8, 20, 26, 48, 66, 114, 154, 240, 326, 490, 656, 940, 1252, 1752, 2306, 3142, 4104, 5500, 7114, 9372, 12030, 15656, 19932, 25628, 32402, 41270, 51816, 65400, 81608, 102226, 126800, 157698, 194550, 240454, 295110, 362600, 442902, 541342, 658230
Offset: 0
a(4) = 6 because in the partitions [1,1,1,1], [1,1,2], [2,2], [1,3], [4] the sums of the even singletons are 0, 2, 0, 0, 4, respectively; their sum is 6.
a(5) = 8 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5] the sums of the even singletons are 0, 2, 0, 0, 2, 4, 0 respectively; their sum is 8.
-
g := 2*x^2*(1+x^2+x^4)/((1-x^4)^2*(product(1-x^i, i = 1 .. 120))): gser := series(g, x = 0, 60): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<1, 0, add((p-> p+`if`(i::even and j=1,
[0, i*p[1]], 0))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50); # Alois P. Heinz, Sep 14 2016
-
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, 0, Sum[Function[p, p + If[EvenQ[i] && j == 1, {0, i*p[[1]]}, 0]][b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 24 2016, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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