A265253
Triangle read by rows: T(n,k) is the number of partitions of n having k even singletons (n,k>=0).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 2, 4, 3, 6, 4, 1, 8, 6, 1, 11, 9, 2, 15, 12, 3, 19, 18, 5, 25, 24, 7, 34, 32, 10, 1, 43, 43, 14, 1, 54, 59, 20, 2, 70, 76, 27, 3, 89, 99, 38, 5, 111, 129, 50, 7, 140, 165, 69, 11, 174, 211, 90, 15, 216, 270, 119, 21, 1, 268, 339, 155, 29, 1, 328, 429, 203, 40, 2
Offset: 0
T(6,1) = 4 because each of the partitions [1,1,1,1,2], [1,2,3], [1,1,4], [6] of n = 6 has 1 even singleton, while the other partitions, namely [1,1,1,1,1,1], [1,1,2,2], [2,2,2], [1,1,1,3], [3,3], [2,4], [1,5], have 0, 0, 0 ,0, 0, 2, 0 even singletons.
Triangle starts:
1;
1;
1, 1;
2, 1;
3, 2;
4, 3;
6, 4, 1.
-
g := mul(((1-x^(2*j))*(1+t*x^(2*j))+x^(4*j))/(1-x^j), j = 1 .. 80): gser := simplify(series(g, x = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, q), q = 0 .. degree(P[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, 1), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30); # Alois P. Heinz, Jan 01 2016
-
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, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[ T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
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.
-
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
-
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 *)
A265256
Number of partitions of n having no odd singletons (n>=0).
Original entry on oeis.org
1, 0, 2, 1, 4, 2, 8, 4, 14, 9, 24, 16, 41, 28, 66, 49, 104, 80, 163, 128, 248, 203, 372, 312, 554, 472, 810, 708, 1172, 1042, 1684, 1516, 2390, 2188, 3364, 3118, 4705, 4404, 6522, 6177, 8980, 8584, 12295, 11844, 16718, 16244, 22604, 22120, 30413, 29944, 40692
Offset: 0
a(5) = 2 because among the 7 partitions of 5 only [1,1,1,1,1] and [1,1,1,2] have no odd singletons (the others are: [1,2,2], [1,1,3], [2,3], [1,4], [5]).
-
g := mul((1-x^(2*j-1)+x^(4*j-2))/(1-x^j), j = 1 .. 80): gser := series(g, x = 0, 60): seq(coeff(gser, x, n), n = 0 .. 55);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
`if`(j=1 and i::odd, 0, b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..60); # Alois P. Heinz, Jan 02 2016
-
nmax = 50; CoefficientList[Series[Product[((1 - x^(2*k-1) + x^(4*k-2))) / (1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 01 2016 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(6*k-3)) / ((1 + x^(2*k-1)) * (1-x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 01 2016 *)
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