A265253 -id:A265253 - cp's OEIS frontend

A265254 Number of partitions of n having no even singletons.

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 19, 25, 34, 43, 54, 70, 89, 111, 140, 174, 216, 268, 328, 402, 495, 601, 727, 883, 1066, 1281, 1540, 1843, 2202, 2627, 3120, 3702, 4392, 5187, 6114, 7206, 8471, 9936, 11644, 13617, 15902, 18548, 21588, 25098, 29156, 33799, 39129

Offset: 0

Emeric Deutsch, Dec 31 2015

nonn

			a(5) = 4 because the partitions [1,1,1,1,1], [1,2,2], [1,1,3], [5] have no even singletons while [1,1,1,2], [2,3], [1,4] do have.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Cristina Ballantine and Amanda Welch, Generalizations of POD and PED partitions, arXiv:2308.06136 [math.CO], 2023. See pp. 15-16.
James A. Sellers, Elementary Proofs of Congruences for POND and PEND Partitions, arXiv:2308.09999 [math.NT], 2023.
James A. Sellers and Nicolas Allen Smoot, Explaining Unforeseen Congruence Relationships Between PEND and POND Partitions via an Atkin--Lehner Involution, arXiv:2503.16019 [math.NT], 2025.

Cf. A265253.

g := mul((1-x^(2*j)+x^(4*j))/(1-x^j), j = 1 .. 80): gser := series(g, x = 0,65): seq(coeff(gser, x, n), n = 0 .. 60);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=1 and i::even, 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) + x^(4*k)) / (1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 01 2016 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(6*k)) / (1 - x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 01 2016 *)

G.f.: g(x) = Product_{j>=1} (1 - x^(2j) + x^(4j))/(1-x^j).
a(n) = A265253(n,0).
G.f.: Product_{k>=1} (1 + x^k) * (1 + x^(6*k)) / (1 - x^(4*k)). - Vaclav Kotesovec, Jan 01 2016
a(n) ~ sqrt(5) * exp(sqrt(5*n)*Pi/3) / (3*2^(5/2)*n). - Vaclav Kotesovec, Jan 01 2016

A265255 Triangle read by rows: T(n,k) is the number of partitions of n having k odd singletons (n, k >=0).

Original entry on oeis.org

1, 0, 1, 2, 1, 2, 4, 0, 1, 2, 5, 8, 1, 2, 4, 11, 14, 3, 5, 9, 20, 0, 1, 24, 8, 10, 16, 37, 1, 2, 41, 15, 21, 28, 65, 3, 5, 66, 30, 39, 49, 108, 9, 10, 104, 57, 69, 0, 1, 80, 178, 19, 20, 163, 99, 120, 1, 2, 128, 286, 39, 37, 248, 170, 201, 3, 5, 203, 448, 73, 68, 372, 284, 327

Offset: 0

Emeric Deutsch, Jan 01 2016

Sum of entries in row n is A000041(n).
T(n,0) = A265256(n).
Sum_{k>=0} k*T(n,k) = A265257(n).

			T(6,2) = 2 because each of the partitions [1,2,3], [1,5] of n = 6 has 2 odd singleton, while the other partitions, namely [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2], [2,2,2], [1,1,1,3], [3,3], [1,1,4], [2,4], [6], have 0, 0, 0, 0, 1, 0, 0, 0, 0  odd singletons.
Triangle starts:
1;
0, 1;
2;
1, 2;
4, 0, 1;
2, 5;
8, 1, 2.

Alois P. Heinz, Rows n = 0..1000, flattened

Cf. A000041, A265253, A265256, A265257.

g := mul(((1-x^(2*j-1))*(1+t*x^(2*j-1))+x^(4*j-2))/(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::odd, 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 && OddQ[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, Dec 10 2016, after Alois P. Heinz *)

G.f.: G(t,x) = Product_{j>=1} ((1 -x^(2j-1))(1+tx^{2j-1}) + x^(4j-2))/ (1-x^j).

A265245 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the squares of the parts is k (n>=0, k>=0).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Emeric Deutsch, Dec 06 2015

Number of entries in row n = 1 + n^2.
Sum of entries in row n = A000041(n).
Sum(k*T(n,k), k>=0) = A066183(n).

			Row 3 is 0,0,0,1,0,1,0,0,0,1 because in the partitions of 3, namely [1,1,1], [2,1], [3], the sums of the squares of the parts are 3, 5, and 9, respectively.
Triangle starts:
1;
0,1;
0,0,1,0,1;
0,0,0,1,0,1,0,0,0,1;
0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,0,1.

Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Cf. A000041, A066183, A229325 - A229332, A264402, A265247 - A265253.

g := 1/(product(1-t^(k^2)*x^k, k = 1 .. 100)): gser := simplify(series(g, x = 0, 15)): for n from 0 to 8 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 8 do seq(coeff(P[n], t, j), j = 0 .. n^2) end do; # yields sequence in triangular form

m = 8; CoefficientList[#, t]& /@ CoefficientList[1/Product[(1 - t^(k^2)* x^k), {k, 1, m}] + O[x]^m, x] // Flatten (* Jean-François Alcover, Feb 19 2019 *)

G.f.: G(t,x) = 1/Product_{k>=1} (1 - t^{k^2}*x^k).

cp's OEIS Frontend

A265254 Number of partitions of n having no even singletons.

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A265255 Triangle read by rows: T(n,k) is the number of partitions of n having k odd singletons (n, k >=0).

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Author

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Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

Formula

A265245 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the squares of the parts is k (n>=0, k>=0).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula