A035457 -id:A035457 - cp's OEIS frontend

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.

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

Emeric Deutsch, Sep 14 2016

T(n,0) = A265254(n).
T(n,n) = A035457(n).
Sum_{k>=0} k*T(n,k) = A276425(n).
Sum_{k>=0} T(n,k) = A000041(n).

			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.

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000041, A035457, A265254, A276422, A276423, A276425.

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 *)

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

A301504 Expansion of Product_{k>=1} (1 + x^(4k))(1 + x^(4*k-3)).

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 1, 0, 1, 3, 2, 0, 2, 5, 4, 1, 2, 7, 7, 2, 3, 10, 11, 4, 4, 14, 17, 8, 6, 19, 25, 13, 8, 25, 36, 21, 12, 33, 50, 33, 18, 43, 69, 49, 26, 56, 93, 71, 38, 72, 124, 102, 55, 92, 163, 142, 79, 118, 212, 195, 112, 151, 273, 265, 157, 193, 350, 354, 217, 246, 444

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 0 or 1 mod 4.

			a(9) = 3 because we have [9], [8, 1] and [5, 4].

Index entries for sequences related to partitions

Cf. A035362, A035451, A035457, A042948, A108932, A132463, A261734, A301505, A301507, A301508.

nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k)) (1 + x^(4 k - 3)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[x^3 QPochhammer[-1, x^4] QPochhammer[-x^(-3), x^4]/(2 (1 + x) (1 - x + x^2)), {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 1}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A042948(k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018

A301505 Expansion of Product_{k>=1} (1 + x^(4k))(1 + x^(4*k-1)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 3, 2, 0, 2, 5, 2, 0, 4, 7, 3, 1, 7, 10, 4, 2, 11, 14, 5, 4, 17, 19, 6, 8, 25, 25, 9, 13, 36, 33, 12, 21, 50, 43, 16, 33, 69, 55, 23, 49, 93, 70, 32, 71, 124, 89, 45, 102, 163, 112, 64, 142, 212, 141, 89, 195, 273, 177, 123, 265, 349

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 0 or 3 mod 4.

			a(11) = 3 because we have [11], [8, 3] and [7, 4].

Index entries for sequences related to partitions

Cf. A014601, A035364, A035457, A131795, A147599, A301504, A301507, A301508.

nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k)) (1 + x^(4 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[x QPochhammer[-1, x^4] QPochhammer[-x^(-1), x^4]/(2 (1 + x)), {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 3}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A014601(k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (2^(5/2)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018

A351593 Number of odd-length integer partitions of n into parts that are alternately equal and strictly decreasing.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 3, 5, 4, 6, 4, 8, 6, 9, 6, 12, 7, 14, 10, 16, 11, 20, 13, 24, 16, 28, 18, 34, 21, 40, 26, 46, 30, 56, 34, 64, 41, 75, 48, 88, 54, 102, 64, 118, 73, 138, 84, 159, 98, 182, 112, 210, 128, 242, 148, 276, 168, 318

Offset: 0

Gus Wiseman, Feb 23 2022

nonn

Also odd-length partitions whose run-lengths are all 2's, except for the last, which is 1.

			The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3  4  5    6  7    8    9    A    B      C    D      E    F
              221     331  332  441  442  443    552  553    554  663
                                          551         661    662  771
                                          33221       44221       44331
                                                                  55221

The even-length ordered version is A003242, ranked by A351010.
The opposite version is A053251, even-length A351007, any length A351006.
This is the odd-length case of A351005, even-length A035457.
With only equalities we get:
  - opposite any length: A351003
  - opposite odd-length: A000009  (except at 0)
  - opposite even-length: A351012
- any length: A351004
- odd-length: A351594
- even-length: A035363
Without equalities we get:
  - opposite any length: A122129 (apparently)
  - opposite odd-length: A122130 (apparently)
  - opposite even-length: A351008
- any length: A122135 (apparently)
- odd-length: A351595
- even-length: A122134 (apparently)
Cf. A000070, A000984, A027383, A053738, A236559, A236914, A350842, A350844.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[If[EvenQ[i],#[[i]]!=#[[i+1]],#[[i]]==#[[i+1]]],{i,Length[#]-1}]&]],{n,0,30}]

A352143 Numbers whose prime indices and conjugate prime indices are all odd.

Original entry on oeis.org

1, 2, 5, 8, 11, 17, 20, 23, 31, 32, 41, 44, 47, 59, 67, 68, 73, 80, 83, 92, 97, 103, 109, 124, 125, 127, 128, 137, 149, 157, 164, 167, 176, 179, 188, 191, 197, 211, 227, 233, 236, 241, 257, 268, 269, 272, 275, 277, 283, 292, 307, 313, 320, 331, 332, 347, 353

Offset: 1

Gus Wiseman, Mar 18 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   5: {3}
   8: {1,1,1}
  11: {5}
  17: {7}
  20: {1,1,3}
  23: {9}
  31: {11}
  32: {1,1,1,1,1}
  41: {13}
  44: {1,1,5}
  47: {15}
  59: {17}
  67: {19}
  68: {1,1,7}
  73: {21}
  80: {1,1,1,1,3}

The restriction to primes is A031368.
These partitions appear to be counted by A053253.
The even version is A066207^2.
For even instead of odd conjugate parts we get A066208^2.
The first condition alone (all odd indices) is A066208, counted by A000009.
The second condition alone is A346635, counted by A000009.
A055922 counts partitions with odd multiplicities, ranked by A268335.
A066207 = indices all even, counted by A035363 (complement A086543).
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162642 counts odd prime exponents, even A162641.
A238745 gives the Heinz number of the conjugate prime signature.
A257991 counts odd indices, even A257992.
A258116 ranks strict partitions with all odd parts, even A258117.
A351979 = odd indices and even multiplicities, counted by A035457.
A352140 = even indices and odd multiplicities, counted by A055922 aerated.
A352141 = even indices and even multiplicities, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000290, A000701, A000720, A028260, A045931, A046682, A055396, A061395, A195017, A241638, A325698, A325700.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]

Intersection of A066208 and A346635.

A116675 Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k odd parts (n>=0, k>=0).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 3, 0, 2, 0, 2, 0, 5, 0, 2, 0, 4, 0, 7, 0, 1, 3, 0, 7, 0, 0, 10, 0, 2, 4, 0, 11, 0, 0, 14, 0, 4, 5, 0, 17, 0, 0, 19, 0, 8, 6, 0, 25, 0, 1, 0, 25, 0, 13, 0, 8, 0, 36, 0, 2, 0, 33, 0, 21, 0, 10, 0, 50, 0, 4, 0, 43, 0, 33, 0, 12, 0, 69, 0, 8, 0, 55, 0, 49, 0, 15, 0, 93, 0, 14

Offset: 0

Emeric Deutsch, Feb 22 2006

Row n contains 1+floor(sqrt(n)) terms (at the end of certain rows there is an extra 0). Row sums yield A000009. T(n,0) = A035457(n) (n>=1); T(2n,0) = A000009(n), T(2n-1,0)=0. T(2n,1)=0, T(2n-1,1) = A036469(n). Sum(k*T(n,k), k>=0) = A116676(n).

			T(8,2) = 4 because we have [7,1], [5,3], [5,2,1] and [4,3,1] ([8] and [6,2] do not qualify).
Triangle starts:
1;
0, 1;
1, 0;
0, 2;
1, 0, 1;
0, 3, 0;

Alois P. Heinz, Rows n = 0..750, flattened

Cf. A000009, A035457, A036469, A116676.

g:=product((1+t*x^(2*j-1))*(1+x^(2*j)),j=1..25): gser:=simplify(series(g,x=0,38)): P[0]:=1: for n from 1 to 26 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 26 do seq(coeff(P[n],t,j),j=0..floor(sqrt(n))) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)->
      x+y, b(n, i-1), `if`(i>n, [], [`if`(irem(i,2)=0, [][], 0),
      b(n-i, i-1)[]]), 0)))
    end:
T:= proc(n) local l; l:= b(n, n); l[], 0$(1+floor(sqrt(n))-nops(l)) end:
seq (T(n), n=0..30);  # Alois P. Heinz, Nov 21 2012

rows = 25; coes = CoefficientList[Product[(1+t*x^(2j-1))(1+x^(2j)), {j, 1, rows}], {x, t}][[1 ;; rows]]; MapIndexed[Take[#1, Floor[Sqrt[#2[[1]]-1]]+1]&, coes] // Flatten (* Jean-François Alcover, May 13 2015 *)

G.f.: product((1+tx^(2j-1))(1+x^(2j)), j=1..infinity).

A351594 Number of odd-length integer partitions y of n that are alternately constant, meaning y_i = y_{i+1} for all odd i.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 4, 2, 7, 3, 9, 4, 13, 6, 19, 6, 26, 10, 35, 12, 49, 16, 64, 20, 87, 27, 115, 32, 151, 44, 195, 53, 256, 69, 328, 84, 421, 108, 537, 130, 682, 167, 859, 202, 1085, 252, 1354, 305, 1694, 380, 2104, 456, 2609, 564, 3218, 676, 3968, 826, 4863

Offset: 0

Gus Wiseman, Feb 24 2022

nonn

These are partitions with all even run-lengths except for the last, which is odd.

			The a(1) = 1 through a(9) = 7 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)    (7)        (8)    (9)
            (111)       (221)    (222)  (331)      (332)  (333)
                        (11111)         (22111)           (441)
                                        (1111111)         (22221)
                                                          (33111)
                                                          (2211111)
                                                          (111111111)

The ordered version (compositions) is A016116 shifted right once.
All odd-length partitions are counted by A027193.
The opposite version is A117409, even-length A351012, any length A351003.
Replacing equal with unequal relations appears to give:
- any length: A122129
- odd length: A122130
- even length: A351008
- opposite any length: A122135
- opposite odd length: A351595
- opposite even length: A122134
This is the odd-length case of A351004, even-length A035363.
The case that is also strict at even indices is:
- any length: A351005
- odd length: A351593
- even length: A035457
- opposite any length: A351006
- opposite odd length: A053251
- opposite even length: A351007
A reverse version is A096441; see also A349060.
Cf. A000009, A000041, A000070, A000984, A003242, A027383, A053738, A236559, A236914, A350842.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

A364891 Triangle read by rows: T(n,k) = (-1)^(k-1)Sum_{j=0..k-1} (-1)^j(p(n - j(2j + 1)) - p(n - (j + 1)(2j + 1))), where p(n) = A000041(n) is the number of partitions of n, and 1 <= k <= n.

Original entry on oeis.org

0, 1, -1, 1, 0, 0, 2, -1, 1, -1, 2, 0, 0, 0, 0, 4, -2, 2, -2, 2, -2, 4, 0, 0, 0, 0, 0, 0, 7, -2, 2, -2, 2, -2, 2, -2, 8, 0, 0, 0, 0, 0, 0, 0, 0, 12, -2, 3, -3, 3, -3, 3, -3, 3, -3, 14, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, -2, 4, -4, 4, -4, 4, -4, 4, -4, 4, -4

Offset: 1

Stefano Spezia, Aug 12 2023

			The triangle begins:
   0;
   1, -1;
   1,  0, 0;
   2, -1, 1, -1;
   2,  0, 0,  0, 0;
   4, -2, 2, -2, 2, -2;
   4,  0, 0,  0, 0,  0, 0;
   7, -2, 2, -2, 2, -2, 2, -2;
   8,  0, 0,  0, 0,  0, 0,  0, 0;
  12, -2, 3, -3, 3, -3, 3, -3, 3, -3;
  14,  1, 0,  0, 0,  0, 0,  0, 0,  0, 0;
  ...

K. Banerjee and M. G. Dastidar, Inequalities for the partition function arising from truncated theta series, RISC Report Series No. 22-20, 2023. See Corollary 1.4 at p. 2.

Cf. A000041, A002865, A035457, A325433, A364892 (row sums), A364893.

T[n_, k_]:=(-1)^(k-1)*Sum[(-1)^j*(PartitionsP[n-j(2j+1)]-PartitionsP[n-(j+1)(2j+1)]), {j, 0, k-1}]; Flatten[Table[T[n, k], {n, 1, 12}, {k, 1, n}]]

1st column: T(n,1) = A002865(n) for n > 0.

abs(T(n,n)) = A035457(n).

A116681 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, in which the sum of the odd parts is k (n>=0, 0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 3, 0, 0, 0, 2, 0, 1, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 1, 0, 2, 0, 2, 4, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 4, 0, 3, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3

Offset: 0

Emeric Deutsch, Feb 22 2006

Row sums yield A000009. T(2n,0)=A000009(n), T(2n-1,0)=0. T(2n,1)=0, T(2n+1,1)=A000009(n), T(n,2)=0. T(n,n)=A000700(n). Sum(k*T(n,k), k=0..n)=A116682(n).

			T(10,4)=2 because we have [6,3,1] and [4,3,2,1].
Triangle starts:
1;
0,1;
1,0,0;
0,1,0,1;
1,0,0,0,1;
0,1,0,1,0,1;

Cf. A000009, A035457, A036469, A116676.

g:=product((1+(t*x)^(2*j-1))*(1+x^(2*j)),j=1..30): gser:=simplify(series(g,x=0,20)): P[0]:=1: for n from 1 to 15 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 15 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

G.f.=product((1+tx^(2j-1))(1+x^(2j)), j=1..infinity).

A307979 Expansion of e.g.f. exp((cosh(x) - cos(x))/2) (even powers only).

Original entry on oeis.org

1, 1, 3, 16, 133, 1576, 24783, 495496, 12245353, 364768576, 12838252443, 526095538816, 24781014246253, 1326767681420416, 80013978835916583, 5392682199766283776, 403287063337529642833, 33261775377836063850496, 3009257393136250807614003, 297176659119237977183973376

Offset: 0

Ilya Gutkovskiy, May 08 2019

nonn

Number of partitions of a 2n-set into blocks congruent to 2 mod 4.

Cf. A003724, A005046, A013469, A016825, A035457, A291975, A306347, A307978.

nmax = 19; Table[(CoefficientList[Series[Exp[(Cosh[x] - Cos[x])/2], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
a[n_] := a[n] = Sum[Boole[MemberQ[{2}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[2 n], {n, 0, 19}]

a(n) = (2*n)! * [x^(2*n)] exp((cosh(x) - cos(x))/2).

cp's OEIS Frontend

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.

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A301504 Expansion of Product_{k>=1} (1 + x^(4*k))*(1 + x^(4*k-3)).

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A301505 Expansion of Product_{k>=1} (1 + x^(4*k))*(1 + x^(4*k-1)).

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A351593 Number of odd-length integer partitions of n into parts that are alternately equal and strictly decreasing.

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A352143 Numbers whose prime indices and conjugate prime indices are all odd.

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A116675 Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k odd parts (n>=0, k>=0).

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A351594 Number of odd-length integer partitions y of n that are alternately constant, meaning y_i = y_{i+1} for all odd i.

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Mathematica

A364891 Triangle read by rows: T(n,k) = (-1)^(k-1)*Sum_{j=0..k-1} (-1)^j*(p(n - j*(2*j + 1)) - p(n - (j + 1)*(2*j + 1))), where p(n) = A000041(n) is the number of partitions of n, and 1 <= k <= n.

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A301504 Expansion of Product_{k>=1} (1 + x^(4k))(1 + x^(4*k-3)).

A301505 Expansion of Product_{k>=1} (1 + x^(4k))(1 + x^(4*k-1)).

A364891 Triangle read by rows: T(n,k) = (-1)^(k-1)Sum_{j=0..k-1} (-1)^j(p(n - j(2j + 1)) - p(n - (j + 1)(2j + 1))), where p(n) = A000041(n) is the number of partitions of n, and 1 <= k <= n.