A055922 -id:A055922 - cp's OEIS frontend

A264399 Triangle read by rows: T(n,k) is the number of partitions of n having k parts with even multiplicities.

1, 1, 1, 1, 3, 2, 3, 5, 2, 6, 4, 1, 9, 6, 9, 11, 2, 16, 13, 1, 20, 15, 7, 25, 28, 3, 32, 33, 11, 1, 40, 52, 9, 54, 55, 24, 2, 69, 82, 25, 84, 101, 40, 6, 101, 148, 46, 2, 136, 163, 73, 13, 156, 239, 89, 6, 202, 274, 127, 23, 1, 244, 364, 170, 14, 306, 437, 211, 46, 2

Offset: 0

Emeric Deutsch, Nov 21 2015

T(n,0) = A055922(n).

Row sums give A000041. - Omar E. Pol, Nov 21 2015

			T(6,1) = 4 because we have [4,1*,1], [3*,3], [2,1*,1,1,1], and [1*,1,1,1,1,1] (parts with even multiplicities are marked).
Triangle starts:
  1;
  1;
  1, 1;
  3;
  2, 3;
  5, 2;
  6, 4, 1;
  ...

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

Cf. A000041, A055922, A264398, A264400.

g := product(1+x^j/(1-x^(2*j))+t*x^(2*j)/(1-x^(2*j)), j = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n from 0 to 28 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 28 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      expand(`if`(j>0 and j::even, x, 1)*b(n-i*j, i-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, Nov 25 2015

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Expand[If[j>0 && EvenQ[ j], x, 1]*b[n-i*j, i-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 25 2015, after Alois P. Heinz *)

T(n) = { Vec(prod(k=1, n, (1+x^k-x^(2*k)+y*x^(2*k))/(1-x^(2*k)) + O(x*x^n))) }
{ my(t=T(10)); for(n=1, #t, print(Vecrev(t[n]))); } \\ Andrew Howroyd, Dec 22 2017

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

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.

A100847 Number of partitions of 2n in which each odd part has even multiplicity and each even part has odd multiplicity.

Original entry on oeis.org

1, 2, 3, 7, 10, 17, 28, 42, 62, 93, 137, 193, 276, 383, 532, 734, 997, 1342, 1807, 2400, 3177, 4190, 5478, 7130, 9245, 11923, 15305, 19591, 24957, 31673, 40075, 50518, 63460, 79523, 99296, 123664, 153616, 190271, 235072, 289776, 356302, 437107, 535112, 653626

Offset: 0

Vladeta Jovovic, Aug 16 2007

			a(3) = 7 because we have 6, 42, 411, 33, 222, 21111 and 111111.

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

Cf. A055922, A117958, A130126, A131942, A102247, A263401.

g:=product((1+x^i-x^(2*i))/(1-x^i),i=1..50): gser:=series(g,x=0,40): seq(coeff(gser,x,n),n=0..35); # Emeric Deutsch, Aug 25 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i+j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)
       +b(n, i-1)))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014

nmax = 50; CoefficientList[Series[Product[(1+x^k-x^(2*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

G.f.: Product_{i>0} (1+x^i-x^(2*i))/(1-x^i).

a(n) ~ sqrt(Pi^2/3 + 4*log(phi)^2) * exp(sqrt((2*Pi^2/3 + 8*log(phi)^2)*n)) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

More terms from Emeric Deutsch, Aug 25 2007

A366853 Number of integer partitions of n into odd, pairwise coprime parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 17, 18, 20, 22, 25, 29, 33, 36, 39, 43, 49, 55, 61, 66, 69, 75, 85, 94, 104, 113, 120, 129, 143, 159, 172, 183, 193, 207, 226, 251, 272, 288, 304, 325, 350, 383, 414, 437, 460, 494, 532, 577, 622, 655, 684

Offset: 0

Gus Wiseman, Nov 01 2023

nonn

			The a(1) = 1 through a(10) = 7 partitions:
1  11  3    31    5      51      7        53        9          73
       111  1111  311    3111    511      71        531        91
                  11111  111111  31111    5111      711        5311
                                 1111111  311111    51111      7111
                                          11111111  3111111    511111
                                                    111111111  31111111
                                                               1111111111

Partitions into odd parts are counted by A000009, ranks A066208.
Allowing even parts gives A051424.
For relatively prime (not pairwise coprime): A366843, with evens A000837.
A000041 counts integer partitions, strict A000009 (also into odds).
A101268 counts pairwise coprime compositions.
A168532 counts partitions by gcd.
Cf. A007359, A018783, A055922, A078374, A337485, A366844, A366852.

pwcop[y_]:=And@@(GCD@@#==1&)/@Subsets[y,{2}]
Table[Length[Select[IntegerPartitions[n],And@@OddQ/@#&&pwcop[#]&]],{n,0,30}]

A102247 Number of partitions of n in which each odd part has odd multiplicity and each even part has even multiplicity.

Original entry on oeis.org

1, 1, 0, 2, 2, 3, 2, 4, 7, 8, 8, 10, 17, 17, 20, 26, 39, 39, 46, 56, 77, 85, 96, 116, 154, 172, 190, 234, 289, 328, 364, 440, 532, 610, 670, 808, 957, 1091, 1204, 1432, 1675, 1905, 2110, 2476, 2867, 3255, 3608, 4184, 4837, 5451, 6050, 6960, 7980, 8961, 9972, 11370

Offset: 0

Vladeta Jovovic, Aug 16 2007

			a(7) = 4 because we have 7, 322, 22111 and 1111111.

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

Cf. A055922, A117958, A130126, A131942, A100847.

g:=product((1+x^(2*i-1)-x^(4*i-2))/(1-x^(2*i)),i=1..40): gser:=series(g,x=0, 60): seq(coeff(gser,x,n),n=0..55); # Emeric Deutsch, Aug 23 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i+j, 2)=0, b(n-i*j, i-1), 0), j=1..n/i)
       +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k-1) - x^(4*k-2))/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

G.f.: Product_{i>0} (1+x^(2*i-1)-x^(4*i-2))/(1-x^(2*i)).

a(n) ~ sqrt(Pi^2/3 + 4*log(phi)^2) * exp(sqrt((Pi^2/3 + 4*log(phi)^2)*n)) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

More terms from Emeric Deutsch, Aug 23 2007

A249389 Decimal expansion of the constant 'B' appearing in the asymptotic expression of the number of partitions of n as (B/(2Pin))exp(2B*sqrt(n)), in case of partitions into integers, each of which occurring only an odd number of times.

Original entry on oeis.org

1, 1, 3, 3, 8, 4, 1, 5, 5, 6, 2, 0, 5, 4, 9, 6, 4, 6, 6, 7, 3, 3, 7, 6, 8, 6, 3, 2, 4, 6, 0, 5, 0, 1, 9, 3, 1, 2, 0, 6, 0, 2, 9, 6, 2, 8, 8, 0, 8, 6, 5, 4, 0, 1, 0, 4, 1, 7, 3, 8, 0, 6, 7, 2, 7, 8, 0, 8, 4, 7, 5, 5, 1, 2, 5, 9, 1, 7, 9, 4, 5, 8, 5, 8, 3, 6, 2, 1, 1, 9, 0, 6, 3, 3, 9, 5, 9, 6, 2

Offset: 1

Jean-François Alcover, Oct 27 2014

			1.133841556205496466733768632460501931206029628808654...

G. C. Greubel, Table of n, a(n) for n = 1..10000
F. C. Auluck, K. S. Singwi and B. K. Agarwala, On a new type of partition, Proc. Nat. Inst. Sci. India 16 (1950) 147-156.
Steven Finch, Integer Partitions, September 22, 2004. [Cached copy, with permission of the author]

Cf. A000041, A002390, A055922.

B = Sqrt[Pi^2/12 + 2*Log[GoldenRatio]^2]; RealDigits[B, 10, 99] // First

sqrt(Pi^2/12 + 2*(log((1+sqrt(5))/2))^2) \\ G. C. Greubel, Apr 06 2017

B = sqrt(Pi^2/12 + 2*log(phi)^2), where phi is the golden ratio.

A366750 Number of strict integer partitions of n into odd parts with a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 3, 1, 0, 2, 0, 1, 3, 1, 0, 3, 2, 1, 4, 1, 1, 5, 0, 1, 5, 1, 2, 5, 1, 1, 5, 2, 2, 6, 0, 1, 9, 1, 0, 9, 0, 3, 9, 1, 1, 9, 5, 1, 11, 1, 0, 15, 1, 2, 13, 1, 5, 14, 0, 1, 18

Offset: 0

Gus Wiseman, Nov 01 2023

nonn

			The a(n) partitions for n = 3, 24, 30, 42, 45, 57, 60:
  (3)  (15,9)  (21,9)  (33,9)   (45)       (57)       (51,9)
       (21,3)  (25,5)  (35,7)   (33,9,3)   (45,9,3)   (55,5)
               (27,3)  (39,3)   (21,15,9)  (27,21,9)  (57,3)
                       (27,15)  (25,15,5)  (33,15,9)  (33,27)
                                (27,15,3)  (33,21,3)  (35,25)
                                           (39,15,3)  (39,21)
                                                      (45,15)
                                                      (27,21,9,3)
                                                      (33,15,9,3)

This is the case of A000700 with a common divisor.
Including evens gives A303280.
The complement is counted by A366844, non-strict version A366843.
The non-strict version is A366852, with evens A018783.
A000041 counts integer partitions, strict A000009 (also into odds).
A051424 counts pairwise coprime partitions, for odd parts A366853.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
Cf. A000837, A007360, A055922, A066208, A078374, A302697, A337485, A366842, A366845, A366848.

Table[Length[Select[IntegerPartitions[n], And@@OddQ/@#&&UnsameQ@@#&&GCD@@#>1&]], {n,0,30}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366750(n): return sum(1 for p in partitions(n) if all(d==1 for d in p.values()) and all(d&1 for d in p) and gcd(*p)>1) # Chai Wah Wu, Nov 02 2023

More terms from Chai Wah Wu, Nov 02 2023

A304393 Expansion of Product_{k>0} (1 + Sum_{m>=0} x^(k*2^m)).

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 8, 10, 17, 19, 27, 33, 48, 56, 76, 92, 126, 146, 192, 228, 298, 352, 444, 528, 667, 783, 969, 1145, 1414, 1658, 2017, 2365, 2878, 3352, 4027, 4703, 5634, 6548, 7773, 9033, 10705, 12381, 14573, 16857, 19790, 22800, 26631, 30655, 35723, 41005

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Also the number of partitions of n in which each part occurs a power of 2 (cf. A000079) of times.

			n | Partitions of n in which each part occurs a power of 2 (cf. A000079) of times
--+------------------------------------------------------------------------------
1 | 1;
2 | 2 = 1+1;
3 | 3 = 2+1;
4 | 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1;
5 | 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1;
6 | 6 = 5+1 = 4+2 = 4+1+1 = 3+2+1 = 3+3 = 2+2+1+1 = 2+1+1+1+1;
7 | 7 = 6+1 = 5+2 = 5+1+1 = 4+3 = 4+2+1 = 3+3+1 = 3+2+2 = 3+2+1+1 = 3+1+1+1+1;

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

Cf. A000079, A055922, A300446, A304332.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*2^j, i-1), j=0..ilog2(n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 13 2018

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
   b[n, i-1] + Sum[b[n-i*2^j, i-1], {j, 0, Floor@Log2[n/i]}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 14 2023, after Alois P. Heinz *)

A309733 Expansion of Product_{k>=1} 1/(1 - x^k/(1 - x^(2*k))).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 36, 62, 104, 174, 286, 478, 780, 1284, 2100, 3433, 5586, 9114, 14798, 24064, 39050, 63376, 102726, 166584, 269835, 437190, 707964, 1146480, 1855966, 3004748, 4863306, 7871798, 12739576, 20617652, 33364524, 53992834, 87369548, 141379728, 228769842

Offset: 0

Seiichi Manyama, Sep 22 2019

nonn

Cf. A006951, A055922, A162891.

nmax = 40; CoefficientList[Series[Product[1/(1 - x^k/(1 - x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 22 2019 *)

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, 1-x^k/(1-x^(2*k))))

a(n) ~ phi^(n+1), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 22 2019

cp's OEIS Frontend

A264399 Triangle read by rows: T(n,k) is the number of partitions of n having k parts with even multiplicities.

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Mathematica

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

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A100847 Number of partitions of 2n in which each odd part has even multiplicity and each even part has odd multiplicity.

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Maple

Mathematica

Formula

Extensions

A366853 Number of integer partitions of n into odd, pairwise coprime parts.

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Examples

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Programs

Mathematica

A102247 Number of partitions of n in which each odd part has odd multiplicity and each even part has even multiplicity.

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Maple

Mathematica

Formula

Extensions

A249389 Decimal expansion of the constant 'B' appearing in the asymptotic expression of the number of partitions of n as (B/(2*Pi*n))*exp(2*B*sqrt(n)), in case of partitions into integers, each of which occurring only an odd number of times.

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Mathematica

PARI

Formula

A366750 Number of strict integer partitions of n into odd parts with a common divisor > 1.

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Examples

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Mathematica

Python

Extensions

A304393 Expansion of Product_{k>0} (1 + Sum_{m>=0} x^(k*2^m)).

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Author

A249389 Decimal expansion of the constant 'B' appearing in the asymptotic expression of the number of partitions of n as (B/(2Pin))exp(2B*sqrt(n)), in case of partitions into integers, each of which occurring only an odd number of times.