A179049 -id:A179049 - cp's OEIS frontend

A376627 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j))^2.

1, 1, 0, 3, 0, 3, 1, 3, 2, 4, 4, 3, 8, 2, 10, 2, 14, 2, 19, 3, 20, 7, 23, 11, 26, 17, 25, 26, 27, 35, 29, 48, 27, 64, 28, 81, 30, 98, 32, 119, 37, 139, 47, 159, 59, 183, 77, 199, 105, 217, 137, 237, 180, 251, 232, 266, 292, 281, 364, 293, 447, 309, 540, 331, 645, 350

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A179049, A376631, A376659.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2)*Product[1+x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k))*(1 + x^(2*k))*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

G.f.: Sum_{k>=0} Product_{j=1..k} (1 + x^(2*j))^2 * x^j.

a(n) ~ c * A376659^sqrt(n) / sqrt(n), where c = sqrt(5/168 + sqrt(11/23) * cosh(arccosh(17*sqrt(23)/(2*11^(3/2)))/3)/21) = 0.2512284115765342169430117...

A097042 G.f. = (1 + 4 * g.f. for A096661)/(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)).

Original entry on oeis.org

1, 2, 0, 4, 2, 4, 4, 8, 8, 10, 12, 16, 20, 24, 28, 36, 42, 48, 60, 72, 84, 100, 116, 136, 160, 186, 216, 252, 292, 336, 388, 448, 512, 588, 672, 768, 878, 1000, 1136, 1292, 1464, 1656, 1876, 2120, 2388, 2696, 3032, 3408, 3832, 4298, 4816, 5396, 6036, 6744, 7532, 8404

Offset: 0

N. J. A. Sloane, Sep 15 2004

nonn

a(0) = 1; for n>0, a(n) = 2*A026832(n) (i.e., essentially Fine's numbers L(n) multiplied by 2).

The number of odd-even overpartitions of n: an odd-even overpartition of n is an overpartition of n with the smallest part odd and such that the difference between successive parts is odd if the smaller part is nonoverlined and even otherwise - see Yang 2017. - Peter Bala, Mar 29 2017

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Min-Joo Jang, Asymptotic behavior of odd-even partitions, arXiv:1703.01837v1 [math.NT], 2017.

Cf. A096661, A026832, A179049.

nmax = 60; Flatten[{1, Rest[CoefficientList[Series[2*Sum[x^(2*k - 1) QPochhammer[-x^(2*k), x], {k, nmax}], {x, 0, nmax}], x]]}] (* Vaclav Kotesovec, Mar 28 2017 *)

a(n) ~ 1/(3^(5/4)*n^(3/4))*exp(Pi*sqrt(n/3)) [Jang 2017]. - Peter Bala, Mar 29 2017

Conjectural g.f.: 1 + 2*Sum_{n >= 1} q^(n*(n+1)/2)/( (1 + q^n) * Product_{k = 1..n} 1 - q^k ). - Peter Bala, Feb 19 2021

Name corrected by Peter Bala, Feb 19 2021

A179080 Number of partitions of n into distinct parts where all differences between consecutive parts are odd.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 4, 2, 6, 4, 7, 5, 9, 8, 12, 10, 14, 15, 17, 19, 22, 26, 26, 32, 32, 42, 40, 52, 48, 66, 59, 79, 73, 98, 89, 118, 108, 143, 133, 170, 160, 204, 194, 241, 236, 286, 283, 336, 339, 396, 407, 464, 483, 544, 575, 634, 681, 740, 803, 862, 944, 1001, 1110, 1162, 1296, 1348, 1512, 1561, 1760, 1805

Offset: 0

Joerg Arndt, Jan 04 2011

nonn

			From _Joerg Arndt_, Oct 27 2012:  (Start)
The a(18) = 15 such partitions of 18 are:
[ 1]  1 2 3 12
[ 2]  1 2 5 10
[ 3]  1 2 7 8
[ 4]  1 2 15
[ 5]  1 4 5 8
[ 6]  1 4 13
[ 7]  1 6 11
[ 8]  1 8 9
[ 9]  2 3 4 9
[10]  2 3 6 7
[11]  3 4 5 6
[12]  3 4 11
[13]  3 6 9
[14]  5 6 7
[15]  18
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Atul Dixit and Gaurav Kumar, The Rogers-Ramanujan dissection of a theta function, arXiv:2411.06412 [math.NT], 2024. See pp. 16, 23.

Cf. A179049 (odd differences and odd minimal part).
Cf. A189357 (even differences, distinct parts), A096441 (even differences).
Cf. A000009 (partitions of 2*n with even differences and even minimal part).

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+2)+b(n-i, i+1)))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)+b(n, 2)):
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 08 2012; revised Feb 24 2020

b[n_, i_, t_] := b[n, i, t] = If[n==0, 1, If[i<1, 0, b[n, i-1, t] + If[i <= n && Mod[i, 2] != t, b[n-i, i-1, Mod[i, 2]], 0]]]; a[n_] := If[n==0, 1, Sum[b[n-i, i-1, Mod[i, 2]], {i, 1, n}]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)
Join[{1},Table[Length[Select[IntegerPartitions[n],Max[Length/@Split[#]]==1 && AllTrue[ Differences[#],OddQ]&]],{n,70}]] (* Harvey P. Dale, Jun 25 2022 *)

N=66; x='x+O('x^N); gf = sum(n=0,N, x^(n*(n+1)/2) / prod(k=1,n+1, 1-x^(2*k) ) ); Vec( gf ) /* Joerg Arndt, Jan 29 2011 */

def A179080(n):
    odd_diffs = lambda x: all(abs(d) % 2 == 1 for d in differences(x))
    satisfies = lambda p: not p or odd_diffs(p)
    def count(pred, iter): return sum(1 for item in iter if pred(item))
    return count(satisfies, Partitions(n, max_slope=-1))
print([A179080(n) for n in range(0, 20)]) # show first terms

# Alternative after Alois P. Heinz:
def A179080(n):
    @cached_function
    def h(n, k):
        if n == 0: return 1
        if k  > n: return 0
        return h(n, k+2) + h(n-k, k+1)
    return h(n, 1) + h(n, 2) if n > 0 else 1
print([A179080(n) for n in range(71)]) # Peter Luschny, Feb 25 2020

G.f.: sum(n>=0, x^(n*(n+1)/2) / prod(k=1..n+1, 1-x^(2*k) ) ). - Joerg Arndt, Jan 29 2011

a(n) = A179049(n) + A218355(n). - Joerg Arndt, Oct 27 2012

A189357 Number of partitions of n into distinct parts where all differences between consecutive parts are even.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 2, 7, 3, 8, 4, 11, 5, 13, 6, 17, 8, 20, 9, 26, 12, 30, 14, 38, 17, 45, 20, 55, 25, 64, 29, 79, 35, 91, 41, 110, 49, 128, 57, 152, 68, 176, 78, 209, 93, 240, 107, 282, 125, 325, 144, 379, 168, 434, 192, 505, 223, 576, 255, 666, 294, 760, 335, 873, 385, 993, 437, 1139

Offset: 0

Joerg Arndt, Apr 20 2011

nonn

Also number of partitions into distinct parts where either all parts are even or all parts are odd.

Also number of symmetric unimodal compositions of n where the maximal part m appears at least m times, see example. [Joerg Arndt, Jun 11 2013]

			a(14)=8 because there are 8 such partitions of 14: 1+13 =2+4+8 =2+12 =3+11 =4+10 =5+9 =6+8 =14
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + x^7 + 4*x^8 + 2*x^9 +...
From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(18)=13 symmetric unimodal compositions of 18 where the maximal part m appears at least m times:
01:  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 ]
03:  [ 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 ]
04:  [ 1 1 1 1 1 2 2 2 2 1 1 1 1 1 ]
05:  [ 1 1 1 1 2 2 2 2 2 1 1 1 1 ]
06:  [ 1 1 1 2 2 2 2 2 2 1 1 1 ]
07:  [ 1 1 1 3 3 3 3 1 1 1 ]
08:  [ 1 1 2 2 2 2 2 2 2 1 1 ]
09:  [ 1 2 2 2 2 2 2 2 2 1 ]
10:  [ 1 2 3 3 3 3 2 1 ]
11:  [ 1 4 4 4 4 1 ]
12:  [ 2 2 2 2 2 2 2 2 2 ]
13:  [ 3 3 3 3 3 3 ]
(End)

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A000009 (partitions of 2*n with even differences and even minimal part), A179080 (odd differences), A179049 (odd differences and odd minimal part).

{a(n)=polcoeff(-1 + prod(m=1,n\1, 1+x^(2*m))+prod(m=1,n\2+1,1+x^(2*m-1))+x*O(x^n),n)} /* Paul D. Hanna */

{a(n)=polcoeff(-1 + sum(m=0,sqrtint(n+1), x^(m^2)*(1+x^m)/prod(k=1,m,1-x^(2*k)+x*O(x^n))),n)} /* Paul D. Hanna */

def A189357(n):
    works = lambda part: all(x % 2 == 0 for x in differences(part))
    def count(pred, iter): return sum(1 for item in iter if pred(item))
    return count(works, Partitions(n, max_slope=-1))
print([A189357(n) for n in range(0, 30)])
# D. S. McNeil, Apr 21 2011 (updated to Python3 by Peter Luschny, Mar 06 2020 )

a(n) = A000009(2*n) + A000700(n) for n>=1, a(0)=1.
G.f.: -1 + prod(n>=1, 1+x^(2*n) ) + prod(n>=1, 1+x^(2*n-1) ).
G.f.: -1 + sum(n>=0, x^(n^2)*(1+x^n) / prod(k=1..n, 1-x^(2*k)) ). [Joerg Arndt, Jan 27 2011]

A218355 Number of partitions into distinct parts where all differences between consecutive parts are odd and the minimal part is even.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 2, 6, 2, 8, 3, 9, 5, 12, 7, 13, 9, 16, 13, 19, 17, 22, 23, 25, 29, 30, 37, 35, 46, 41, 58, 49, 70, 57, 85, 68, 103, 81, 123, 97, 145, 115, 172, 139, 201, 164, 236, 197, 274, 234, 318, 280, 368, 330, 425, 394, 488, 463, 561, 548, 644, 642, 738, 755, 844, 879, 965, 1029

Offset: 0

Joerg Arndt, Oct 27 2012

nonn

Parts are even, odd, even, odd, ... .

			The a(23) = 13 such partitions of 23 are:
[ 1]  2 3 18
[ 2]  2 5 16
[ 3]  2 7 14
[ 4]  2 9 12
[ 5]  2 21
[ 6]  4 5 14
[ 7]  4 7 12
[ 8]  4 9 10
[ 9]  4 19
[10]  6 7 10
[11]  6 17
[12]  8 15
[13]  10 13

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A179049 (parts are odd, even, odd, even, ...).

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+2)+b(n-i, i+1)))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 08 2012; revised Feb 24 2020

b[n_, i_, t_] := b[n, i, t] = If[n==0, 1-Mod[t, 2], If[i<1, 0, b[n, i-1, t] + If[i <= n && Mod[i, 2] != t, b[n-i, i-1, Mod[i, 2]], 0]]]; a[n_] := If[n==0, 1, Sum[ b[n-i, i-1, Mod[i, 2]], {i, 1, n}]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 02 2015, after Alois P. Heinz *)

N=76; x='x+O('x^N);
gf179080 = sum(n=0, N, x^(n*(n+1)/2) / prod(k=1, n+1, 1-x^(2*k) ) );
gf179049 = sum(n=0, N, x^(n*(n+1)/2) / prod(k=1, n, 1-x^(2*k) ) );
gf = gf179080 - gf179049;
Vec( gf )

N=75; x='x+O('x^N); gf = sum(n=0, N, x^((n+1)*(n+4)/2) / prod(k=1, n+1, 1-x^(2*k) ) ); v2=Vec( gf )

# After Alois P. Heinz.
def A218355(n):
    @cached_function
    def h(n, k):
        if n == 0: return 1
        if k  > n: return 0
        return h(n, k+2) + h(n-k, k+1)
    return h(n, 2)
print([A218355(n) for n in range(76)]) # Peter Luschny, Feb 25 2020

G.f.: sum(n>=0, x^((n+1)*(n+4)/2) / prod(k=1..n+1, 1-x^(2*k) ) ).

a(n) = A179080(n) - A179049(n).

A376625 G.f.: Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 - x^(2j))^2.

Original entry on oeis.org

1, 1, 0, 3, 0, 5, 1, 9, 2, 13, 6, 20, 12, 27, 23, 39, 40, 51, 69, 70, 108, 92, 169, 125, 252, 166, 370, 227, 527, 307, 743, 425, 1021, 586, 1393, 816, 1867, 1132, 2481, 1577, 3256, 2184, 4247, 3019, 5479, 4149, 7036, 5670, 8966, 7698, 11377, 10386, 14356, 13915, 18060

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000700, A072223, A179049, A340647, A376542, A376658.

nmax=80; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} Product_{j=1..k} x^j/(1 - x^(2*j))^2.

a(n) ~ (r^(1/4) * sqrt(log(r)^2 + 2*polylog(2, sqrt(r))) / (2*Pi*sqrt(1 + 3*r^2))) * A376658^sqrt(n) / n, where r = A072223 = 0.52488859865640479389948613854128391569... is the smallest real root of the equation (1 - r^2)^2 = r.

cp's OEIS Frontend

A376627 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j))^2.

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A097042 G.f. = (1 + 4 * g.f. for A096661)/(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)).

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A179080 Number of partitions of n into distinct parts where all differences between consecutive parts are odd.

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A189357 Number of partitions of n into distinct parts where all differences between consecutive parts are even.

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A218355 Number of partitions into distinct parts where all differences between consecutive parts are odd and the minimal part is even.

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A376625 G.f.: Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^(2*j))^2.

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A376627 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j))^2.

A376625 G.f.: Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 - x^(2j))^2.