A292043 -id:A292043 - cp's OEIS frontend

A278399 G.f.: Re((i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

1, -1, -1, -2, -2, -3, -2, -3, -2, -2, 0, 0, 3, 4, 8, 9, 14, 16, 22, 24, 30, 32, 39, 40, 46, 46, 51, 48, 52, 46, 46, 36, 32, 16, 8, -15, -30, -60, -82, -122, -151, -200, -238, -296, -342, -412, -464, -542, -602, -686, -750, -841, -904, -996, -1058, -1146, -1198

Offset: 0

Vladimir Reshetnikov, Nov 20 2016

The q-Pochhammer symbol (a; q)inf = Product{k>=0} (1 - a*q^k).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A278400, A278401, A278402, A278420, A292042, A292043.

G := add((-1)^n*x^(n*(2*n-1))/mul(1 - x^k, k = 1..2*n), n = 0..7):
S := series(G, x, 101):
seq(coeff(S, x, j), j = 0..100); # Peter Bala, Feb 04 2021

Mathematica
```
Re[(QPochhammer[I, x] + O[x]^60)[[3]]]
```

(i; x)_inf is the g.f. for a(n) + i*A278400(n).

G.f.: Sum_{n >= 0} (-1)^n^x^(n*(2*n-1))/Product_{k = 1..2*n} 1 - x^k. - Peter Bala, Feb 04 2021

A278400 G.f.: Im((i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

-1, -1, -1, 0, 0, 1, 2, 3, 4, 6, 6, 8, 9, 10, 10, 11, 10, 10, 8, 6, 2, 0, -7, -12, -20, -28, -39, -48, -62, -74, -90, -104, -122, -136, -156, -171, -190, -204, -222, -232, -247, -252, -260, -258, -258, -244, -232, -204, -176, -130, -84, -15, 54, 148, 244, 368

Offset: 0

Vladimir Reshetnikov, Nov 20 2016

The q-Pochhammer symbol (a; q)inf = Product{k>=0} (1 - a*q^k).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A278399, A278401, A278402, A278420, A292042, A292043.

with(gfun): series(add((-1)^(n+1)*x^(n*(2*n+1))/mul(1 - x^k, k = 1..2*n+1), n = 0..6), x, 100): seriestolist(%); # Peter Bala, Feb 06 2021

Mathematica
```
Im[(QPochhammer[I, x] + O[x]^60)[[3]]]
```

(i; x)_inf is the g.f. for A278399(n) + i*a(n).

G.f.: Sum_{n >= 0} (-1)^(n+1)*x^(n*(2*n+1))/Product_{k = 1..2*n+1} 1 - x^k. - Peter Bala, Feb 06 2021

A292042 G.f.: Re((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -3, -4, -3, -3, -1, -1, 2, 3, 7, 9, 14, 16, 23, 26, 33, 37, 45, 48, 57, 60, 68, 70, 77, 76, 82, 78, 80, 72, 70, 55, 48, 26, 11, -19, -42, -84, -116, -169, -213, -278, -333, -413, -479, -572, -651, -757, -846, -965, -1062

Offset: 0

Seiichi Manyama, Sep 08 2017

sign

			Product_{k>=1} (1 - i*x^k) = 1 + (0-1i)*x + (0-1i)*x^2 + (-1-1i)*x^3 + (-1-1i)*x^4 + (-2-1i)*x^5 + (-2+0i)*x^6 + (-3+0i)*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A035294, A278399, A278400, A278420, A292043, A292052.

N:= 100:
S := convert(series( add( (-1)^n*x^(n*(2*n+1))/(mul(1 - x^k,k = 1..2*n)), n = 0..floor(sqrt(N/2)) ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021

Re[CoefficientList[Series[QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 08 2017 *)

( i*x; x)_inf is the g.f. for a(n) + i*A292043(n).
(-i*x; x)_inf is the g.f. for a(n) + i*A292052(n).
a(n)^2 + A292043(n)^2 = A278420(n). - Vaclav Kotesovec, Sep 08 2017
From Peter Bala, Jan 15 2021: (Start)
G.f.: A(x) = Sum_{n >= 0} (-1)^n*x^(n*(2*n+1))/Product_{k = 1..2*n}  (1 - x^k). Cf. A035294.
Conjectural g.f.: A(x) = (1/2)*Sum_{n >= 0} (-x)^(n*(n-1)/2)/Product_{k = 1..n} (1 - x^k). (End)

A300574 Coefficient of x^n in 1/((1-x)(1+x^3)(1-x^5)(1+x^7)(1-x^9)...).

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 2, 1, 0, 0, 2, 2, 1, 0, 2, 3, 2, 0, 2, 4, 4, 0, 1, 4, 6, 2, 1, 4, 8, 4, 2, 4, 10, 6, 2, 3, 12, 10, 4, 2, 13, 14, 8, 2, 14, 18, 12, 2, 14, 22, 18, 3, 14, 26, 26, 6, 14, 29, 34, 10, 14, 32, 44, 16, 14, 34, 56, 26, 16, 34, 67, 38, 20, 34, 78, 52, 26

Offset: 0

Gus Wiseman, Mar 08 2018

By Theorem 1 of Craig, the values a(n) in this list are known to be nonnegative. Combined with Theorem 2 of Seo and Yee, this shows that a(n) = |number of partitions of n into odd parts with an odd index minus the number of partitions of n into odd parts with an even index|. - William Craig, Dec 31 2021

Seunghyun Seo and Ae Ja Yee, Index of seaweed algebras and integer partitions, Electronic Journal of Combinatorics, 27:1 (2020), #P1.47. See Conjecture 1 and Theorem 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Gus Wiseman)
Vincent E. Coll, Andrew W. Mayers, and Nick W. Mayers, Statistics on integer partitions arising from seaweed algebras, arXiv:1809.09271 [math.CO], 2018.
William Craig, Seaweed Algebras and the Index Statistic for Partitions, arXiv:2112.09269 [math.CO], 2021.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000009, A010815, A027193, A067659, A078408, A081362, A099323, A220418, A290261, A292043, A292137, A298118, A300575.

CoefficientList[Series[1/QPochhammer[x, -x^2], {x, 0, 100}], x]
nmax = 100; CoefficientList[Series[Product[1/((1+x^(4*k-1))*(1-x^(4*k-3))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 04 2019 *)

O.g.f.: Product_{n >= 0} 1/(1 - (-1)^n x^(2n+1)).
a(n) = Sum (-1)^k where the sum is over all integer partitions of n into odd parts and k is the number of parts not congruent to 1 modulo 4.
a(n) has average order Gamma(1/4) * exp(sqrt(n/3)*Pi/2) / (2^(9/4) * 3^(1/8) * Pi^(3/4) * n^(5/8)). - Vaclav Kotesovec, Jun 04 2019

A278420 a(n) = (A278399(n)^2 + A278400(n)^2)/2.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 4, 9, 10, 20, 18, 32, 45, 58, 82, 101, 148, 178, 274, 306, 452, 512, 785, 872, 1258, 1450, 2061, 2304, 3274, 3796, 5108, 6056, 7954, 9376, 12200, 14733, 18500, 22608, 28004, 34354, 41905, 51752, 62122, 77090, 91764, 114640, 134560, 167690

Offset: 0

Vladimir Reshetnikov, Nov 21 2016

nonn

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

Cf. A278399, A278400, A292042, A292043.

Abs[(QPochhammer[I, x] + O[x]^60)[[3]]]^2 / 2

a(n) = A292042(n)^2 + A292043(n)^2. - Vaclav Kotesovec, Sep 08 2017

A292137 G.f.: Im(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, -1, -2, -2, -2, -2, -3, -3, -2, -2, -2, -1, 1, 2, 2, 4, 6, 7, 8, 10, 13, 14, 14, 15, 17, 17, 15, 15, 16, 14, 10, 8, 6, 1, -5, -10, -14, -21, -31, -38, -43, -53, -64, -71, -77, -86, -97, -104, -108, -115, -124, -127, -125, -127, -130, -125, -116

Offset: 0

Seiichi Manyama, Sep 09 2017

			Product_{k>=1} 1/(1 - i*x^k) = 1 + (0+1i)*x + (-1+1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2-1i)*x^6 + (-1-2i)*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A000108, A010815, A027193, A063834, A067659, A081362, A099323, A196545, A220418, A290261, A292042, A292043, A292136, A292138, A298118, A300355.

N:= 100:
S := convert(series( add( (-1)^n*x^(2*n+1)/(mul(1 - x^k,k = 1..2*n+1)), n = 0..N ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021

Im[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 17 2017 *)

1/(i*x; x)_inf is the g.f. for A292136(n) + i*a(n).
a(n) = Sum (-1)^((k - 1)/2) where the sum is over all integer partitions of n into an odd number of parts and k is the number of parts. - Gus Wiseman, Mar 08 2018
G.f.: Sum_{n >= 0} (-1)^n * x^(2*n+1)/Product_{k = 1..2*n+1} (1 - x^k). - Peter Bala, Jan 15 2021

A292136 G.f.: Re(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

1, 0, -1, -1, -1, -1, -2, -1, 0, 0, 0, 1, 2, 3, 3, 4, 6, 6, 5, 6, 7, 6, 5, 5, 5, 3, 0, -2, -3, -6, -11, -13, -14, -19, -24, -27, -29, -33, -38, -40, -40, -43, -47, -46, -43, -43, -43, -38, -30, -26, -22, -12, 1, 11, 20, 36, 56, 71, 85, 106, 130, 149, 166, 190, 217

Offset: 0

Seiichi Manyama, Sep 09 2017

			Product_{k>=1} 1/(1 - i*x^k) = 1 + (0+1i)*x + (-1+1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2-1i)*x^6 + (-1-2i)*x^7 + ...
Product_{k>=1} 1/(1 + i*x^k) = 1 + (0-1i)*x + (-1-1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2+1i)*x^6 + (-1+2i)*x^7 + ...
From _Peter Bala_, Jan 19 2021: (Start)
The number of partitions of n = 13 into an even number of parts is:
# parts (2*k)   2    4    6   8   10   12
# partitions    6   18   14   7    3    1
Hence a(13) = Sum (-1)^k = -6 + 18 - 14 + 7 - 3 + 1 = 3. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A292042, A292043, A292137, A292138.

N:= 100:
S := convert(series( add( (-1)^n*x^(2*n)/(mul(1 - x^k,k = 1..2*n)), n = 0..N ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021

Re[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 17 2017 *)

1/( i*x; x)_inf is the g.f. for a(n) + i*A292137(n).
1/(-i*x; x)_inf is the g.f. for a(n) + i*A292138(n).
From Peter Bala, Jan 19 2021: (Start)
a(n) = Sum (-1)^k, where the sum is over all integer partitions of n into an even number of parts and 2*k is the number of parts in a partition. An example is given below.
G.f.: Sum_{n >= 0} (-1)^n * x^(2*n)/Product_{k = 1..2*n} (1 - x^k). (End)

A292052 G.f.: Im((-i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 0, 0, -1, -2, -3, -4, -6, -7, -9, -10, -12, -13, -15, -15, -16, -16, -16, -14, -13, -9, -6, 0, 5, 14, 22, 34, 45, 60, 74, 93, 110, 132, 152, 177, 199, 226, 249, 277, 300, 328, 348, 373, 389, 408, 417, 428, 425, 424, 407, 389, 352, 314, 252

Offset: 0

Seiichi Manyama, Sep 08 2017

			Product_{k>=1} (1 + i*x^k) = 1 + (0+1i)*x + (0+1i)*x^2 + (-1+1i)*x^3 + (-1+1i)*x^4 + (-2+1i)*x^5 + (-2+0i)*x^6 + (-3+0i)*x^7 + ...

Robert Israel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A278399, A278400, A278420, A292042, A292043.

N:= 100: # to get a(0)..a(N)
P:= mul(1+I*x^k, k=1..N):
S:= series(P, x, N+1):
seq(evalc(Im(coeff(S,x,j))),j=0..N); # Robert Israel, Sep 08 2017

Im[CoefficientList[Series[QPochhammer[-I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 09 2017 *)

(-i*x; x)_inf is the g.f. for A292042(n) + i*a(n).

a(n) = -A292043(n).

A300575 Coefficient of x^n in (1+x)(1-x^3)(1+x^5)(1-x^7)(1+x^9)...

Original entry on oeis.org

1, 1, 0, -1, -1, 1, 1, -1, -2, 0, 2, 0, -3, -1, 3, 2, -3, -3, 3, 4, -3, -6, 2, 7, -1, -8, 0, 10, 2, -11, -4, 12, 7, -13, -10, 13, 13, -13, -17, 13, 22, -11, -26, 9, 31, -6, -36, 2, 41, 3, -46, -9, 51, 17, -55, -26, 59, 36, -62, -48, 63, 61, -64, -75, 64, 92, -60, -109, 55, 127, -48, -147, 37, 167

Offset: 0

Gus Wiseman, Mar 09 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000009, A010815, A027193, A067659, A078408, A081362, A220418, A290261, A292043, A292137, A300574.

CoefficientList[Series[QPochhammer[-x,-x^2],{x,0,100}],x]

O.g.f.: Product_{n >= 0} (1 + (-1)^n x^(2n+1)).

a(n) = Sum (-1)^k where the sum is over all strict integer partitions of n into odd parts and k is the number of parts not congruent to 1 modulo 4.

cp's OEIS Frontend

A278399 G.f.: Re((i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

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A278400 G.f.: Im((i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

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A292042 G.f.: Re((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

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A300574 Coefficient of x^n in 1/((1-x)(1+x^3)(1-x^5)(1+x^7)(1-x^9)...).

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A278420 a(n) = (A278399(n)^2 + A278400(n)^2)/2.

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A292137 G.f.: Im(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

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A292136 G.f.: Re(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

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A292052 G.f.: Im((-i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

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