A081362 -id:A081362 - cp's OEIS frontend

A284312 Expansion of Product_{k>=0} (1 - x^(3*k+1)) in powers of x.

1, -1, 0, 0, -1, 1, 0, -1, 1, 0, -1, 2, -1, -1, 2, -1, -1, 3, -2, -1, 3, -3, 0, 4, -4, 0, 4, -5, 1, 5, -7, 2, 5, -8, 4, 5, -10, 5, 5, -12, 8, 5, -14, 10, 4, -16, 14, 3, -19, 17, 1, -20, 22, -1, -23, 26, -4, -25, 33, -8, -27, 38, -13, -28, 46, -19, -30, 53, -26, -29

Offset: 0

Seiichi Manyama, Mar 24 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), this sequence (m=3), A284313 (m=4), A284314 (m=5).

Cf. A078181, A261612.

CoefficientList[Series[Product[1 - x^(3k + 1), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, 1 - x^(3*k + 1)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n)*Sum_{k=1..n} A078181(k)*a(n-k), a(0) = 1.

A284317 Expansion of Product_{k>=0} (1 - x^(5*k+4)) in powers of x.

Original entry on oeis.org

1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 3, -1, 0, 0, -2, 3, -1, 0, 0, -3, 4, -1, 0, 1, -4, 4, -1, 0, 1, -5, 5, -1, 0, 2, -7, 5, -1, 0, 3, -8, 6, -1, 0, 5, -10, 6, -1, -1, 6, -12, 7, -1, -1, 9, -14

Offset: 0

Seiichi Manyama, Mar 25 2017

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Robert Israel, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(m*k+m-1)): A081362 (m=2), A284315 (m=3), A284316 (m=4), this sequence (m=5).

Cf. A281243, A284103.

S:= series(mul(1-x^(5*k+4),k=0..200),x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Mar 27 2017

CoefficientList[Series[Product[1 - x^(5k + 4), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, 1 - x^(5*k + 4)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n)*Sum_{k=1..n} A284103(k)*a(n-k), a(0) = 1.

G.f. is the QPochhammer symbol (x^4;x^5)infinity. - _Robert Israel, Mar 27 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

A292520 Expansion of Product_{k>=1} 1/(1 + x^(k^2)).

Original entry on oeis.org

1, -1, 1, -1, 0, 0, 0, 0, 1, -2, 2, -2, 1, 0, 0, 0, 0, -1, 2, -2, 2, -1, 0, 0, 0, -1, 2, -3, 3, -2, 1, 0, 1, -2, 3, -4, 3, -2, 1, 0, 1, -2, 3, -4, 3, -2, 1, 0, 0, -2, 4, -5, 6, -4, 2, -1, 0, -2, 5, -7, 8, -6, 3, -1, 0, -1, 3, -6, 7, -6, 4, -1, 1, -1, 3, -6, 7, -8, 6, -3, 2, -4, 6, -9, 11, -9, 7, -4, 1, -3, 7

Offset: 0

Ilya Gutkovskiy, Sep 18 2017

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Convolution inverse of A033461.

The difference between the number of partitions of n into an even number of squares and the number of partitions of n into an odd number of squares.

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Index entries for related partition-counting sequences

Cf. A001156, A033461, A081362, A243148, A276516, A279225, A279226.

nmax = 100; CoefficientList[Series[Product[1/(1 + x^(k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 19 2017 *)

G.f.: Product_{k>=1} 1/(1 + x^(k^2)).
a(n) ~ (-1)^n * exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2^(7/3)) * Zeta(3/2)^(1/3) / (2^(5/3) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 19 2017
a(n) = Sum_{k=0..n} (-1)^k * A243148(n,k). - Alois P. Heinz, Jul 25 2022

A359936 Expansion of Product_{k>=0} (1 - x^(k^2+1)) in powers of x.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jan 19 2023

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A002522, A081362, A276516, A284312, A284313, A284314, A284585, A284499, A357911, A359937.

my(N=100, x='x+O('x^N)); Vec(prod(k=0, sqrtint(N), 1-x^(k^2+1)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, sumdiv(j, d, issquare(d-1)*d)*v[i-j+1])/i); v;

a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A359937(k) * a(n-k).

A022598 Expansion of Product_{m>=1} (1+q^m)^(-3).

Original entry on oeis.org

1, -3, 3, -4, 9, -12, 15, -21, 30, -43, 54, -69, 94, -123, 153, -193, 252, -318, 391, -486, 609, -754, 918, -1119, 1376, -1680, 2019, -2432, 2946, -3540, 4220, -5034, 6015, -7157, 8463, -9999, 11835, -13956, 16374, -19206, 22542, -26376, 30750, -35829, 41745

Offset: 0

N. J. A. Sloane

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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - 3*x + 3*x^2 - 4*x^3 + 9*x^4 - 12*x^5 + 15*x^6 - 21*x^7 + 30*x^8 + ...
G.f. = 1/q - 3*q^7 + 3*q^15 - 4*q^23 + 9*q^31 - 12*q^39 + 15*q^47 - 21*q^55 + ...

T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^3.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 13.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A022568, A081362.

Column k=3 of A286352.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[x^2])^3, {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^3, n))};

Expansion of chi(-x)^3 = phi(-x) / psi(x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Aug 09 2015
Expansion of q^(1/8) * (eta(q) / eta(q^2))^3 in powers of q. - Michael Somos, Apr 24 2015
Euler transform of period 2 sequence [ -3, 0, ...]. - Michael Somos, Aug 09 2015
Convolution cube of A081362. - Michael Somos, Apr 24 2015
Convolution inverse of A022568. - Michael Somos, Aug 09 2015
a(n) ~ (-1)^n * exp(Pi*sqrt(n/2)) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(3/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(-3*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A284316 Expansion of Product_{k>=0} (1 - x^(4*k+3)) in powers of x.

Original entry on oeis.org

1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 2, -1, 0, -1, 2, -1, 0, -1, 3, -1, 0, -2, 3, -1, 0, -3, 4, -1, 1, -4, 4, -1, 1, -5, 5, -1, 2, -7, 5, -1, 3, -8, 6, -1, 5, -10, 6, -2, 6, -12, 7, -2, 9, -14, 7, -3, 11, -16, 8, -4, 15, -19, 8, -6, 18, -21, 9

Offset: 0

Seiichi Manyama, Mar 25 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(m*k+m-1)): A081362 (m=2), A284315 (m=3), this sequence (m=4), A284317 (m=5).

Cf. A050452, A147599, A284313, A111330.

CoefficientList[Series[Product[1 - x^(4k + 3), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, 1 - x^(4*k+3)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n)*Sum_{k=1..n} A050452(k)*a(n-k), a(0) = 1.

O.g.f.: Sum_{n >= 0} (-1)^n*x^(n*(2*n+1)) / Product_{k = 1..n} ( 1 - x^(4*k) ). Cf. A284313. - Peter Bala, Nov 28 2020

A290217 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji)).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 0, 0, 1, -1, -2, -1, 0, 1, -1, -2, 3, 1, 0, 1, -1, -2, 0, -1, -1, 0, 1, -1, -2, 0, 5, -5, 1, 0, 1, -1, -2, 0, 1, 1, 9, -1, 0, 1, -1, -2, 0, 1, 9, -12, 3, 2, 0, 1, -1, -2, 0, 1, 4, -4, -3, -20, -2, 0, 1, -1, -2, 0, 1, 4, 6, -15, 31, 16, 2

Offset: 0

Seiichi Manyama, Oct 06 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -1, -1, -1, ...
   0,  0, -2, -2, -2, ...
   0, -1,  3,  0,  0, ...
   0,  1, -1,  5,  1, ...
   0, -1, -5,  1,  9, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000007, A081362, A293287, A290395.
Rows n=0..1 give A000012, (-1)*A057427.
Main diagonal gives A258210.
Cf. A290216, A293307.

A298993 Expansion of Product_{n>=1} 1/sqrt(1 + (4*x)^n).

Original entry on oeis.org

1, -2, -2, -36, 54, -476, 556, -6088, 35878, -156844, 444164, -1734648, 11948604, -35313048, 156354328, -864527760, 4733447686, -12692853452, 54065039380, -226098757912, 1278838329812, -5257771138376, 19455009120232, -76455773381360, 453306681446748

Offset: 0

Seiichi Manyama, Jan 31 2018

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Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000984, A081362, A271235, A298994.

CoefficientList[Series[Sqrt[2/QPochhammer[-1, 4*x]], {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 18 2018 *)

a(n) ~ (-1)^n * exp(Pi*sqrt(n/12)) * 4^(n-1) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 18 2018

A307060 Expansion of 1/(2 - Product_{k>=1} 1/(1 + x^k)).

Original entry on oeis.org

1, -1, 1, -2, 4, -7, 12, -21, 38, -68, 120, -212, 377, -670, 1188, -2107, 3740, -6638, 11778, -20898, 37084, -65808, 116775, -207212, 367696, -652478, 1157815, -2054524, 3645730, -6469316, 11479734, -20370656, 36147506, -64143372, 113821732, -201975429, 358403220, -635982680, 1128544452, -2002589998

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

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Invert transform of A081362.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A081362, A299208, A304969, A307058.

Cf. A307057, A307058, A307059, A307062, A307063.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1-x^(2*j-1): j in [1..m+2]])) )); // G. C. Greubel, Jan 24 2024

nmax = 39; CoefficientList[Series[1/(2 - Product[1/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

m=80;
def f(x): return 1/( 2 - product(1-x^(2*j-1) for j in range(1,m+3)) )
def A307060_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307060_list(m) # G. C. Greubel, Jan 24 2024

G.f.: 1/(2 - Product_{k>=1} (1 - x^(2*k-1))).
a(0) = 1; a(n) = Sum_{k=1..n} A081362(k)*a(n-k).
From G. C. Greubel, Jan 24 2024: (Start)
G.f.: 1/(2 - QPochhammer(x)/QPochhammer(x^2)).
G.f.: 1/(2 - x^(1/24)*eta(x)/eta(x^2)), where eta(x) is the Dedekind eta function. (End)

cp's OEIS Frontend

A284312 Expansion of Product_{k>=0} (1 - x^(3*k+1)) in powers of x.

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A284317 Expansion of Product_{k>=0} (1 - x^(5*k+4)) in powers of x.

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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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A292520 Expansion of Product_{k>=1} 1/(1 + x^(k^2)).

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A359936 Expansion of Product_{k>=0} (1 - x^(k^2+1)) in powers of x.

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A022598 Expansion of Product_{m>=1} (1+q^m)^(-3).

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A284316 Expansion of Product_{k>=0} (1 - x^(4*k+3)) in powers of x.

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A290217 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} j*x^(j*i)).

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A290217 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji)).