A022567 -id:A022567 - cp's OEIS frontend

A022597 Expansion of Product_{m >= 1} (1 + q^m)^(-2).

1, -2, 1, -2, 4, -4, 5, -6, 9, -12, 13, -16, 21, -26, 29, -36, 46, -54, 62, -74, 90, -106, 122, -142, 171, -200, 227, -264, 311, -358, 408, -470, 545, -626, 709, -810, 933, -1062, 1198, -1362, 1555, -1760, 1980, -2238, 2536, -2858, 3205, -3602, 4063, -4560, 5092, -5704, 6400, -7150, 7966

Offset: 0

N. J. A. Sloane

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

McKay-Thompson series of class 24J for the Monster group.

			G.f. = 1 - 2*x + x^2 - 2*x^3 + 4*x^4 - 4*x^5 + 5*x^6 - 6*x^7 + 9*x^8 + ...
T24J = 1/q - 2*q^11 + q^23 - 2*q^35 + 4*q^47 - 4*q^59 + 5*q^71 - 6*q^83 + ...

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
D. Foata and G.-N. Han, Jacobi and Watson Identities Combinatorially Revisited
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
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, Jacobi Theta Functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group

Cf. A022567, A073252, A081362.
Cf. A089814 (expansion of Product_{k>=1}(1-q^(10k-5))^2).
Column k=2 of A286352.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2, {x, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, n}]^-2, {x, 0, n}]; (* Michael Somos, Jul 11 2011 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^2, n))}; /* Michael Somos, Sep 10 2005 */

Expansion of q^(1/12) * (eta(q) / eta(q^2))^2 in powers of q.
Euler transform of period 2 sequence [ -2, 0, ...]. - Michael Somos, Sep 10 2005
Expansion of chi(-x)^2 in powers of x where chi() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A022567.
G.f.: Product_{k>0} (1 + x^k)^-2.
Convolution square of A081362. Convolution inverse of A022567.
a(n) = (-1)^n * A073252(n).
a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(-2*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A261616 Expansion of Product_{k>=0} 1/(1 - x^(3*k+1))^2.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 13, 18, 26, 34, 44, 58, 76, 96, 123, 156, 196, 244, 304, 374, 461, 566, 690, 836, 1015, 1224, 1470, 1762, 2110, 2512, 2987, 3542, 4191, 4944, 5825, 6842, 8025, 9392, 10971, 12788, 14891, 17300, 20068, 23242, 26883, 31034, 35787, 41204

Offset: 0

Vaclav Kotesovec, Aug 26 2015

nonn

Self-convolution of A035382.

In general, if a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} 1/(1 - x^(a*k+b))^2, then a(n) ~ Gamma(b/a)^2 * a^(b/a - 3/4) * exp(2*Pi*sqrt(n/(3*a))) * Pi^(2*b/a - 2) / (4 * 3^(b/a - 1/4) * n^(b/a + 1/4)).

Cf. A022567, A035382, A261610, A261612, A261615.

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(3*k+1))^2, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2*Pi*sqrt(n)/3) * Gamma(1/3)^2 / (4 * sqrt(3) * Pi^(4/3) * n^(7/12)).

A263002 Expansion of (f(-x^5) / f(-x))^2 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 5, 10, 20, 34, 61, 100, 165, 260, 408, 620, 940, 1390, 2045, 2960, 4257, 6040, 8525, 11900, 16522, 22738, 31130, 42300, 57210, 76872, 102834, 136800, 181230, 238900, 313725, 410160, 534330, 693330, 896655, 1155420, 1484274, 1900420, 2426215, 3088100

Offset: 0

Michael Somos, Oct 07 2015

nonn

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

Number of 5-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019

			G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 34*x^5 + 61*x^6 + 100*x^7 + ...
G.f. = q + 2*q^4 + 5*q^7 + 10*q^10 + 20*q^13 + 34*q^16 + 61*q^19 + 100*q^22 + ...

Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Vaclav Kotesovec, 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
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A058511.

Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.

f:=(k,M) -> mul(1-q^(k*j),j=1..M);
LRBP := (L,M) -> (f(L,M)/f(1,M))^2;
S := L -> seriestolist(series(LRBP(L,80),q,60));
S(5); # N. J. A. Sloane, Oct 20 2019

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^5] / QPochhammer[ x])^2, {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^5 + A) / eta(x + A))^2, n))};

Expansion of q^(-1/3) * (eta(q^5) / eta(q))^2 in powers of q.
Euler transform of period 5 sequence [ 2, 2, 2, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (45 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A058511.
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u - v^2) * (v - u^2) - 4*u^2*v^2.
Convolution inverse is A058511.
a(n) ~ exp(4*Pi*sqrt(n/15)) / (sqrt(2) * 3^(1/4) * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
See Maple code for a simple g.f. - N. J. A. Sloane, Oct 20 2019

A285928 Expansion of (Product_{k>0} (1 - x^(5*k)) / (1 - x^k))^5 in powers of x.

Original entry on oeis.org

1, 5, 20, 65, 190, 501, 1240, 2890, 6440, 13775, 28502, 57205, 111880, 213670, 399620, 733128, 1321850, 2345340, 4100700, 7072520, 12045005, 20272465, 33746060, 55595635, 90706390, 146638756, 235016940, 373580735, 589238640, 922537655, 1434232510, 2214817165

Offset: 0

Seiichi Manyama, Apr 28 2017

nonn

In general, if m > 1 and g.f. = Product_{k>=1} ((1 - x^(m*k)) / (1 - x^k))^m, then a(n, m) ~ exp(Pi*sqrt(2*(m-1)*n/3)) * (m-1)^(1/4) / (2^(5/4) * 3^(1/4) * m^(m/2) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017

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

(Product_{k>0} (1 - x^(m*k)) / (1 - x^k))^m: A022567 (m=2), A285927 (m=3), A093160 (m=4), this sequence (m=5).

Cf. A035959, A285932.

nmax = 40; CoefficientList[Series[Product[((1 - x^(5*k)) / (1 - x^k))^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 30 2017 *)

a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A116073(k)*a(n-k) for n > 0.

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(1/4) * 5^(5/2) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017

A358369 Euler transform of 2^floor(n/2), (A016116).

Original entry on oeis.org

1, 1, 3, 5, 12, 20, 43, 73, 146, 250, 475, 813, 1499, 2555, 4592, 7800, 13761, 23253, 40421, 67963, 116723, 195291, 332026, 552882, 932023, 1544943, 2585243, 4267081, 7094593, 11662769, 19281018, 31575874, 51937608, 84753396, 138772038, 225693778, 368017636

Offset: 0

Peter Luschny, Nov 17 2022

nonn

Cf. A002513, A016116.

Sequences that can be represented as a EulerTransform(BinaryRecurrenceSequence()) include A000009, A000041, A000712, A001970, A002513, A010054, A015128, A022567, A034691, A111317, A111335, A117410, A156224, A166861, A200544, A261031, A261329, A358449.

BinaryRecurrenceSequence := proc(b, c, u0:=0, u1:=1) local u;
u := proc(n) option remember; if n < 2 then return [u0, u1][n + 1] fi;
b*u(n - 1) + c*u(n - 2) end; u end:
EulerTransform := proc(a) local b;
b := proc(n) option remember; if n = 0 then return 1 fi; add(add(d * a(d),
d = NumberTheory:-Divisors(j)) * b(n-j), j = 1..n) / n end; b end:
a := EulerTransform(BinaryRecurrenceSequence(0, 2, 1)): seq(a(n), n=0..36);

from typing import Callable
from functools import cache
from sympy import divisors
def BinaryRecurrenceSequence(b:int, c:int, u0:int=0, u1:int=1) -> Callable:
    @cache
    def u(n: int) -> int:
        if n < 2:
            return [u0, u1][n]
        return b * u(n - 1) + c * u(n - 2)
    return u
def EulerTransform(a: Callable) -> Callable:
    @cache
    def b(n: int) -> int:
        if n == 0:
            return 1
        s = sum(sum(d * a(d) for d in divisors(j)) * b(n - j)
            for j in range(1, n + 1))
        return s // n
    return b
b = BinaryRecurrenceSequence(0, 2, 1)
a = EulerTransform(b)
print([a(n) for n in range(37)])

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 2, 1)
a = EulerTransform(b)
print([a(n) for n in range(37)])

A261615 Expansion of Product_{k>=0} (1 + x^(3*k+1))^2.

Original entry on oeis.org

1, 2, 1, 0, 2, 4, 2, 2, 5, 4, 3, 8, 10, 6, 9, 14, 11, 14, 22, 18, 17, 30, 32, 28, 41, 46, 39, 54, 68, 60, 73, 94, 85, 96, 131, 128, 130, 170, 175, 176, 229, 246, 237, 294, 330, 320, 386, 446, 430, 492, 582, 578, 642, 762, 763, 818, 977, 1008, 1061, 1254, 1311

Offset: 0

Vaclav Kotesovec, Aug 26 2015

nonn

Self-convolution of A261612.

In general, if a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} (1 + x^(a*k+b))^2, then a(n) ~ exp(Pi*sqrt(2*n/(3*a))) / (2^(2*b/a + 1/4) * 3^(1/4) * a^(1/4) * n^(3/4)).

Cf. A022567, A035382, A261610, A261612, A261616.

nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k+1))^2, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(2*n)/3) / (2^(11/12) * sqrt(3) * n^(3/4)).

A285927 Expansion of (Product_{k>0} (1 - x^(3*k)) / (1 - x^k))^3 in powers of x.

Original entry on oeis.org

1, 3, 9, 19, 42, 81, 155, 276, 486, 821, 1368, 2214, 3541, 5544, 8586, 13082, 19740, 29403, 43414, 63423, 91935, 132075, 188418, 266733, 375232, 524331, 728514, 1006216, 1382604, 1889739, 2570719, 3480420, 4691682, 6297102, 8418252, 11209347, 14870970

Offset: 0

Seiichi Manyama, Apr 28 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Mohammed L. Nadji and Moussa Ahmia, Congruences for L-regular tripartitions for L in {2, 3}, Integers (2024) Vol. 24, Art. No. A86. See p. 2.

(Product_{k>0} (1 - x^(m*k)) / (1 - x^k))^m: A022567 (m=2), this sequence (m=3), A093160 (m=4), A285928 (m=5).

Cf. A000726, A199659.

nmax = 40; CoefficientList[Series[Product[((1 - x^(3*k)) / (1 - x^k))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 30 2017 *)

a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A046913(k)*a(n-k) for n > 0.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(7/4) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017

A328547 Number of 3-regular bipartitions of n.

Original entry on oeis.org

1, 2, 5, 8, 16, 26, 44, 68, 108, 162, 245, 356, 521, 740, 1053, 1468, 2045, 2804, 3836, 5184, 6988, 9326, 12409, 16376, 21546, 28154, 36674, 47492, 61317, 78764, 100880, 128628, 163553, 207134, 261630, 329288, 413395, 517316, 645803, 803844, 998282

Offset: 0

N. J. A. Sloane, Oct 19 2019

nonn

Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.

Cf. A000726.

f:=(k,M) -> mul(1-q^(k*j),j=1..M);
LRBP := (L,M) -> (f(L,M)/f(1,M))^2;
S := L -> seriestolist(series(LRBP(L,80),q,60));
S(3);

nmax = 40; CoefficientList[Series[Product[1 + x^j + x^(2*j), {j, 1, nmax}]^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2024 *)

a(n) ~ exp(Pi*sqrt(8*n)/3) / (2^(3/4) * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 08 2024

A328548 Number of 6-regular bipartitions of n.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 63, 106, 175, 280, 441, 680, 1034, 1548, 2290, 3346, 4840, 6930, 9837, 13844, 19337, 26810, 36925, 50530, 68741, 92984, 125113, 167490, 223155, 295960, 390825, 513954, 673214, 878480, 1142190, 1479892, 1911051, 2459896, 3156602

Offset: 0

N. J. A. Sloane, Oct 19 2019

nonn

Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.

Cf. A219601.

f:=(k,M) -> mul(1-q^(k*j),j=1..M);
LRBP := (L,M) -> (f(L,M)/f(1,M))^2;
S := L -> seriestolist(series(LRBP(L,80),q,60));
S(6);

nmax = 40; CoefficientList[Series[Product[(1 - x^(6*k))/(1 - x^k), {k, 1, nmax}]^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2024 *)

a(n) ~ 5^(1/4) * exp(Pi*sqrt(10*n)/3) / (2^(9/4) * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 08 2024

A022600 Expansion of Product_{m>=1} (1+q^m)^(-5).

Original entry on oeis.org

1, -5, 10, -15, 30, -56, 85, -130, 205, -315, 465, -665, 960, -1380, 1925, -2651, 3660, -5020, 6775, -9070, 12126, -16115, 21220, -27765, 36235, -47101, 60810, -78115, 100105, -127825, 162391, -205530, 259475, -326565

Offset: 0

N. J. A. Sloane

sign

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

Cf. Related to Expansion of Product_{m>=1} (1+q^m)^k: A022627 (k=-32), A022626 (k=-31), A022625 (k=-30), A022624 (k=-29), A022623 (k=-28), A022622 (k=-27), A022621 (k=-26), A022620 (k=-25), A007191 (k=-24), A022618 (k=-23), A022617 (k=-22), A022616 (k=-21), A022615 (k=-20), A022614 (k=-19), A022613 (k=-18), A022612 (k=-17), A022611 (k=-16), A022610 (k=-15), A022609 (k=-14), A022608 (k=-13), A007249 (k=-12), A022606 (k=-11), A022605 (k=-10), A022604 (k=-9), A007259 (k=-8), A022602 (k=-7), A022601 (k=-6), this sequence (k=-5), A022599 (k=-4), A022598 (k=-3), A022597 (k=-2), A081362 (k=-1), A000009 (k=1), A022567 (k=2), A022568 (k=3), A022569 (k=4), A022570 (k=5), A022571 (k=6), A022572 (k=7), A022573 (k=8), A022574 (k=9), A022575 (k=10), A022576 (k=11), A022577 (k=12), A022578 (k=13), A022579 (k=14), A022580 (k=15), A022581 (k=16), A022582 (k=17), A022583 (k=18), A022584 (k=19), A022585 (k=20), A022586 (k=21), A022587 (k=22), A022588 (k=23), A014103 (k=24), A022589 (k=25), A022590 (k=26), A022591 (k=27), A022592 (k=28), A022593 (k=29), A022594 (k=30), A022595 (k=31), A022596 (k=32), A025233 (k=48).

Column k=5 of A286352.

nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

x='x+O('x^50); Vec(prod(m=1, 50, (1 + x^m)^(-5))) \\ Indranil Ghosh, Apr 05 2017

a(n) ~ (-1)^n * 5^(1/4) * exp(Pi*sqrt(5*n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-5*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

cp's OEIS Frontend

A022597 Expansion of Product_{m >= 1} (1 + q^m)^(-2).

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A261616 Expansion of Product_{k>=0} 1/(1 - x^(3*k+1))^2.

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A263002 Expansion of (f(-x^5) / f(-x))^2 in powers of x where f() is a Ramanujan theta function.

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A285928 Expansion of (Product_{k>0} (1 - x^(5*k)) / (1 - x^k))^5 in powers of x.

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A358369 Euler transform of 2^floor(n/2), (A016116).

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A261615 Expansion of Product_{k>=0} (1 + x^(3*k+1))^2.

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A285927 Expansion of (Product_{k>0} (1 - x^(3*k)) / (1 - x^k))^3 in powers of x.

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A328547 Number of 3-regular bipartitions of n.

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