A286352 -id:A286352 - cp's OEIS frontend

A286335 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^k.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 2, 0, 1, 5, 10, 13, 9, 3, 0, 1, 6, 15, 24, 24, 14, 4, 0, 1, 7, 21, 40, 51, 42, 22, 5, 0, 1, 8, 28, 62, 95, 100, 73, 32, 6, 0, 1, 9, 36, 91, 162, 206, 190, 120, 46, 8, 0, 1, 10, 45, 128, 259, 384, 425, 344, 192, 66, 10, 0

Offset: 0

Ilya Gutkovskiy, May 07 2017

A(n,k) is the number of partitions of n into distinct parts (or odd parts) with k types of each part.

			A(3,2) = 6 because we have [3], [3'], [2, 1], [2', 1], [2, 1'] and [2', 1'] (partitions of 3 into distinct parts with 2 types of each part).
Also A(3,2) = 6 because we have [3], [3'], [1, 1, 1], [1, 1, 1'], [1, 1', 1'] and [1', 1', 1'] (partitions of 3 into odd parts with 2 types of each part).
Square array begins:
  1,  1,  1,   1,   1,   1,  ...
  0,  1,  2,   3,   4,   5,  ...
  0,  1,  3,   6,  10,  15,  ...
  0,  2,  6,  13,  24,  40,  ...
  0,  2,  9,  24,  51,  95,  ...
  0,  3, 14,  42, 100, 206,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
N. J. A. Sloane, Transforms

Columns k=0-32 give: A000007, A000009, A022567-A022596.
Rows n=0-2 give: A000012, A001477, A000217.
Main diagonal gives A270913.
Antidiagonal sums give A299106.
Cf. A144064, A286352, A308680.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..n/i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 29 2019

Table[Function[k, SeriesCoefficient[Product[(1 + x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 + x^j)^k.

A(n,k) = Sum_{i=0..k} binomial(k,i) * A308680(n,k-i). - Alois P. Heinz, Aug 29 2019

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

Original entry on oeis.org

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

A299208 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

Original entry on oeis.org

1, 1, 0, -1, -2, -1, 1, 3, 3, 1, -3, -6, -5, 1, 9, 12, 5, -9, -20, -18, 1, 26, 38, 21, -21, -61, -62, -9, 72, 120, 81, -44, -177, -205, -64, 186, 366, 293, -63, -496, -657, -304, 445, 1084, 1014, 33, -1341, -2053, -1238, 959, 3132, 3378, 770, -3474, -6260, -4619, 1656, 8809, 10929, 4306, -8520

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Antidiagonal sums of A286352.
Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A318582, A331484.
Cf. A081362, A299108, A299162, A299164, A299166, A299167, A299209, A299210, A299211, A299212.

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

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

a(0) = 1; a(n) = Sum_{k=1..n} A081362(k-1)*a(n-k).

A007249 McKay-Thompson series of class 4D for the Monster group.

Original entry on oeis.org

1, -12, 66, -232, 639, -1596, 3774, -8328, 17283, -34520, 66882, -125568, 229244, -409236, 716412, -1231048, 2079237, -3459264, 5677832, -9200232, 14729592, -23325752, 36567222, -56778888, 87369483, -133315692

Offset: 0

N. J. A. Sloane

sign

The convolution square root of A007191, and also the left and right borders of the triangle A161196. - Gary W. Adamson, Jun 06 2009

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

			1 - 12*x + 66*x^2 - 232*x^3 + 639*x^4 - 1596*x^5 + 3774*x^6 + ...
T4D = 1/q - 12*q + 66*q^3 - 232*q^5 + 639*q^7 - 1596*q^9 + 3774*q^11 - ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group

Cf. A007191, A022577, A112142.

Column k=12 of A286352.

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m) / (m / 16 / q)^(1/2), {q, 0, n}]] (* Michael Somos, Jul 22 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(1/2) / (m / 16 / q), {q, 0, 2 n}]] (* Michael Somos, Jul 22 2011 *)
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
QP = QPochhammer; s = (QP[q]/QP[q^2])^12 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015, adapted from PARI *)
eta[q_]:=q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(1/2)*(eta[q]/eta[q^2])^12, {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Feb 13 2018 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^12, n))} /* Michael Somos, Jul 22 2011 */

G.f.: Product_{m>=1} (1 + x^m)^(-12).
Expansion of chi(-x)^12 in powers of x where chi() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 64 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022577. - Michael Somos, Jul 22 2011
a(n) = (-1)^n * A112142(n). (class 8B). Convolution inverse of A022577. - Michael Somos, Jul 22 2011
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (2^(5/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(12/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-12*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
Expansion of q^(1/2)*(eta(q)/eta(q^2))^12 in powers of q. - G. C. Greubel, Feb 13 2018

A007259 Expansion of Product_{m>=1} (1 + q^m)^(-8).

Original entry on oeis.org

1, -8, 28, -64, 134, -288, 568, -1024, 1809, -3152, 5316, -8704, 13990, -22208, 34696, -53248, 80724, -121240, 180068, -264448, 384940, -556064, 796760, -1132544, 1598789, -2243056, 3127360, -4333568, 5971922, -8188096, 11170160, -15163392, 20491033, -27572936

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 6F for the Monster group.

			1 - 8*x + 28*x^2 - 64*x^3 + 134*x^4 - 288*x^5 + 568*x^6 - 1024*x^7 + ...
T6F = 1/q - 8q^2 + 28q^5 - 64q^8 + 134q^11 - 288q^14 + 568q^17 + ...

T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 118, Problem 24.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group

Cf. A007263, A014705, A022573.

Column k=8 of A286352.

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^8, {q, 0, n}] (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - q^k, {k, 1, n, 2}]^8, {q, 0, n}] (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^-8, {q, 0, n}] (* Michael Somos, Jul 11 2011 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^8, n))}

Expansion of chi(-q)^8 in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Aug 18 2007
Expansion of q^(-1/3) * (eta(q) / eta(q^2))^8 in powers of q. - Michael Somos, Aug 18 2007
Euler transform of period 2 sequence [ -8, 0, ...]. - Michael Somos, Aug 18 2007
Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = v^2 - u^2 * v - 16 * u. - Michael Somos, Aug 18 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 16 / f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 18 2007
G.f.: Product_{k>0} (1 + x^k)^(-8).
a(2*n) = A014705(n). a(2*n + 1) = -8 * A022573(n). a(n) = A007263(3*n - 1).
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(8/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-8*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

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

sign

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

A022599 Expansion of Product_{m>=1} (1+q^m)^(-4).

Original entry on oeis.org

1, -4, 6, -8, 17, -28, 38, -56, 84, -124, 172, -232, 325, -448, 594, -784, 1049, -1388, 1796, -2320, 3005, -3864, 4912, -6216, 7877, -9940, 12430, -15488, 19309, -23972, 29580, -36408, 44766, -54876, 66978, -81536, 99150, -120272, 145374, -175344, 211242

Offset: 0

N. J. A. Sloane

sign

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

McKay-Thompson series of class 12J for the Monster group.

			G.f. = 1 - 4*x + 6*x^2 - 8*x^3 + 17*x^4 - 28*x^5 + 38*x^6 - 56*x^7 + ...
T12J = 1/q - 4*q^5 + 6*q^11 - 8*q^17 + 17*q^23 - 28*q^29 + 38*q^35 + ...

T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
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.
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group

Cf. A007259, A022597, A022569.

Column k=4 of A286352.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      -4*irem(d, 2)*d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 02 2014

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[x^2])^4, {x, 0, n}]; (* Michael Somos, Jul 05 2014 *)
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^4, {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))^4, n))};

Expansion of chi(-x)^4 in powers of x where chi() is a Ramanujan theta function.
Expansion of q^(1/6) * (eta(q) / eta(q^2))^4 in powers of q.
Euler transform of period 2 sequence [ -4, 0, ...]. - Michael Somos, Apr 26 2008
Given G.f. A(x) then B(q) = (A(q^6) / q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u * (16 + u * v) - v^2. - Michael Somos, Apr 26 2008
Given G.f. A(x) then B(q) = A(q^6) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 4 * v * (v + u^2) - w^2 * (v - u^2). - Michael Somos, Apr 26 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A022569.
Convolution inverse is A022569. Convolution square of A022597. Convolution square is A007259.
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (2 * 6^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(-4*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A302017 Expansion of 1/(1 - xProduct_{k>=1} (1 + x^(2k-1))).

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 21, 39, 73, 137, 257, 482, 903, 1693, 3173, 5948, 11149, 20899, 39174, 73430, 137641, 258002, 483614, 906513, 1699219, 3185111, 5970352, 11191163, 20977346, 39321116, 73705711, 138158128, 258971363, 485430483, 909918190, 1705601814, 3197075934, 5992778881, 11233201667

Offset: 0

Ilya Gutkovskiy, Mar 30 2018

nonn

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Self-Conjugate Partition

Antidiagonal sums of absolute values of A286352.

Cf. A000700, A299106, A299208.

nmax = 38; CoefficientList[Series[1/(1 - x Product[(1 + x^(2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 38; CoefficientList[Series[1/(1 - x QPochhammer[x^2]^2/(QPochhammer[x] QPochhammer[x^4])), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + (-x)^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A000700(k-1)*a(n-k).
a(n) ~ c / r^n, where r = 0.5334880525001986092393688937248506539793821912... is the root of the equation 1 + r - r^2 * QPochhammer(-1/r, r^2) = 0 and c = 0.48000092330632206397886602198643227268597451507794232644772186731542555975... = (2*(1 + r)*Log[r])/(2*(2 + r)*Log[r] + (1 + r)*Log[1 - r^2] + (1 + r) * QPolyGamma[Log[-1/r] / Log[r^2], r^2] + 4*r^4*Log[r] * Derivative[0,1][QPochhammer][-1/r, r^2]). - Vaclav Kotesovec, Mar 31 2018

A022601 Expansion of Product_{m>=1} (1+q^m)^(-6).

Original entry on oeis.org

1, -6, 15, -26, 51, -102, 172, -276, 453, -728, 1128, -1698, 2539, -3780, 5505, -7882, 11238, -15918, 22259, -30810, 42438, -58110, 78909, -106392, 142770, -190698, 253179, -334266, 439581, -575784, 750613, -974316, 1260336, -1624702, 2086530, -2670162

Offset: 0

N. J. A. Sloane

sign

McKay-Thompson series of class 8F for the Monster group.

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

			G.f. = 1 - 6*x + 15*x^2 - 26*x^3 + 51*x^4 - 102*x^5 + 172*x^6 - 276*x^7 + ...
T8F = 1/q - 6*q^3 + 15*q^7 - 26*q^11 + 51*q^15 - 102*q^19 + 172*q^23 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
Cf. A022571, A058088, A081362, A112145, A112150.
Column k=6 of A286352.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^6, {x, 0, n}]; (* Michael Somos, Jul 01 2014 *)
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^6, n))}; /* Michael Somos, Jul 01 2014 */

Expansion of chi(-x)^6 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Jul 01 2014
Expansion of q^(1/4) * 2 * k'(q) / k(q)^(1/2) in powers of q where k() is the elliptic modulus. - Michael Somos, Jul 01 2014
Expansion of q^(1/4) * (eta(q) / eta(q^2))^6 in powers of q. - Michael Somos, Jul 01 2014
Euler transform of period 2 sequence [ -6, 0, ...]. - Michael Somos, Jul 01 2014
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (21 + 6*u*v). - Michael Somos, Jul 01 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022571. - Michael Somos, Jul 01 2014
Convolution inverse of A022571. Convolution sixth power of A081362. - Michael Somos, Jul 01 2014
a(n) = (-1)^n * A112150(n) = A058088(2*n) = A112145(2*n). - Michael Somos, Jul 01 2014
a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(6/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-6*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

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

A286335 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^k.

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

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A299208 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

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A007249 McKay-Thompson series of class 4D for the Monster group.

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A007259 Expansion of Product_{m>=1} (1 + q^m)^(-8).

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

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A022599 Expansion of Product_{m>=1} (1+q^m)^(-4).

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A302017 Expansion of 1/(1 - xProduct_{k>=1} (1 + x^(2k-1))).