A286335 -id:A286335 - cp's OEIS frontend

A022567 Expansion of Product_{m>=1} (1+x^m)^2.

1, 2, 3, 6, 9, 14, 22, 32, 46, 66, 93, 128, 176, 238, 319, 426, 562, 736, 960, 1242, 1598, 2048, 2608, 3306, 4175, 5248, 6570, 8198, 10190, 12622, 15589, 19190, 23552, 28830, 35190, 42842, 52034, 63040, 76198, 91904, 110604, 132832, 159216, 190464, 227417

Offset: 0

N. J. A. Sloane, Jun 14 1998

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of partitions of n into distinct parts, with 2 types of each part. E.g., for n=4, we consider k and k* to be different versions of k and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*, thus a(4)=9. - Jon Perry, Apr 04 2004
Number of partitions of n into odd parts, each part being of two kinds. E.g., a(3)=6 because we have 3, 3', 1+1+1, 1+1+1', 1+1'+1', 1'+1'+1'. - Emeric Deutsch, Mar 22 2005
Euler transform of period 2 sequence [2,0,2,0,...]. - Emeric Deutsch, Mar 22 2005
Equals A000041 convolved with A010054. - Gary W. Adamson, Jun 11 2009
The sum of the least gaps in all partitions of n. The "least gap" of a partition is the least positive integer that is not a part of the partition. Example: a(4) = 9 because the least gaps in [4], [3,1], [2,2], [2,1,1], and [1,1,1,1] are 1, 2, 1, 3, and 2, respectively. - Emeric Deutsch, May 18 2015
Number of 2-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019
The least gap is also known as the minimal excludant or mex; see Andrews and Newman. - George Beck, Dec 10 2020

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 9*x^4 + 14*x^5 + 22*x^6 + 32*x^7 + 46*x^8 + ...
G.f. = q + 2*q^13 + 3*q^25 + 6*q^37 + 9*q^49 + 14*q^61 + 22*q^73 + 32*q^85 + ...

P. J. Grabner, A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
George E. Andrews, David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Cristina Ballantine, Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.
Helena Bergold, Lukas Egeling, and Hung. P. Hoang, Signotopes with few plus signs, arXiv:2411.19208 [math.CO], 2024. See p. 14.
J. Currie, N. Rampersad, Binary words avoiding xx^Rx and strongly unimodal sequences, JIS 18 (2015) #15.10.3.
Alejandro Erickson, Frank Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.
Alejandro Erickson and Mark Schurch, Monomer-dimer tatami tilings of square regions, arXiv preprint arXiv:1110.5103 [math.CO], 2011.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 852
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. 8.
Mircea Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See q(n)'.
Mbavhalelo Mulokwe and Konstantinos Zoubos, Free fermions, neutrality and modular transformations, arXiv:2403.08531 [hep-th], 2024.
Michael Somos, Introduction to Ramanujan theta functions
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A022597, A285221, A304048.
Cf. A010054. - Gary W. Adamson, Jun 11 2009
Column k=2 of A286335.
Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.

Coefficients(&*[(1+x^m)^2:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

A022567 := proc(n)
    local x,m;
    product((1+x^m)^2,m=1..n) ;
    expand(%) ;
    coeff(%,x,n) ;
end proc: # R. J. Mathar, Jun 18 2016

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^-2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
(QPochhammer[-1, x]^2/4 + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)
nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 2; poly[[3]] = 1; Do[Do[Do[poly[[j+1]] += poly[[j-k+1]], {j, nmax, k, -1}]; , {p, 1, 2}], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 14 2017 *)

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^2, n))}; /* Michael Somos, Mar 21 2004 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^2, n))}; /* Michael Somos, Jun 03 2005 */

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 0, 2)
a = EulerTransform(b)
print([a(n) for n in range(45)]) # Peter Luschny, Nov 11 2020

a(n) = p(n)+p(n-1)+p(n-3)+p(n-6)+...+p(n-k*(k+1)/2)+..., where p() is A000041(). E.g. a(8) = p(8)+p(7)+p(5)+p(2) = 22+15+7+2 = 46. - Vladeta Jovovic, Aug 09 2004
Expansion of q^(-1/12) * (eta(q^2) / eta(q))^2 in powers of q. - Michael Somos, Apr 27 2008
Expansion of chi(-q)^(-2) in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Apr 27 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022597. - Michael Somos, Apr 27 2008
G.f.: Product_{k>0} (1 + x^k)^2.
Convolution square of A000009. Convolution inverse of A022597. - Michael Somos, Apr 27 2008
Parity result: a(n) is even except when n is twice a generalized pentagonal number (i.e., of the form 2*A001318(m) for some m). - Peter Bala, Mar 19 2009
a(n) ~ exp(Pi * sqrt(2*n/3)) / (4 * 6^(1/4) * n^(3/4)) * (1 + (Pi/(12*sqrt(6)) - 3*sqrt(3/2)/(8*Pi)) / sqrt(n) + (Pi^2/1728 - 45/(256*Pi^2) - 5/64)/n). - Vaclav Kotesovec, Mar 05 2015, extended Jan 22 2017
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

A299106 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)).

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 41, 88, 189, 405, 869, 1864, 3998, 8575, 18392, 39448, 84610, 181475, 389235, 834848, 1790617, 3840591, 8237462, 17668057, 37895195, 81279216, 174331098, 373912708, 801983781, 1720128713, 3689404772, 7913191304, 16972547194, 36403436640

Offset: 0

Ilya Gutkovskiy, Feb 02 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000
N. J. A. Sloane, Transforms

Antidiagonal sums of A286335.

Cf. A000009, A067687, A299105, A299108.

nmax = 33; CoefficientList[Series[1/(1 - x Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[1/(1 - x/QPochhammer[x, x^2]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A000009(k-1)*a(n-k).
a(n) ~ c * d^n, where d = 2.14484226934608840026733598736202689102117985119507858808036465196716739... is the root of the equation QPochhammer(1/d, 1/d^2)*d = 1 and c = 0.4217892515709863296976217395517853732959704351198250451894928058439... = 2/(2+Derivative[0, 1][QPochhammer][-1, 1/d]/d^2). - Vaclav Kotesovec, Feb 03 2018, updated Mar 31 2018

A308680 Number T(n,k) of colored integer partitions of n such that all colors from a k-set are used and parts differ by size or by color; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 3, 8, 9, 4, 1, 0, 4, 14, 19, 14, 5, 1, 0, 5, 22, 39, 36, 20, 6, 1, 0, 6, 34, 72, 85, 60, 27, 7, 1, 0, 8, 50, 128, 180, 160, 92, 35, 8, 1, 0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1, 0, 12, 104, 354, 680, 845, 720, 434, 184, 54, 10, 1

Offset: 0

Alois P. Heinz, Aug 29 2019

For fixed k > 0, T(n,k) ~ exp(Pi*sqrt(k*n/3)) * k^(1/4) / (3^(1/4) * 2^((k+3)/2) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

T is the convolution triangle of A000009 (see A357368). - Peter Luschny, Oct 19 2022

			T(4,1) = 2: 3a1a, 4a.
T(4,2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
T(4,3) = 3: 2a1b1c, 2b1a1c, 2c1a1b.
T(4,4) = 1: 1a1b1c1d.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,   1;
  0,  2,  5,   3,   1;
  0,  3,  8,   9,   4,   1;
  0,  4, 14,  19,  14,   5,   1;
  0,  5, 22,  39,  36,  20,   6,   1;
  0,  6, 34,  72,  85,  60,  27,   7,  1;
  0,  8, 50, 128, 180, 160,  92,  35,  8, 1;
  0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Partition (number theory)

Columns k=0-10 give: A000007, A000009 (for n>0), A327380, A327381, A327382, A327383, A327384, A327385, A327386, A327387, A327388.
Main diagonal and lower diagonals give: A000012, A001477, A000096.
Row sums give A304969.
T(2n,n) gives A324595.
Cf. A060642, A286335, A325915.

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..min(k, n/i))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
T:= proc(n, k) option remember;
      `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, b(n)),
          (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Jan 31 2021
# Uses function PMatrix from A357368.
PMatrix(10, A000009); # Peter Luschny, Oct 19 2022

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t,    b[t, Min[t, i - 1], k]*Binomial[k, j]][n - i*j], {j, 0, Min[k, n/i]}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * binomial(k,i) * A286335(n,k-i).
Sum_{k=1..n} k * T(n,k) = A325915(n).
G.f. of column k: (-1 + Product_{j>=1} (1 + x^j))^k. - Alois P. Heinz, Jan 29 2021

A286352 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + x^j)^k.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 1, -1, 0, 1, -4, 3, -2, 1, 0, 1, -5, 6, -4, 4, -1, 0, 1, -6, 10, -8, 9, -4, 1, 0, 1, -7, 15, -15, 17, -12, 5, -1, 0, 1, -8, 21, -26, 30, -28, 15, -6, 2, 0, 1, -9, 28, -42, 51, -56, 38, -21, 9, -2, 0, 1, -10, 36, -64, 84

Offset: 0

Seiichi Manyama, May 08 2017

			Square array begins:
   1,  1,  1,  1,  1,   1, ...
   0, -1, -2, -3, -4,  -5, ...
   0,  0,  1,  3,  6,  10, ...
   0, -1, -2, -4, -8, -15, ...
   0,  1,  4,  9, 17,  30, ...

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

Columns k=0-32 give: A000007, A081362, A022597-A022627.
Main diagonal gives A255526.
Antidiagonal sums give A299208.
Cf. A286335.

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

A022577 Expansion of Product_{m>=1} (1+x^m)^12.

Original entry on oeis.org

1, 12, 78, 376, 1509, 5316, 16966, 50088, 138738, 364284, 913824, 2203368, 5130999, 11585208, 25444278, 54504160, 114133296, 234091152, 471062830, 931388232, 1811754522, 3471186596, 6556994502, 12222818640, 22502406793, 40944396120, 73680871326, 131211105208, 231355524048, 404110659732

Offset: 0

N. J. A. Sloane

nonn

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

			G.f. = 1 + 12*x + 78*x^2 + 376*x^3 + 1509*x^4 + 5316*x^5 + 16966*x^6 + ...
G.f. = q + 12*q^3 + 78*q^5 + 376*q^7 + 1509*q^9 + 5316*q^11 + 16966*q^13 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Column k=12 of A286335.

Cf. A000122, A000700, A007096, A007249, A010054, A121373, A124863.

Coefficients(&*[(1+x^m)^12:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018

N:= 50:
G:= mul(1+x^m,m=1..N+1)^12:
S:= series(G,x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Feb 26 2018

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16 / q)^(1/2) / (1 - m), {q, 0, n}]]; (* Michael Somos, Jul 22 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16 / q) /(1-m)^(1/2), {q, 0, 2 n}]]; (* Michael Somos, Jul 22 2011 *)
CoefficientList[QPochhammer[-1, q]^12/4096+O[q]^30, q] (* Jean-François Alcover, Nov 27 2015 *)
With[{nmax=50}, CoefficientList[Series[Product[(1+q^k)^12, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 25 2018 *)

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 + x^k, 1 + x * O(x^n))^12, n))}; /* Michael Somos, Jul 16 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^12, n))}; /* Michael Somos, Jul 16 2005 */

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^12)) \\ G. C. Greubel, Feb 25 2018

G.f.: Product_{k>=1} ( 1 + x^k )^12.
Expansion of chi(-x)^-12 in powers of x where chi() is a Ramanujan theta function.
Expansion of k^2 / (16 * q * k') in powers of q^2. - Michael Somos, Jul 22 2011
Expansion of q^(-1/2) * (k/4) / (1 - k^2) in powers of q. - Michael Somos, Jul 16 2005
Expansion of q^(-1/2) * (eta(q^2) / eta(q))^12 in powers of q. - Michael Somos, Jul 16 2005
Euler transform of period 2 sequence [12, 0, ...]. - Michael Somos, Jul 16 2005
Given g.f. A(x), then B(q) = (q * A(q^2))^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (4096*u*v + 48*u + 1)*v - u^2 . - Michael Somos, Jul 16 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/64 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007249. - Michael Somos, Jul 22 2011
A124863(n) = (-1)^n * a(n). A007096(4*n + 2) = 8 * a(n). Convolution inverse of A007249.
a(n) ~ exp(2 * Pi * sqrt(n)) / (128 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (12/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

More terms added by G. C. Greubel, Feb 25 2018

A022568 Expansion of Product_{m>=1} (1+x^m)^3.

Original entry on oeis.org

1, 3, 6, 13, 24, 42, 73, 120, 192, 302, 465, 702, 1046, 1536, 2226, 3195, 4536, 6378, 8896, 12306, 16896, 23045, 31224, 42048, 56310, 75000, 99384, 131072, 172071, 224910, 292774, 379608, 490338, 631104, 809472, 1034814, 1318707, 1675344, 2122176, 2680602, 3376728, 4242432, 5316562, 6646272

Offset: 0

N. J. A. Sloane

The g.f. Product_{m >= 1} (1 + x^m)^3 = Product_{m >= 1} (1 - x^m + 2*x^m)^3 == Product_{m >= 1} (1 - x^m)^3 == Sum_{m >= 0} (-1)^m*(2*m + 1)*q^(m*(m+1)/2) (mod 2) by an identity of Jacobi. It follows that a(n) is odd iff n = m*(m + 1)/2 for some nonnegative integer m. - Peter Bala, Jan 07 2025

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. 8.
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.

Cf. A000009, A001935.

Column k=3 of A286335.

Coefficients(&*[(1+x^m)^3:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

nmax=50; CoefficientList[Series[Product[(1+q^m)^3,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)
nmax = 50; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 3; poly[[3]] = 3; poly[[4]] = 1; Do[Do[Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];, {p, 1, 3}], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Mar 31 2018 *)

x='x+O('x^51); Vec(prod(m=1, 50, (1 + x^m)^3)) \\ Indranil Ghosh, Apr 03 2017

a(n) ~ exp(Pi * sqrt(n)) / (8 * n^(3/4)) * (1 + (Pi/16 - 3/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Mar 05 2015, extended Jan 16 2017
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
G.f.: Sum_{n >= 0} q^(n*(n+1)/2) / Sum_{n in Z} (-1)^n * q^(n^2). - Peter Bala, Jan 07 2025

A022569 Expansion of Product_{m>=1} (1+x^m)^4.

Original entry on oeis.org

1, 4, 10, 24, 51, 100, 190, 344, 601, 1024, 1702, 2768, 4422, 6948, 10752, 16424, 24782, 36972, 54602, 79872, 115805, 166540, 237664, 336720, 473856, 662596, 920934, 1272728, 1749407, 2392268, 3255410, 4409344, 5945730, 7983388, 10675712, 14220240, 18870672, 24951740, 32878114

Offset: 0

N. J. A. Sloane

nonn

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

			G.f. = 1 + 4*x + 10*x^2 + 24*x^3 + 51*x^4 + 100*x^5 + 190*x^6 + 344*x^7 + ...
G.f. = q + 4*q^7 + 10*q^13 + 24*q^19 + 51*q^25 + 100*q^31 + 190*q^37 + 344*q^43 + ...

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. 8.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A022599.

Column k=4 of A286335.

Coefficients(&*[(1+x^m)^4:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^-4, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^4, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^4, n))}; /* Michael Somos, Apr 26 2008 */

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^4)) \\ G. C. Greubel, Feb 26 2018

Expansion of q^(-1/6) * (eta(q^2) / eta(q))^4 in powers of q.
Expansion of chi(-q)^(-4) in powers of q where chi() is a Ramanujan theta function.
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) = v * (1 + 16 * u * v) - u^2. - Michael Somos, Apr 26 2008
Given G.f. A(x) then B(x) = A(q^6) * q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v * (u^2 - v) - 4 * w^2 * (u^2 + v). - Michael Somos, Apr 26 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022599.
G.f.: Product_{k>0} (1 + x^k)^4.
Convolution inverse of A022599.
G.f.: T(0)/x, where T(k) = 1 - 1/(1 - (1+(x)^(k+1))^4/((1+(x)^(k+1))^4 - 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 07 2013
a(n) ~ exp(2 * Pi * sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)) * (1 + (Pi/(6*sqrt(3)) - 3*sqrt(3)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Mar 05 2015, extended Jan 16 2017
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

A022571 Expansion of Product_{m>=1} (1+x^m)^6.

Original entry on oeis.org

1, 6, 21, 62, 162, 384, 855, 1806, 3648, 7110, 13434, 24702, 44361, 78006, 134592, 228302, 381300, 627840, 1020394, 1638528, 2601849, 4088780, 6363354, 9813504, 15005458, 22760262, 34261248, 51204222, 76005906, 112092438, 164296989, 239404860, 346898496, 499971968, 716906394

Offset: 0

N. J. A. Sloane

nonn

A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", New York, Gordon and Breach Science Publishers, 1986-1992, p. 755, Eq. 6.2.2.2. MR0874986 (88f:00013)

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

Cf. A000009.

Column k=6 of A286335.

Coefficients(&*[(1+x^m)^6:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

nmax=50; CoefficientList[Series[Product[(1+q^m)^6,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)

a(n)=if(n<0, 0, polcoeff( prod(k=1,n, 1+x^k, 1+x*O(x^n))^6, n)) /* Michael Somos, Jul 09 2005 */

Euler transform of period 2 sequence [6, 0, ...]. - Michael Somos, Jul 09 2005
Expansion of q^(-1/4)(eta(q^2)/eta(q))^6 in powers of q. - Michael Somos, Jul 09 2005
Expansion of q^(-1/4)(1/2)k^(1/2)/k' in powers of q. - Michael Somos, Jul 09 2005
Given g.f. A(x), then B(x)=(x*A(x^4))^4 satisfies 0=f(B(x), B(x^2)) where f(u, v)=(4096uv+48u+1)v-u^2 . - Michael Somos, Jul 09 2005
Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^3)) where f(u, v)=(u^2-v^2)^2 -uv(1+8uv)^2 . - Michael Somos, Jul 09 2005
G.f.: Product_{k>0} (1+x^k)^6.
a(n) ~ exp(Pi * sqrt(2*n)) / (16 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(6*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A022573 Expansion of Product_{m>=1} (1+x^m)^8.

Original entry on oeis.org

1, 8, 36, 128, 394, 1088, 2776, 6656, 15155, 33056, 69508, 141568, 280382, 541696, 1023512, 1895424, 3446617, 6163536, 10854400, 18846592, 32296742, 54673920, 91506000, 151523840, 248403014, 403396288, 649286724, 1036287744, 1640796160, 2578305024, 4022351720, 6232177664, 9592906446

Offset: 0

N. J. A. Sloane

nonn

			G.f. = 1 + 8*x + 36*x^2 + 128*x^3 + 394*x^4 + 1088*x^5 + 2776*x^6 + ...
G.f. = q + 8*q^4 + 36*q^7 + 128*q^10 + 394*q^13 + 1088*q^16 + 2776*q^19 + ...

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

Cf. A000009.

Column k=8 of A286335.

Coefficients(&*[(1+x^m)^8:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^8, {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)
With[{nmax=50}, CoefficientList[Series[Product[(1+q^k)^8, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 26 2018 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^8, n))}; /* Michael Somos, May 29 2004 */

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^8)) \\ G. C. Greubel, Feb 26 2018

G.f.: Product_{k>0} (1 + x^k)^8.
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^2 + v + 16*u*v^2. - Michael Somos, May 29 2004
Expansion of q^(-1/3) * (eta(q^2) / eta(q))^8 in powers of q.
Euler transform of period 2 sequence [ 8, 0, ...]. - Michael Somos, Jun 06 2005
Expansion of chi(-x)^-8 in powers of x where chi() is a Ramanujan theta function.
a(n) ~ exp(2 * Pi * sqrt(2*n/3)) / (16 * 2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (8/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(8*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A022596 Expansion of Product_{m>=1} (1+q^m)^32.

Original entry on oeis.org

1, 32, 528, 6016, 53384, 393920, 2517824, 14329600, 74059812, 352722720, 1565583648, 6533812352, 25823152256, 97218393280, 350348856704, 1213526698240, 4054279504266, 13103911398400, 41081428394096, 125210147216000, 371754750363712, 1077136199182976, 3050503922469440

Offset: 0

N. J. A. Sloane

nonn

In general, for k > 0, if g.f. = Product_{m>=1} (1+q^m)^k, then a(n) ~ k^(1/4) * exp(Pi * sqrt(k*n/3)) / (2^((k+3)/2) * 3^(1/4) * n^(3/4)) * (1 + (Pi*k^(3/2) / (48*sqrt(3)) - 3^(3/2) / (8*Pi*sqrt(k))) / sqrt(n)). - Vaclav Kotesovec, Mar 05 2015, extended Jan 16 2017

G. C. Greubel, Table of n, a(n) for n = 0..5000

Column k=32 of A286335.

Coefficients(&*[(1+x^m)^32:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Mar 20 2018

nmax=50; CoefficientList[Series[Product[(1+q^m)^32,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^32)) \\ G. C. Greubel, Mar 20 2018

a(n) ~ exp(Pi * 4 * sqrt(2*n/3)) / (65536 * 6^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015

Terms a(19) onward added by G. C. Greubel, Mar 20 2018

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