A073252 -id:A073252 - 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

A338463 Expansion of g.f.: (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^2.

Original entry on oeis.org

1, 0, 2, 2, 3, 4, 5, 8, 9, 12, 15, 20, 23, 28, 36, 44, 52, 62, 76, 90, 106, 124, 149, 176, 203, 236, 279, 324, 372, 430, 499, 576, 657, 752, 867, 992, 1124, 1280, 1463, 1662, 1876, 2124, 2410, 2722, 3061, 3446, 3889, 4374, 4896, 5490, 6166, 6900, 7700, 8600

Offset: 2

Ilya Gutkovskiy, Jan 31 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1002

Cf. A000700, A022597, A047654, A048574, A073252, A327380, A338223.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( (-1 + (&*[1+x^(2*j+1): j in [0..m+2]]) )^2 )); // G. C. Greubel, Sep 07 2023

nmax = 55; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
With[{k=2}, Drop[CoefficientList[Series[(2/QPochhammer[-1,-x] -1)^k, {x,0,80}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

m=80
def f(x): return (-1 + product(1+x^(2*j-1) for j in range(1,m+3)) )^2
def A338463_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
a=A338463_list(m); a[2:] # G. C. Greubel, Sep 07 2023

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^2.
a(n) = Sum_{k=1..n-1} A000700(k) * A000700(n-k).
a(n) = A073252(n) - 2 * A000700(n) for n > 0.
a(n) = [x^n]( (2/QPochhammer(-1,-x) - 1)^2 ). - G. C. Greubel, Sep 07 2023

A226635 Expansion of psi(x^4) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 13, 18, 27, 37, 53, 71, 100, 132, 179, 235, 313, 405, 531, 681, 880, 1119, 1429, 1801, 2280, 2852, 3575, 4444, 5529, 6827, 8436, 10357, 12716, 15530, 18958, 23036, 27978, 33839, 40896, 49254, 59265, 71083, 85180, 101781, 121494, 144659

Offset: 0

Michael Somos, Aug 31 2013

nonn

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

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 13*x^6 + 18*x^7 + 27*x^8 + 37*x^9 + ...
G.f. = q^11 + q^35 + 2*q^59 + 3*q^83 + 6*q^107 + 8*q^131 + 13*q^155 + 18*q^179 + ...

G. C. Greubel, 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

Cf. A022597, A073252.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^2] / (2 q^(1/2) QPochhammer[ q]), {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^8 + A)^2 / (eta(x + A) * eta(x^4 + A)), n))};

Expansion of q^(-11/24) * eta(q^8)^2 / (eta(q) * eta(q^4)) in powers of q.
Euler transform of period 8 sequence [1, 1, 1, 2, 1, 1, 1, 0, ...].
G.f.: (Sum_{k>=1} x^(2*k*(k-1))) / (Product_{k>=1} (1 - x^k)).
2 * a(n) = A073252(2*n + 1).  -2 * a(n) = A022597(2*n + 1).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(13/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
Expansion of (chi(q)^2 - chi(-q)^2)/(4*q) in powers of q^2 where chi() is a Ramanujan theta function. - Michael Somos, Nov 02 2019

A226622 Expansion of phi(x^2) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 4, 5, 9, 13, 21, 29, 46, 62, 90, 122, 171, 227, 311, 408, 545, 709, 933, 1198, 1555, 1980, 2536, 3205, 4063, 5092, 6400, 7966, 9928, 12281, 15198, 18684, 22979, 28097, 34346, 41789, 50813, 61527, 74453, 89757, 108114, 129809, 155704, 186221, 222503

Offset: 0

Michael Somos, Aug 31 2013

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

			1 + x + 4*x^2 + 5*x^3 + 9*x^4 + 13*x^5 + 21*x^6 + 29*x^7 + 46*x^8 + 62*x^9 + ...
1/q + q^23 + 4*q^47 + 5*q^71 + 9*q^95 + 13*q^119 + 21*q^143 + 29*q^167 + ...

G. C. Greubel, 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

Cf. A000700, A022597, A073252, A226635.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] / QPochhammer[ q], {q, 0, n}]

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^8 + A)^2), n))}

Expansion of q^(1/24) * eta(q^4)^5 / (eta(q) * eta(q^2)^2 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ 1, 3, 1, -2, 1, 3, 1, 0, ...].
G.f.: (Sum_{k in Z} x^(2*k^2)) / (Product_{k>0} (1 - x^k)).
a(n) = A022597(2*n) = A073252(2*n).
G.f. A(x) satisfies A(x^2) = ( chi(x)^2 + chi(-x)^2 )/2, where chi(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700. Cf. A226635. - Peter Bala, Sep 29 2023
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 29 2025

A382345 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where n unlabeled objects are distributed into k containers of two kinds. Containers may be left empty.

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 4, 2, 0, 5, 6, 7, 2, 0, 6, 8, 12, 8, 2, 0, 7, 10, 17, 18, 11, 2, 0, 8, 12, 22, 28, 26, 12, 2, 0, 9, 14, 27, 38, 46, 34, 15, 2, 0, 10, 16, 32, 48, 66, 64, 46, 16, 2, 0, 11, 18, 37, 58, 86, 100, 94, 56, 19, 2, 0, 12, 20, 42, 68, 106, 136, 152, 124, 70, 20, 2, 0

Offset: 0

Peter Dolland, Mar 29 2025

			Array starts:
 0 : [1, 2,  3,   4,   5,   6,   7,    8,    9,   10,   11]
 1 : [0, 2,  4,   6,   8,  10,  12,   14,   16,   18,   20]
 2 : [0, 2,  7,  12,  17,  22,  27,   32,   37,   42,   47]
 3 : [0, 2,  8,  18,  28,  38,  48,   58,   68,   78,   88]
 4 : [0, 2, 11,  26,  46,  66,  86,  106,  126,  146,  166]
 5 : [0, 2, 12,  34,  64, 100, 136,  172,  208,  244,  280]
 6 : [0, 2, 15,  46,  94, 152, 217,  282,  347,  412,  477]
 7 : [0, 2, 16,  56, 124, 214, 316,  426,  536,  646,  756]
 8 : [0, 2, 19,  70, 167, 302, 464,  640,  825, 1010, 1195]
 9 : [0, 2, 20,  84, 212, 406, 648,  922, 1212, 1512, 1812]
10 : [0, 2, 23, 100, 271, 542, 899, 1314, 1766, 2236, 2717]
...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Antidiagonal sums give A000712.
Alternating antidiagonal sums give A073252.
Without empty containers: A381895.
Cf. A382342.

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(x^j*b(n-i*j, min(n-i*j, i-1))*(j+1), j=0..n/i))))
    end:
A:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(A(n, d-n), n=0..d), d=0..11);  # Alois P. Heinz, Mar 29 2025

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
   Sum[x^j*b[n - i*j, Min[n - i*j, i - 1]]*(j + 1), {j, 0, n/i}]]]];
A[n_, k_] := Coefficient[b[n + k, n + k], x, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Apr 07 2025, after Alois P. Heinz *)

from sympy.utilities.iterables import partitions
def a_row(n, length=11) -> list[int]:
    if n == 0 : return list(range(1, length + 1))
    t = [0] * length
    for p in partitions(n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= p[k] + 1
        if s > 0 :
            t[s] += fact
    for i in range(1, length - 1):
        t[i+1] += t[i] * 2 - t[i-1]
    return t
for n in range(11): print(a_row(n))

A(0,k) = k + 1.
A(1,k) = 2*k.
A(2,k+1) = 2 + 5 * k.
A(n,1) = 2.
A(n,k) = Sum_{i=0..k} (k + 1 - i) * A382342(n,i) for k <= n.
A(n,n+k) = A(n,n) + k * A000712(n).
A(n,k) = A382342(n,k) + 2 * A(n,k-1) - A(n,k-2) for 2 <= k <= n.
A(n,k) = A382342(n+k,k). - Alois P. Heinz, Mar 31 2025

A304626 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(n*k)))^n.

Original entry on oeis.org

1, 0, 1, 10, 47, 201, 849, 3578, 15147, 64516, 276268, 1188342, 5130987, 22226036, 96543989, 420368843, 1834203939, 8018057328, 35107961157, 153950675566, 675978772306, 2971700764920, 13078268135661, 57613905606250, 254038914924767, 1121081799217206, 4951199308679965

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Number of partitions of n into 2 or more distinct parts, with n types of each part. - Ilya Gutkovskiy, May 16 2018

Index entries for sequences related to partitions

Cf. A058576, A073252, A111133, A255526, A270913, A304625.

Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(n k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[(1 + x^k)^n, {k, 1, n - 1}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[(QPochhammer[-1, x, 1 + n]/QPochhammer[-1, x^n, 1 + n])^n, {x, 0, n}], {n, 0, 26}]

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.502476747617354487738... and c = 0.2605422331424384694... - Vaclav Kotesovec, May 16 2018

A073253 Table of expansion of Product (1+(xy)^n/y)(1+(xy)^n/x), n>0 by antidiagonals.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 2, 5, 2, 0, 0, 0, 0, 0, 0, 1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0, 3, 7, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 7, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 11, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 11, 11, 2, 0

Offset: 0

Michael Somos, Jul 23 2002

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Combinatorial interpretation is number of partitions of Gaussian integer n+ki into distinct parts of form a+(a-1)i and (b-1)+bi, a,b>0.
Jacobi triple product identity implies the g.f. equals the Ramanujan theta function divided by Product (1-(xy)^m), m>0.

			{1}; {1, 1}; {0, 1 ,0}; {0, 1, 1, 0}; {0, 1, 2, 1, 0}; {0, 0, 2, 2, 0, 0}; {0, 0, 1, 3, 1, 0, 0}; ...

J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge Univ. Press, 1992. p. 141.

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for Gaussian integers and primes

A073252 gives antidiagonal sums.

{T(n, k) = if( n<0 || k<0, 0, polcoeff( polcoeff( prod( i=1, max(n, k), (1 + x^i * y^(i-1)) * (1 + x^(i-1) *y^i), 1 + x * O(x^n) + y * O(y^k)), n), k))}

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

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A338463 Expansion of g.f.: (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^2.

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A226635 Expansion of psi(x^4) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

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A226622 Expansion of phi(x^2) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

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A382345 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where n unlabeled objects are distributed into k containers of two kinds. Containers may be left empty.

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A304626 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(n*k)))^n.

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A073253 Table of expansion of Product (1+(xy)^n/y)(1+(xy)^n/x), n>0 by antidiagonals.

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