A226622 -id:A226622 - cp's OEIS frontend

A069910 Expansion of Product_{i in A069908} 1/(1 - x^i).

1, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 11, 12, 16, 18, 23, 26, 33, 37, 46, 52, 63, 72, 87, 98, 117, 133, 157, 178, 209, 236, 276, 312, 361, 408, 471, 530, 609, 686, 784, 881, 1004, 1126, 1279, 1433, 1621, 1814, 2048, 2286, 2574, 2871, 3223, 3590, 4022, 4472, 5000

Offset: 0

N. J. A. Sloane, May 05 2002

Number 39 of the 130 identities listed in Slater 1952.

Number of partitions of 2*n into distinct odd parts. - Vladeta Jovovic, May 08 2003

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + ...
G.f. = q^-1 + q^95 + q^143 + 2*q^191 + 2*q^239 + 3*q^287 + 3*q^335 + ...

M. D. Hirschhorn, The Power of q, Springer, 2017. Chapter 19, Exercises p. 173.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
G. E. Andrews et al., q-Engel series expansions and Slater's identities, Quaestiones Math., 24 (2001), 403-416.
George E. Andrews, Jethro van Ekeren and Reimundo Heluani, The singular support of the Ising model, arXiv:2005.10769 [math.QA], 2020. See (1.4.2) p. 2.
T. Gannon, G. Hoehn, H. Yamauchi, et. al., VOA Unitary Minimal Model m=1, character.
M. D. Hirschhorn, Some partition theorems of the Rogers-Ramanujan type, J. Combin. Theory Ser. A 27 (1979), no. 1, 33-37. MR0541341 (80j:05010). See Theorem 4. [From _N. J. A. Sloane_, Mar 19 2012]
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Lucy Joan Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., Series 2, vol.s2-54, no.2, pp. 147-167, (1952).
Eric Weisstein's World of Mathematics, Jackson-Slater Identity
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - _N. J. A. Sloane_, Dec 25 2012

Cf. A000700, A027356, A035457, A069908, A069909, A069911, A122129, A122135.

a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*[0$2, 1$4, 0$5, 1$4, 0][irem(d, 16)+1],
      d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

max = 56; p = Product[1/(1-x^i), {i, Select[Range[max], MemberQ[{2, 3, 4, 5, 11, 12, 13, 14}, Mod[#, 16]]&]}]; s = Series[p, {x, 0, max}]; a[n_] := Coefficient[s, x, n]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Apr 09 2014 *)
nmax=60; CoefficientList[Series[Product[(1-x^(8*k-1))*(1-x^(8*k-7))*(1-x^(8*k))*(1-x^(16*k-6))*(1-x^(16*k-10))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 04 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0 }[[ Mod[k, 16] + 1]], {k, n}], {x, 0, n}]; (* Michael Somos, Apr 14 2016 *)

{a(n) = my(A); if( n<0,0, n=2*n; A = x * O(x^n); polcoeff( eta(-x + A) / eta(x^2 + A), n))}; /* Michael Somos, Apr 11 2004 */

N=66;  q='q+O('q^N);  S=1+sqrtint(N);
gf=sum(n=0, S, q^(2*n^2) / prod(k=1, 2*n, 1-q^k ) );
Vec(gf)  \\ Joerg Arndt, Apr 01 2014

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^-[ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0][k%16 + 1]), n))}; /* Michael Somos, Apr 14 2016 */

Euler transform of period 16 sequence [0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, ...]. - Michael Somos, Apr 11 2004
G.f.: Sum_{n>=0} q^(2*n^2) / Product_{k=1..2*n} (1 - q^k). - Joerg Arndt, Apr 01 2014
a(n) ~ exp(sqrt(n/3)*Pi) / (2^(5/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 04 2015
Expansion of f(x^3, x^5) / f(-x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Apr 14 2016
a(n) = A000700(2*n).
a(n) = A027356(4n+1,2n+1). - Alois P. Heinz, Oct 28 2019
From Peter Bala, Feb 08 2021: (Start)
G.f.: A(x) = Product_{n >= 1} (1 + x^(4*n))^2*(1 + x^(4*n-2))*(1 + x^(8*n-3))*(1 + x^(8*n-5)).
The 2 X 2 matrix Product_{k >= 0} [1, x^(2*k+1); x^(2*k+1), 1] = [A(x^2), x*B(x^2); x*B(x)^2, A(x^2)], where B(x) is the g.f. of A069911.
A(x^2) + x*B(x^2) = A^2(-x) + x*B^2(-x) = Product_{k >= 0} 1 + x^(2*k+1), the g.f. of A000700.
A^2(x) + x*B^2(x) is the g.f. of A226622.
(A^2(x) + x*B^2(x))/(A^2(x) - x*B^2(x)) is the g.f. of A208850.
A^4(sqrt(x)) - x*B^4(sqrt(x)) is the g.f. of A029552.
A(x)*B(x) is the g.f. of A226635; A(-x)/B(-x) is the g.f. of A111374; B(-x)/A(-x) is the g.f. of A092869. (End)

A073252 Coefficients of replicable function number "48g".

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

Michael Somos, Jul 22 2002

Old name was: McKay-Thompson series of class 48g for the Monster group.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Combinatorial interpretation of sequence: [ X1, X2 ] = 2 strictly increasing sequences (possibly null) of odd positive integers; a(n) = #pairs with sum of entries = n.

			a(4) = 4: [ (1),(3) ],[ (3),(1) ],[ (),(1,3) ],[ (1,3),() ]
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 + 12*x^9 + ...
G.f. = 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..1000
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).
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. A000122, A000700, A010054, A022597, A121373, A226622, A226635.

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

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^2, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
QP = QPochhammer; s = (QP[q^2]^2 / (QP[q] * QP[q^4]))^2 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^2, {x, 0, n}]; (* Michael Somos, Nov 03 2019 *)

{a(n) = if( n<0, 0, polcoeff( prod( i=1, (1+n)\2, 1 + x^(2*i - 1), 1 + x * O(x^n))^2, n))};

{a(n) = if( n<0, 0, polcoeff( 1 / prod( i=1, n, 1 + (-x)^i, 1 + x * O(x^n))^2, n))};

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

from sage.modular.etaproducts import qexp_eta
m=80
def f(x): return qexp_eta(QQ[['q']], m+2).subs(q=x)
def A073252_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (f(x^2)^2/(f(x)*f(x^4)))^2 ).list()
A073252_list(m) # G. C. Greubel, Sep 07 2023

G.f.: 1 / (Prod_{k>0} 1 + (-x)^k)^2 = (Prod_{k>0} 1 + x^(2*k - 1))^2.
Expansion of q^(1/12) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^2 in powers of q.
Expansion of chi(q)^2 = phi(q) / f(-q^2) = f(q) / psi(-q) = (phi(q) / f(q))^2 = (psi(q) / f(-q^4))^2 = (f(-q^2) / psi(-q))^2 = (phi(-q^2) / f(-q))^2 = (f(q) / f(-q^2))^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [2, -2, 2, 0, ...].
Equals the convolution square of A000700.
a(n) = (-1)^n * A022597(n).
a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018
a(2*n) = A226622(n). a(2*n + 1) = 2 * A226635(n). - Michael Somos, Nov 03 2019

Comments from Len Smiley.

New name from Michael Somos, Nov 03 2019

A381899 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1} such that I(x) + W(x)*(n-W(x)) = k, where I(x) is the number of inversions in x and W(x) is the number of 1's in x, n >= 0, 0 <= k <= floor(n^2/2).

Original entry on oeis.org

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

Offset: 0

Geoffrey Critzer, Mar 09 2025

Sum_{k>=0} T(n,k)*2^k = A132186(n).
Sum_{k>=0} T(n,k)*3^k = A053846(n).
Sum_{k>=0} T(n,k)*q^k = the number of idempotent n X n matrices over GF(q).
It appears that if n is even the n-th row converges to 2,0,0,...,21,13,9,5,4,1,1 which is A226622 reversed, and if n is odd the sequence is twice A226635.
From Alois P. Heinz, Mar 09 2025: (Start)
Sum_{k>=0} k * T(n,k) = 3*A001788(n-1) for n>=1.
Sum_{k>=0} (-1)^k * T(n,k) = A060546(n). (End)

			Triangle T(n,k) begins:
  1;
  2;
  2, 1, 1;
  2, 0, 2, 2, 2;
  2, 0, 0, 2, 3, 3, 4, 1, 1;
  2, 0, 0, 0, 2, 2, 4, 4, 6, 4, 4, 2, 2;
  ...
T(4,5) = 3 because we have: {0, 1, 0, 0}, {0, 1, 0, 1}, {1, 1, 0, 1}.

Alois P. Heinz, Rows n = 0..50, flattened

Row sums give A000079.

Cf. A001788, A060546, A226635, A226622, A132186, A053846, A083906.

b:= proc(i, j) option remember; expand(`if`(i+j=0, 1,
     `if`(i=0, 0, b(i-1, j))+`if`(j=0, 0, b(i, j-1)*z^i)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(
         expand(add(b(n-j, j)*z^(j*(n-j)), j=0..n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 09 2025

nn = 7; B[n_] := FunctionExpand[QFactorial[n, q]]*q^Binomial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}]; Map[CoefficientList[#, q] &, Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^2, {z, 0, nn}],z]]

Sum_{n>=0} Sum_{k>=0} T(n,k)*q^k*x^n/(n_q!*q^binomial(n,2)) = e(x)^2 where e(x) = Sum_{n>=0} x^n/(n_q!*q^binomial(n,2)) where n_q! = Product{i=1..n} (q^n-1)/(q-1).

A366104 G.f. ( Chi(sqrt(x))^4 + Chi(-sqrt(x))^4 )/2, where Chi(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700.

Original entry on oeis.org

1, 6, 17, 38, 84, 172, 325, 594, 1049, 1796, 3005, 4912, 7877, 12430, 19309, 29580, 44766, 66978, 99150, 145374, 211242, 304382, 435194, 617674, 870651, 1219352, 1697283, 2348888, 3232919, 4426546, 6030872, 8177986, 11039633, 14838518, 19862613, 26482878, 35175989, 46552818, 61393694

Offset: 0

Peter Bala, Sep 29 2023

Compare with A224916 with g.f. ( Chi(sqrt(x))^4 - Chi(-sqrt(x))^4 )/(8*sqrt(x)),
A069910 with g.f. ( Chi(sqrt(x)) + Chi(-sqrt(x)) )/2,
A069911 with g.f. ( Chi(sqrt(x)) - Chi(-sqrt(x)) )/2,
A226622 with g.f. ( Chi(sqrt(x))^2 + Chi(-sqrt(x))^2 )/2 and
A226635 with g.f. ( Chi(sqrt(x))^2 - Chi(-sqrt(x))^2 )/(4*sqrt(x)),
Jacobi's "aequatio identica satis abstrusa" is the identity ( Chi(sqrt(x))^8 - Chi(-sqrt(x))^8 )/(16*sqrt(x)) =  Product_{k >= 1} (1 + x^k)^8.

Cf. A000700, A014705, A069910, A069911, A224916, A226622, A226635.

with(QDifferenceEquations):
 seq(coeff((1/2)*expand(QPochhammer(-q,q^2,40)^4 + QPochhammer(q,q^2,40)^4), q, 2*n), n = 0..40);
#alternative program
seq(coeff(expand(QPochhammer(-q^2, q^2, 20)^2 * QPochhammer(-q, q^2, 20)^6), q, n), n = 0..40);

nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k))^2 * (1 + x^(2*k-1))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 29 2025 *)

G.f.: Product_{k >= 1} (1 + x^(2*k))^2*(1 + x^(2*k-1))^6.
G.f.: x^(1/12) * eta(x^2)^10 * eta(x^4)^2 / ( eta(x) * eta(x^4) )^6.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 29 2025

cp's OEIS Frontend

A069910 Expansion of Product_{i in A069908} 1/(1 - x^i).

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A073252 Coefficients of replicable function number "48g".

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A381899 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1} such that I(x) + W(x)*(n-W(x)) = k, where I(x) is the number of inversions in x and W(x) is the number of 1's in x, n >= 0, 0 <= k <= floor(n^2/2).

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A366104 G.f. ( Chi(sqrt(x))^4 + Chi(-sqrt(x))^4 )/2, where Chi(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700.

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