A097566 -id:A097566 - cp's OEIS frontend

A085261 Expansion of chi(x) / phi(x^2) in powers of x where phi(), chi() are Ramanujan theta functions.

1, 1, -2, -1, 5, 3, -9, -5, 18, 10, -30, -16, 53, 29, -85, -44, 139, 73, -215, -110, 335, 172, -502, -253, 755, 382, -1104, -550, 1614, 805, -2312, -1142, 3305, 1631, -4650, -2277, 6525, 3193, -9041, -4395, 12486, 6063, -17070, -8247, 23255, 11218, -31414, -15090, 42289, 20285

Offset: 0

Michael Somos, Jun 23 2003

sign

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

			G.f. = 1 + x - 2*x^2 - x^3 + 5*x^4 + 3*x^5 - 9*x^6 - 5*x^7 + 18*x^8 + 10*x^9 - ...
G.f. = 1/q + q^23 - 2*q^47 - q^71 + 5*q^95 + 3*q^119 - 9*q^143 - 5*q^167 + 18*q^191 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
George E. Andrews, On a Partition Function of Richard Stanley, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), #R1.
M. Ishikawa and J. Zeng, The Andrews-Stanley partition function and Al-Salam-Chihara polynomials, Disc. Math., 309 (2009), 151-175. (See t(n) p. 151. Note that there is a typo in the g.f. for f(n) - see A144558.) [Added by _N. J. A. Sloane_, Jan 25 2009.]
O. P. Lossers, Comparing Odd Parts in Conjugate Partitions: Solution 10969, Amer. Math. Monthly, 111 (Jun-Jul 2004), pp. 536-539.
Michael Somos, Introduction to Ramanujan theta functions
R. P. Stanley, Problem 10969, Amer. Math. Monthly, 109 (2002), 760.

Cf. A000041, A097566, A190101, A246712.

t1:=mul( (1+q^(2*n-1))/((1-q^(4*n))*(1+q^(4*n-2))^2), n=1..100): t2:=series(t1,q,100): f:=n->coeff(t2,q,n); # N. J. A. Sloane, Jan 25 2009

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / EllipticTheta[ 3, 0, x^2], {x, 0, n}]; (* Michael Somos, Jun 01 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8) QPochhammer[ -x^2]^2), {x, 0, n}]; (* Michael Somos, Sep 02 2014 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^8 + A)^2 / eta(x + A) / eta(x^4 + A)^6, n))};

{a(n) = polcoeff( prod( k=1,( n+1)\2, 1 + x^(2*k - 1), 1 + x * O(x^n)) / prod(k=1, (n+2)\4, (1 - x^(4*k)) * (1 + x^(4*k - 2))^2, 1 + x * O(x^n)), n)};

Expansion of psi(x) / f(x^2)^2 in powers of x where psi(), f() are Ramanujan theta functions. - Michael Somos, Sep 02 2014
Expansion of q^(1/24) * eta(q^2)^4 * eta(q^8)^2 / (eta(q) * eta(q^4)^6) in powers of q.
Euler transform of period 8 sequence [1, -3, 1, 3, 1, -3, 1, 1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 24^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246712. - Michael Somos, Sep 02 2014
G.f.: Product_{k>0} (1 + x^(2*k - 1)) / ((1 - x^(4*k)) * (1 + x^(4*k - 2))^2).

A190101 Number of transpose partition pairs of order n whose number of odd parts differ by numbers of the form 4*k + 2.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 5, 5, 1, 5, 18, 18, 6, 18, 55, 55, 23, 56, 150, 150, 73, 155, 376, 377, 205, 394, 885, 890, 526, 940, 1979, 1996, 1261, 2128, 4240, 4290, 2863, 4611, 8764, 8895, 6213, 9630, 17561, 17877, 12980, 19479, 34243, 34961, 26246, 38310, 65187

Offset: 0

Michael Somos, May 04 2011

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Denoted u(n)/2 by Lossers. t(n) is A097566(n). Stanley's f(n) is A085261(n). Partitions p(n) is A000041(n).
As noted in the solution the number of odd parts of a partition and its conjugate are of the same parity as n. Hence the difference in the number of odd parts must be even and if it is not divisible by 4 then it is of the form 4*k + 2 and the partition is not self conjugate.

			G.f. = x^2 + x^3 + x^5 + 5*x^6 + 5*x^7 + x^8 + 5*x^9 + 18*x^10 + 18*x^11 + ...
G.f. = q^47 + q^71 + q^119 + 5*q^143 + 5*q^167 + q^191 + 5*q^215 + ...
a(6) = 5 because ([6], [1,1,1,1,1,1]), ([5,1], [2,1,1,1,1]), ([4,2], [2,2,1,1]), ([4,1,1], [3,1,1,1]), ([3,3], [2,2,2]) are the 5 pairs of partitions of 6 where each partition and its transpose number of odd parts differ by 6, 2, 2, 2, 2 which are of the form 4*k + 2.

G. C. Greubel, Table of n, a(n) for n = 0..1000
O. P. Lossers, Comparing Odd Parts in Conjugate Partitions: Solution 10969, Amer. Math. Monthly, 111 (Jun-Jul 2004), pp. 536-539.
R. P. Stanley, Problem 10969, Amer. Math. Monthly, 109 (2002), 760.

Cf. A000041, A085261, A097566.

a[n_] := SeriesCoefficient[EllipticTheta[2, 0, q^8]/( 2*QPochhammer[q] * EllipticTheta[3, 0, q^2]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 18 2017 *)
nmax = 100; CoefficientList[Series[x^2 * Product[(1 + x^(4*k))^3 * (1 + x^(8*k)) * (1 + x^(16*k))^2 / ((1 + x^(2*k))^2 * (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 05 2025 *)

{a(n) = local(A); if( n<2, 0, n = n-2; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^8 + A) * eta(x^32 + A))^2 / (eta(x + A) * eta(x^4 + A)^5 * eta(x^16 + A)), n))};

Expansion of x^2 * psi(x^16) / (f(-x) * phi(x^2)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Expansion of q^(1/24) * (eta(q^2) * eta(q^8) * eta(q^32))^2 / (eta(q) * eta(q^4)^5 * eta(q^16)) in powers of q.
Euler transform of period 32 sequence [ 1, -1, 1, 4, 1, -1, 1, 2, 1, -1, 1, 4, 1, -1, 1, 3, 1, -1, 1, 4, 1, -1, 1, 2, 1, -1, 1, 4, 1, -1, 1, 1, ...].
p(n) = t(n) + u(n). f(n) = t(n) - u(n). u(n) = 2*a(n).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*n*sqrt(3)). - Vaclav Kotesovec, Jul 05 2025

A097567 T(n,k)= count of partitions p such that Abs( Odd(p)-Odd(p') ) = k, where p' is the transpose of p and Odd(p) counts the odd elements in p. Related to Stanley's 'f'.

Original entry on oeis.org

1, 1, 0, 0, 0, 2, 1, 0, 2, 0, 3, 0, 0, 0, 2, 3, 0, 2, 0, 2, 0, 1, 0, 8, 0, 0, 0, 2, 3, 0, 8, 0, 2, 0, 2, 0, 10, 0, 2, 0, 8, 0, 0, 0, 2, 10, 0, 8, 0, 8, 0, 2, 0, 2, 0, 4, 0, 26, 0, 2, 0, 8, 0, 0, 0, 2, 10, 0, 26, 0, 8, 0, 8, 0, 2, 0, 2, 0, 27, 0, 10, 0, 28, 0, 2, 0, 8, 0, 0, 0, 2, 27, 0, 26, 0, 28, 0, 8, 0, 8

Offset: 0

Wouter Meeussen, Aug 28 2004

Table starts {1}, {1,0}, {0,0,2}, {1,0,2,0}, {3,0,0,0,2}, .. where the odd columns are 0. Row sums are A000041 by definition.

George E. Andrews, On a Partition Function of Richard Stanley, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), Research Paper #R1.
Andrew V. Sills, A Combinatorial proof of a partition identity of Andrews and Stanley, International Journal of Mathematics and Mathematical Sciences, Volume 2004, Issue 47, Pages 2495-2501.

Cf. A097566.

Table[par=Partitions[n];Table[Count[par, q_/;Abs[Count[q, ?OddQ]-Count[TransposePartition[q], ?OddQ]]===k], {k, 0, n}], {n, 0, 16}]

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A085261 Expansion of chi(x) / phi(x^2) in powers of x where phi(), chi() are Ramanujan theta functions.

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A190101 Number of transpose partition pairs of order n whose number of odd parts differ by numbers of the form 4*k + 2.

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A097567 T(n,k)= count of partitions p such that Abs( Odd(p)-Odd(p') ) = k, where p' is the transpose of p and Odd(p) counts the odd elements in p. Related to Stanley's 'f'.

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