A132970 -id:A132970 - cp's OEIS frontend

A085140 Expansion of q^(-1/6) * eta(q^2)^3 / eta(q)^2 in powers of q.

1, 2, 2, 4, 5, 6, 10, 12, 15, 20, 26, 32, 40, 50, 60, 76, 92, 110, 134, 160, 191, 230, 272, 320, 380, 446, 522, 612, 715, 830, 966, 1120, 1292, 1494, 1720, 1976, 2272, 2602, 2974, 3400, 3876, 4412, 5020, 5700, 6460, 7322, 8282, 9352, 10559, 11900, 13396

Offset: 0

Michael Somos, Jun 20 2003

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
In the notation of Dragonette on page 498 Lemma 6, the generating function is G_2(q^(1/2))/2.
Equals A000009 convolved with A010054. [Gary W. Adamson, Mar 16 2010]

			G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 12*x^7 + 15*x^8 + ...
G.f. = q + 2*q^7 + 2*q^13 + 4*q^19 + 5*q^25 + 6*q^31 + 10*q^37 + 12*q^43 + 15*q^49 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
L. A. Dragonette, Some Asymptotic Formulae for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500.
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. 16.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A010054. [Gary W. Adamson, Mar 16 2010]

Cf. A053254, A132970.

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k) * (1 + x^k)^3, {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 2, n, 2}] / Product[ 1 - x^k, {k, 1, n, 2}]^2, {x, 0, n}];
a[ n_] := With[ {t = Log[q]/(2 Pi I)}, SeriesCoefficient[ q^(-1/6) DedekindEta[ 2 t]^3 / DedekindEta[ t]^2, {q, 0, n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ -x, x]^3, {x, 0, n}]; (* Michael Somos, Jul 11 2015 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(j j + j) / Product[ 1 + x^k, {k, 1, 2 j + 1, 2}], {j, 0, Sqrt[8 n + 1]/2}], {x, 0, 2 n}]]; (* Michael Somos, Jul 11 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / eta(x + A)^2, n))};

Expansion of psi(x) / chi(-x) = f(-x^2) / chi(-x)^2 = f(-x) / chi(-x)^3 = phi(-x) / chi(-x)^4 = phi(x) / chi(-x^2)^2 = f(-x^2)^2 / phi(-x) = f(-x)^4 / phi(-x)^3 = psi(x)^2 / f(-x^2) = chi(x)^2 * psi(x^2) = f(-x^2)^3 / f(-x)^2 in powers of x where f(), phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Feb 18 2006
Euler transform of period 2 sequence [ 2, -1, ...].
G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k - 1))^2 = Product_{k>0} (1 - x^k) * (1 + x^k)^3.
a(n) = b(n)+b(n-1)+b(n-3)+b(n-6)+...+b(n-k*(k+1)/2)+..., where b() is A000009(). E.g. a(8) = b(8)+b(7)+b(5)+b(2) = 6+5+3+1 = 15. - Vladeta Jovovic, Aug 18 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = (3/4)^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132970. - Michael Somos, Jul 11 2015
a(n) = A053254(2*n). - Michael Somos, Jul 11 2015
a(n) ~ exp(Pi*sqrt(n/3))/(4*sqrt(n)). - Vaclav Kotesovec, Sep 07 2015

A132969 Expansion of phi(q) * chi(q) in powers of q where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 3, 2, 1, 5, 5, 3, 5, 6, 10, 10, 8, 13, 15, 15, 16, 23, 27, 25, 30, 35, 40, 42, 45, 55, 66, 68, 70, 86, 95, 100, 110, 125, 141, 150, 161, 185, 207, 215, 235, 266, 293, 310, 335, 375, 410, 438, 470, 521, 575, 610, 653, 725, 785, 835, 900, 983, 1070, 1140

Offset: 0

Michael Somos, Sep 04 2007

nonn

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

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

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; top of p. 60.

G. C. Greubel, Table of n, a(n) for n = 0..1000
George E. Andrews, Integer partitions with even parts below odd parts and the mock theta functions, to appear in Annals of Combinatorics, 2017.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A053250, A053251, A124226, A132970.

nmax = 60; CoefficientList[Series[Product[(1 - x^(2*k)) * ( (1 + x^k) / (1 + x^(2*k)) )^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A) * eta(x^4 + A))^3, n))};

Expansion of phi(q) + 2 * psi(q) in powers of q where phi(), psi() are Ramanujan 3rd order mock theta functions.
Expansion of q^(1/24) * eta(q^2)^7 / (eta(q) * eta(q^4))^3 in powers of q.
Euler transform of period 4 sequence [ 3, -4, 3, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 48^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t).
G.f.: ( Sum_{k in Z} x^k^2 ) * ( Product_{k>0} (1 + x^(2*k-1)) ).
G.f.: Product_{k>0} (1 - x^(2*k)) * ((1 + x^k) / (1 + x^(2*k)))^3.
a(n) = (-1)^n * A132970(n). a(n) = (-1)^n * A124226(n) unless n=1.
a(n) ~ exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - Vaclav Kotesovec, Sep 08 2015

A124226 Number of partitions of n with even crank minus number of partitions of n with odd crank.

Original entry on oeis.org

1, -1, 2, -1, 5, -5, 3, -5, 6, -10, 10, -8, 13, -15, 15, -16, 23, -27, 25, -30, 35, -40, 42, -45, 55, -66, 68, -70, 86, -95, 100, -110, 125, -141, 150, -161, 185, -207, 215, -235, 266, -293, 310, -335, 375, -410, 438, -470, 521, -575, 610, -653, 725, -785, 835, -900, 983, -1070, 1140, -1220, 1331, -1443, 1532

Offset: 0

Vladeta Jovovic, Oct 20 2006

For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).

			G.f. = 1 - x + 2*x^2 - x^3 + 5*x^4 - 5*x^5 + 3*x^6 - 5*x^7 + 6*x^8 + ...

Cf. A124227, A124228, A132970.

p:=2*q + product((1-q^i)/(1+q^i)^2, i=1..200): s:=series(p, q, 200): for j from 0 to 199 do printf(`%d,`,coeff(s,q, j)) od: # James Sellers, Nov 30 2006

G.f.: 2*x + Product_{i>=1} (1-x^i)/(1+x^i)^2.
a(n) = A132970(n) unless n=1. - Michael Somos, Jul 27 2015
a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - Vaclav Kotesovec, Oct 14 2017

More terms from James Sellers, Nov 30 2006

A124227 Number of partitions of n with even crank.

Original entry on oeis.org

1, 0, 2, 1, 5, 1, 7, 5, 14, 10, 26, 24, 45, 43, 75, 80, 127, 135, 205, 230, 331, 376, 522, 605, 815, 946, 1252, 1470, 1902, 2235, 2852, 3366, 4237, 5001, 6230, 7361, 9081, 10715, 13115, 15475, 18802, 22145, 26742, 31463, 37775, 44362, 52998, 62142

Offset: 0

Vladeta Jovovic, Oct 20 2006

For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).

Cf. A000041, A124226, A124228.

A000041 := proc(n) combinat[numbpart](n) ; end: A124226 := proc(n) local x,gf,i ; gf := 1; for i from 1 to n+1 do gf := taylor(gf*(1-x^i)/(1+x^i)^2,x=0,n+1) ; od ; coeftayl(2*x+gf,x=0,n) ; end: A124227 := proc(n) (A000041(n)+A124226(n))/2 ; end: for n from 0 to 60 do printf("%a, ",A124227(n)) ; od ; # R. J. Mathar, May 18 2007

A132970[n_] := SeriesCoefficient[EllipticTheta[4, 0, x] QPochhammer[x, x^2], {x, 0, n}];
a[n_] := If[n == 1, 0, (PartitionsP[n] + A132970[n])/2];
Table[a[n], {n, 0, 47}] (* Jean-François Alcover, Oct 26 2023, after Michael Somos in A124226 *)

a(n) = (A000041(n) + A124226(n))/2.

More terms from R. J. Mathar, May 18 2007

A124228 Number of partitions of n with odd crank.

Original entry on oeis.org

0, 1, 0, 2, 0, 6, 4, 10, 8, 20, 16, 32, 32, 58, 60, 96, 104, 162, 180, 260, 296, 416, 480, 650, 760, 1012, 1184, 1540, 1816, 2330, 2752, 3476, 4112, 5142, 6080, 7522, 8896, 10922, 12900, 15710, 18536, 22438, 26432, 31798, 37400, 44772, 52560, 62612

Offset: 0

Vladeta Jovovic, Oct 20 2006

For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).

Cf. A000041, A124226, A124227.

A000041 := proc(n) combinat[numbpart](n) ; end: A124226 := proc(n) local x,gf,i ; gf := 1; for i from 1 to n+1 do gf := taylor(gf*(1-x^i)/(1+x^i)^2,x=0,n+1) ; od ; coeftayl(2*x+gf,x=0,n) ; end: A124228 := proc(n) (A000041(n)-A124226(n))/2 ; end: for n from 0 to 60 do printf("%a, ",A124228(n)) ; od ; # R. J. Mathar, May 18 2007

A132970[n_] := SeriesCoefficient[EllipticTheta[4, 0, x] QPochhammer[x, x^2], {x, 0, n}];
a[n_] := If[n < 2, n, (PartitionsP[n] - A132970[n])/2];
Table[a[n], {n, 0, 47}] (* Jean-François Alcover, Oct 26 2023, after Michael Somos in A124226 *)

a(n) = (A000041(n)-A124226(n))/2.

More terms from R. J. Mathar, May 18 2007

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A085140 Expansion of q^(-1/6) * eta(q^2)^3 / eta(q)^2 in powers of q.

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A132969 Expansion of phi(q) * chi(q) in powers of q where phi(), chi() are Ramanujan theta functions.

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A124226 Number of partitions of n with even crank minus number of partitions of n with odd crank.

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A124227 Number of partitions of n with even crank.

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A124228 Number of partitions of n with odd crank.

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