A007325 -id:A007325 - cp's OEIS frontend

A327688 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A007325.

1, -1, 0, 0, -1, 0, 1, 0, 0, 0, 0, -2, 2, 1, 0, 1, -1, -1, -1, -1, 2, 1, 0, 1, -1, -3, 1, 2, -1, 0, 4, -6, -2, 3, -1, 1, 4, -1, -2, -1, 2, -4, 4, 0, -3, 1, -3, 4, 2, -1, 3, -1, -3, -1, 2, -3, 1, 2, -6, -3, 12, -7, 3, 11, -7, -4, 7, -10, -1, 7, 2, -16, 11, 2, -10, 14, -4, 3, -3

Offset: 0

Seiichi Manyama, Sep 22 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.

Cf. A007325, A035187, A307757, A327686, A327690, A327691, A327716.

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^sumdiv(k, d, kronecker(5, d))))

G.f.: Product_{i>=1} Product_{j>=1} (1-x^(i*(5*j-1))) * (1-x^(i*(5*j-4))) / ((1-x^(i*(5*j-2))) * (1-x^(i*(5*j-3)))).

G.f.: Product_{k>=1} (1-x^k)^A035187(k).

A167683 Hankel transform of A007325.

Original entry on oeis.org

1, 0, -1, 1, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 0

Paul Barry, Nov 09 2009

sign

Nonzero elements appear to be indexed by abs(A131723). -1 elements appear to be indexed by 2*A014493(n).

A258040 Expansion of f(x) / f(-x) in powers of x where f() is the g.f. for A007325.

Original entry on oeis.org

1, -2, 2, 0, -2, 2, 0, 0, -2, 2, 2, -8, 8, 0, -8, 8, -2, 0, -6, 8, 6, -24, 24, 0, -24, 22, -4, 0, -16, 20, 16, -64, 62, 0, -60, 56, -10, 0, -40, 48, 38, -148, 144, 0, -136, 126, -24, 0, -88, 106, 82, -320, 308, 0, -288, 264, -48, 0, -180, 216, 168, -652, 624

Offset: 0

Michael Somos, May 16 2015

sign

			G.f. = 1 - 2*x + 2*x^2 - 2*x^4 + 2*x^5 - 2*x^8 + 2*x^9 + 2*x^10 - 8*x^11 + ...

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. A007325.

a[ n_] := SeriesCoefficient[ Product[(1 - x^k)^{ 2, -1, -2, 0, 0, 1, -2, 0, 2, 0, 2, 0, -2, 1, 0, 0, -2, -1, 2, 0}[[ Mod[k, 20, 1]]], {k, 1, n}], {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^ [ 0, 2, -1, -2, 0, 0, 1, -2, 0, 2, 0, 2, 0, -2, 1, 0, 0, -2, -1, 2][k%20 + 1]) ,n))};

Expansion of f(-x, -x^4) * f(-x^2, +x^3) / (f(+x, -x^4) * f(-x^2, -x^3)) = f(-x, -x^9) * f(+x^3, +x^7) / (f(+x, +x^9) * f(-x^3, -x^7)) in powers of x where f(,) is the Ramanujan general theta function.
Euler transform of period 20 sequence [ -2, 1, 2, 0, 0, -1, 2, 0, -2, 0, -2, 0, 2, -1, 0, 0, 2, 1, -2, 0, ...].
a(10*n + 3) = a(10*n + 7) = 0.

A092872 Expansion of r(q^9) / (r(q) r(q^3)) in powers of q where r() is the Rogers-Ramanujan continued fraction function (A007325).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, -1, -1, -1, -1, 0, 0, -2, -1, 2, 3, 0, -1, 2, 3, 0, -1, -1, -2, -2, 0, 0, -4, -3, 5, 7, 0, -2, 4, 6, 0, -4, -3, -5, -6, 0, 0, -8, -6, 10, 14, 0, -5, 9, 13, 0, -7, -5, -9, -10, 0, 0, -16, -12, 20, 28, 0, -8, 17, 24, 0, -14, -11, -18, -20, 0, 0, -30, -21, 36, 50, 0, -16, 30, 44, 0, -23, -18, -31, -34, 0, 0, -52, -38, 63

Offset: 1

Michael Somos, Mar 09 2004, Dec 11 2008

sign

Euler transform of period 45 sequence [1, -1, 0, 1, 0, 0, -1, -1, -1, 0, 1, 0, -1, 1, 0, 1, -1, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, -1, 1, 0, 1, -1, 0, 1, 0, -1, -1, -1, 0, 0, 1, 0, -1, 1, 0,...].

			q + q^2 + q^5 + q^6 - q^8 - q^9 - q^10 - q^11 - 2*q^14 - q^15 + 2*q^16 + ...

{a(n)=if(n<1, 0, n--; polcoeff( prod(k=1, n, (1-x^k+x*O(x^n))^-(kronecker(5, k\9-k%9)*if(k%9==0,-1, k%3>0))), n))}

{a(n)=local(A, u, v); if(n<0, 0, A=x; for(k=2, n, u=A+x*O(x^k); v=subst(u, x, x^2); A-=x^k*polcoeff(u^2-v+u*v^3+u^3*v^2-2*u*v*(1-u+v+u*v), k+1)/2); polcoeff(A, n))}

G.f. A(x) satisfies 0=f(A(x),A(x^2)) where f(u,v) = u^2 - v + u*v^3 + u^3*v^2 - 2*u*v*(1 - u + v + u*v).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w + 2*u*w + 2*u*v*w - u*w^2 - u^2*v -u*v^2*w. - Michael Somos, Dec 11 2008

A295703 Expansion of R(x*R(x)), where R(x) = 1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))), a continued fraction (g.f. for A007325).

Original entry on oeis.org

1, -1, 2, -3, 2, 4, -18, 43, -80, 123, -148, 78, 287, -1364, 3858, -8627, 15901, -23076, 20061, 18294, -140623, 420241, -930040, 1655753, -2293975, 1872682, 1835066, -12983537, 37871888, -83222132, 149287250, -212064236, 186932259, 131172644, -1139053896, 3449157957, -7710640256

Offset: 0

Ilya Gutkovskiy, Nov 29 2017

sign

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A003823, A007325, A055101, A055102, A055103, A127632, A286509, A291651.

nmax = 36; CoefficientList[Series[1/(1 + ContinuedFractionK[(x/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}]))^k, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
g[x_] := g[x] = QPochhammer[x, x^5] QPochhammer[x^4, x^5]/(QPochhammer[x^2, x^5] QPochhammer[x^3, x^5]); a[n_] := a[n] = SeriesCoefficient[g[x g[x]], {x, 0, n}];  Table[a[n], {n, 0, 36}]

G.f.: 1/(1 + x/(1 + x/(1 + x^2/(1 + x^3/(1 + ...))))/(1 + x^2/(1 + x/(1 + x^2/(1 + x^3/(1 + ...))))^2/(1 + x^3/(1 + x/(1 + x^2/(1 + x^3/(1 + ...))))^3/(1 + ...)))), a continued fraction.

A003823 Power series expansion of the Rogers-Ramanujan continued fraction 1+x/(1+x^2/(1+x^3/(1+x^4/(1+...)))).

Original entry on oeis.org

1, 1, 0, -1, 0, 1, 1, -1, -2, 0, 2, 2, -1, -3, -1, 3, 3, -2, -5, -1, 6, 5, -3, -8, -2, 8, 7, -5, -12, -2, 13, 12, -7, -18, -4, 18, 16, -11, -26, -5, 27, 24, -14, -37, -8, 37, 33, -21, -52, -10, 53, 47, -29, -72, -15, 71, 63, -40, -98, -19, 99, 88, -53, -133, -27, 131, 115, -73, -178, -35, 177, 156, -95, -236, -48, 232, 204, -127, -311

Offset: 0

N. J. A. Sloane

This is the q-expansion of the Gamma(5)-modular function (or automorphic function) Lambda given, for example, in Erdelyi et al., Higher Transcendental Functions eq. 44 volume 3 page 24 sec. 14.6.3 - Warren Smith.
Number 14 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Aug 07 2014
A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma(5). [Yang 2004] - Michael Somos, Aug 07 2014

			G.f. = 1 + x - x^3 + x^5 + x^6 - x^7 - 2*x^8 + 2*x^10 + 2*x^11 - x^12 - ...
G.f. = 1/q + q^4 - q^14 + q^24 + q^29 - q^34 - 2*q^39 + 2*q^49 + 2*q^54 - q^59 + ...

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 81.
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 24.
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 404.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
G. E. Andrews, Ramanujan's "lost" notebook, III, the Rogers-Ramanujan continued fraction, Adv. Math. 41 (1981), 186-208.
S.-D. Chen and S.-S. Huang, On the series expansion of the Göllnitz-Gordon continued fraction, Internat. J. Number Theory, 1 (2005), 53-63.
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eq. (6.5).
J. Malenfant, Generalizing Ramanujan's J Functions, arXiv preprint arXiv:1109.5957 [math.NT], 2011.
Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.

Cf. A007325.

M := 100: a[ M ] := 1+z; for n from M-1 by -1 to 1 do a[ n ] := series( 1 + z^n/a[ n+1 ], z, M+1); od: a[ 1 ];
M:=100; qf:=(a,q)->mul(1-a*q^j,j=0..M); t1:=qf(q^2,q^5)*qf(q^3,q^5)/(qf(q,q^5)*qf(q^4,q^5)); series(%,q,M); seriestolist(%);

kmax = 16; f[x_] := Product[(1-x^(5k-2))*(1-x^(5k-3))/((1-x^(5k-1))*(1-x^(5k-4))), {k, 1, kmax}]; CoefficientList[ Series[f[x], {x, 0, 5*kmax}], x] (* Jean-François Alcover, Nov 02 2011, after g.f. *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; (* Michael Somos, Jul 09 2014 *)
a[ n_] := If[n < 0, 0, SeriesCoefficient[ 1 / ContinuedFractionK[ x^k, 1, {k, 0, n}], {x, 0, n}]]; (* Michael Somos, Jul 09 2014 *)

{a(n) = local(k); if( n<0, 0, k = (3 + sqrtint(9 + 40*n)) \ 10; polcoeff( sum( i=-k, k, (-1)^i * x^((5*i^2 + i)/2), x * O(x^n)) / sum( i=-k, k, (-1)^i * x^((5*i^2 + 3*i)/2), x * O(x^n)), n))}; /* Michael Somos, Dec 13 2002 */

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, if( k%5, (1 - x^k)^( -(-1)^binomial( k%5, 2)), 1), 1 + x * O(x^n)), n))}; /* Michael Somos, Dec 13 2002 */

{a(n) = local(cf); if( n<0, 0, cf = contfracpnqn( matrix(2, (sqrtint(8*n + 1) + 1)\2, i, j, if( i==1, x^(j-1), 1))); polcoeff( cf[1, 1] / cf[2, 1] + x * O(x^n), n))}; /* Michael Somos, Dec 13 2002 */

G.f.: Prod_{k>0} (1-x^{5k-2})(1-x^{5k-3})/((1-x^{5k-1})(1-x^{5k-4})).
G.f.: (Sum_{k in Z} (-1)^k * x^((5*k + 1) * k/2)) / (Sum_{k in Z} (-1)^k * x^((5*k + 3) * k/2)). - Michael Somos, Dec 13 2002
Euler transform of period 5 sequence [1, -1, -1, 1, 0, ...]. - Michael Somos, Dec 13 2002
G.f. is reciprocal of that for the Rogers-Ramanujan continued fraction r(tau) - see A007325.
Expansion of f(-x^2, -x^3) / f(-x, -x^4) in powers of x where f(,) is Ramanujan's two-variable theta function. - Michael Somos, Aug 07 2014
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109091(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 01 2017

A078905 The q expansion of Lambda^5, a Hauptmodul for Gamma_1(5).

Original entry on oeis.org

1, -5, 15, -30, 40, -26, -30, 125, -220, 245, -124, -180, 615, -1010, 1085, -550, -705, 2415, -3850, 3980, -1926, -2460, 8090, -12550, 12715, -6074, -7500, 24360, -37150, 36930, -17251, -21155, 67380, -101210, 99295, -45924, -55305, 174500, -259140, 251275, -114750

Offset: 1

Michael Somos, Dec 12 2002

Denoted by r^5(tau) by Duke (2005). - Michael Somos, Jul 09 2014

			G.f. = q - 5*q^2 + 15*q^3 - 30*q^4 + 40*q^5 - 26*q^6 - 30*q^7 + 125*q^8 + ...

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 24.
B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 12, Entry 1(ii).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1001 from T. D. Noe)
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 150.

Cf. A007325, A109064.

QP = QPochhammer; s = (QP[q, q^5]*(QP[q^4, q^5]/(QP[q^2, q^5]*QP[q^3, q^5]) ))^5 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, from g.f. of A007325 *)

{a(n) = local(k); if( n<1, 0, k = (7 + sqrtint(40*n - 32)) \ 10; polcoeff( x * (sum(i=-k, k, (-1)^i * x^((5*i^2 + 3*i) / 2), O(x^n)) / sum(i=-k, k, (-1)^i * x^((5*i^2 + i) / 2), O(x^n)))^5, n))};

{a(n) = local(A); if( n<1, 0, A=O(x^n); A = (eta(x + A) / eta(x^5 + A))^6 / x; polcoeff( 2 / (11 + A + sqrt(125 + 22*A + A^2)), n))};

{a(n) = local(A, u, v); if( n<0, 0, A=x; for(k=2, n, u = A + x*O(x^k); v = subst(u, x, x^2); A -= x^k * polcoeff( u^2 - v + u*v^3 + u^3*v^2 + 10*u*v * (1 - u + v + u*v), k+1) / 2); polcoeff(A, n))};

G.f.: x * ( Product_{k>0} (1 - x^{5*k - 1}) * (1 - x^{5*k - 4}) / ((1 - x^{5*k - 2}) * (1 - x^{5*k - 3})) )^5
G.f.: x * ((Sum_{k in Z} (-1)^k * x^((5*k + 3) * k/2)) / (Sum_{k in Z} (-1)^k * x^((5*k + 1) * k/2)))^5.
G.f. A(x) = x * B(x)^5 where B(x) is the g.f. of A007325.
Euler transform of period 5 sequence [ -5, 5, 5, -5, 0, ...].
G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u,v) = u^2 - v + u*v^3 + u^3*v^2 + 10*u*v * (1 - u + v + u*v). - Michael Somos, Mar 09 2004
Given g.f. A(q), then q * A'(q) / A(q) = g.f. of A109064. [Duke (2005)] - Michael Somos, Jul 09 2014
a(1) = 1, a(n) = -(5/(n-1))*Sum_{k=1..n-1} A109091(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017

A055101 Expansion of square of continued fraction 1/ ( 1+q/ ( 1+q^2/ ( 1+q^3/ ( 1+q^4/... )))).

Original entry on oeis.org

1, -2, 3, -2, -1, 4, -6, 6, -3, -2, 9, -16, 17, -10, -5, 24, -36, 36, -21, -10, 46, -74, 77, -42, -22, 94, -144, 142, -78, -38, 172, -266, 266, -146, -73, 312, -471, 464, -251, -122, 534, -814, 801, -432, -213, 910, -1364, 1328, -713, -344, 1485, -2234, 2178

Offset: 0

N. J. A. Sloane, Jun 14 2000

Seiichi Manyama, Table of n, a(n) for n = 0..10000
For the third power see G. E. Andrews, Simplicity and surprise in Ramanujan's "Lost" Notebook, Amer. Math. Monthly, 104 (No. 10, Dec. 1997), 918-925.

Product_{k>0} ((1-x^{5k-1}) * (1-x^{5k-4})/((1-x^{5k-2}) * (1-x^{5k-3})))^m: A285444 (m=-4), A285443 (m=-3), A285442 (m=-2), A003823 (m=-1), A007325 (m=1), this sequence (m=2), A055102 (m=3), A055103 (m=4).

Cf. A109091, A340453, A340454, A340455, A340456.

a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A109091(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 16 2017
Euler transform of period 5 sequence [-2, 2, 2, -2, 0, ...]. - Georg Fischer, Aug 18 2020
From Seiichi Manyama, Jul 29 2024: (Start)
G.f.: ( Sum_{k in Z} x^(3*k) / (1 - x^(5*k+1)) ) / ( Sum_{k in Z} x^k / (1 - x^(5*k+1)) ).
G.f.: ( Sum_{k in Z} x^(2*k) / (1 - x^(5*k+2)) ) / ( Sum_{k in Z} x^k / (1 - x^(5*k+2)) ). (End)

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000

A109091 Expansion of (1 - eta(q)^5 / eta(q^5)) / 5 in powers of q.

Original entry on oeis.org

1, -1, -2, 3, 1, 2, -6, -5, 7, -1, 12, -6, -12, 6, -2, 11, -16, -7, 20, 3, 12, -12, -22, 10, 1, 12, -20, -18, 30, 2, 32, -21, -24, 16, -6, 21, -36, -20, 24, -5, 42, -12, -42, 36, 7, 22, -46, -22, 43, -1, 32, -36, -52, 20, 12, 30, -40, -30, 60, -6, 62, -32, -42, 43, -12, 24, -66, -48, 44, 6, 72, -35, -72, 36, -2, 60, -72

Offset: 1

Michael Somos, Jun 18 2005

			G.f. = q - q^2 - 2*q^3 + 3*q^4 + q^5 + 2*q^6 - 6*q^7 - 5*q^8 + 7*q^9 - q^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. This is the expression B^5/C in the notation of p. 106. [Added by _N. J. A. Sloane_, Nov 13 2009]

Cf. A003823, A007325, A078905, A109064, A229793.

Cf. A284097, A284103, A284150, A284152, A284280, A284281.

a[ n_] := If[ n < 1, 0, Sum[ d KroneckerSymbol[ 5, d], {d, Divisors@n}]]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ q]^5 / QPochhammer[ q^5]) / 5, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)

{a(n) = my(A); if( n<1, 0, A = x * O(x^n); -1/5 * polcoeff( eta(x + A)^5 / eta(x^5 + A), n))};

{a(n) = if( n<1, 0, sumdiv(n, d, d * kronecker(5, d)))} /* Michael Somos, Mar 21 2008 */

def s(k, m, n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0 && i % k == m}
  s
end
def A109091(n)
  (1..n).map{|i| s(5, 1, i) + s(5, 4, i) - s(5, 2, i) - s(5, 3, i)}
end # Seiichi Manyama, Apr 01 2017

G.f.: (1 - Product_{k>0} (1 - x^k)^5 / (1 - x^(5*k))) / 5 = Sum_{k>0} x^k * (1 - x^k)^2 * (1 + x^(6*k) - 4*x^(2*k) * (1 + x^k +x^(2*k))) / (1 - x^(5*k))^2.
-5*a(n) = A109064(n) unless n = 0.
a(n) = A284097(n) + A284103(n) - A284280(n) - A284281(n) = A284150(n) - A284152(n). - Seiichi Manyama, Apr 01 2017
L.g.f.: -log(1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(15*sqrt(5)) = 0.294254... . - Amiram Eldar, Jan 29 2024

A055102 Expansion of cube of continued fraction 1/ ( 1+q/ ( 1+q^2/ ( 1+q^3/ ( 1+q^4/... )))).

Original entry on oeis.org

1, -3, 6, -7, 3, 6, -17, 24, -21, 6, 21, -54, 77, -72, 24, 64, -159, 216, -190, 57, 159, -392, 534, -468, 144, 381, -924, 1220, -1044, 312, 833, -1992, 2625, -2244, 669, 1746, -4138, 5382, -4530, 1332, 3474, -8184, 10591, -8886, 2607, 6724, -15711

Offset: 0

N. J. A. Sloane, Jun 14 2000

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293. see p. 282.
G. E. Andrews, Simplicity and surprise in Ramanujan's "Lost" Notebook, Amer. Math. Monthly, 104 (No. 10, Dec. 1997), 918-925.

Product_{k>0} ((1-x^{5k-1}) * (1-x^{5k-4})/((1-x^{5k-2}) * (1-x^{5k-3})))^m: A285444 (m=-4), A285443 (m=-3), A285442 (m=-2), A003823 (m=-1), A007325 (m=1), A055101 (m=2), this sequence (m=3), A055103 (m=4).

Cf. A109091, A340455, A340456.

a(0) = 1, a(n) = -(3/n)*Sum_{k=1..n} A109091(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 16 2017

G.f.: ( Sum_{k in Z} x^(2*k) / (1 - x^(5*k+2)) ) / ( Sum_{k in Z} x^k / (1 - x^(5*k+1)) ). - Seiichi Manyama, Jul 29 2024

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000

cp's OEIS Frontend

A327688 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A007325.

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A167683 Hankel transform of A007325.

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A258040 Expansion of f(x) / f(-x) in powers of x where f() is the g.f. for A007325.

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A092872 Expansion of r(q^9) / (r(q) r(q^3)) in powers of q where r() is the Rogers-Ramanujan continued fraction function (A007325).

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A295703 Expansion of R(x*R(x)), where R(x) = 1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))), a continued fraction (g.f. for A007325).

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A003823 Power series expansion of the Rogers-Ramanujan continued fraction 1+x/(1+x^2/(1+x^3/(1+x^4/(1+...)))).

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Maple

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A078905 The q expansion of Lambda^5, a Hauptmodul for Gamma_1(5).

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A055101 Expansion of square of continued fraction 1/ ( 1+q/ ( 1+q^2/ ( 1+q^3/ ( 1+q^4/... )))).

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