A340456 -id:A340456 - cp's OEIS frontend

A097195 Expansion of s(12)^3s(18)^2/(s(6)^2s(36)), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815. Then replace q^6 with q.

1, 2, 2, 2, 1, 2, 2, 2, 3, 0, 2, 2, 2, 2, 0, 4, 2, 2, 2, 0, 1, 2, 4, 2, 0, 2, 2, 2, 3, 2, 2, 0, 2, 2, 0, 2, 4, 2, 2, 0, 2, 4, 0, 4, 0, 2, 2, 2, 1, 0, 4, 2, 2, 0, 2, 2, 2, 4, 2, 0, 3, 2, 2, 2, 0, 0, 2, 4, 2, 0, 2, 4, 2, 2, 0, 0, 2, 2, 4, 2, 4, 2, 0, 2, 0, 4, 0, 2, 1, 0, 2, 2, 4

Offset: 0

N. J. A. Sloane, Sep 16 2004

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

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.38).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A004016, A010815, A093602.

Cf. A008441, A033687, A340456, A374900.

a[n_] := DivisorSum[6n+1, KroneckerSymbol[-3, #]&]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Nov 23 2015, after Michael Somos *)
QP = QPochhammer; s = QP[q^2]^3*(QP[q^3]^2/QP[q]^2/QP[q^6]) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# < 2, 0^#2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger@(6 n + 1))]; (* Michael Somos, Mar 05 2016 *)

{a(n) = if( n<0, 0, sumdiv(6*n+1, d, kronecker(-3, d)))}; /* Michael Somos, Nov 03 2005 */

{a(n) = my(A, p, e); if( n<0, 0, n = 6*n+1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p>3, if( p%6==1, e+1, !(e%2)))))}; /* Michael Somos, Nov 03 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)), n))}; /* Michael Somos, Nov 03 2005 */

Fine gives an explicit formula for a(n) in terms of the divisors of n.
a(n) = b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
From Michael Somos, Nov 03 2005: (Start)
G.f.: Sum_{k in Z} x^k / (1 - x^(6*k + 1)).
G.f.: Sum_{k>=0} a(k) * x^(6*k + 1) = Sum_{k>0} x^(2*k-1) * (1 - x^(4*k - 2)) * (1 - x^(8*k - 4)) * (1 - x^(20*k - 10)) / (1 - x^(36*k - 18)). (End)
From Michael Somos, Mar 05 2016: (Start)
Expansion of q^(-1/6) * eta(q^2)^3 * eta(q^3)^2 / (eta(q)^2 * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ 2, -1, 0, -1, 2, -2, ...].
6 * a(n) = A004016(6*n + 1). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(3) = 1.813799... (A093602). - Amiram Eldar, Nov 24 2023

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

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

A340455 G.f.: Sum_{n>=0} x^(2n)/(1 - x^(5n+2)) - xSum_{n>=0} x^(3n)/(1 - x^(5*n+3)).

Original entry on oeis.org

1, -1, 2, 0, 0, 0, 2, -2, 2, 1, 0, 0, 1, -2, 2, 0, 2, 0, 2, -2, 0, 0, 0, 2, 2, -2, 2, 0, -1, 0, 4, -2, 2, -1, 0, 0, 0, 0, 2, 0, 2, 0, 2, -2, 2, 0, -2, 0, 2, -2, 2, 2, 0, 0, 2, -2, 2, 1, 2, -2, 0, -2, 2, 0, 1, 2, 2, -2, 0, 0, 0, 0, 2, -2, 4

Offset: 0

Paul D. Hanna, Jan 20 2021

sign

The g.f. of this sequence equals the numerator of George E. Andrews' expression for the cube of Ramanujan's continued fraction. See references given in A007325.

			G.f.: P(q) = 1 - q + 2*q^2 + 2*q^6 - 2*q^7 + 2*q^8 + q^9 + q^12 - 2*q^13 + 2*q^14 + 2*q^16 + 2*q^18 - 2*q^19 + 2*q^23 + 2*q^24 - 2*q^25 + 2*q^26 - q^28 + ...
Given the g.f. of this sequence,
P(q) = Sum_{n>=0} q^(2*n)/(1 - q^(5*n+2)) - q*Sum_{n>=0} q^(3*n)/(1 - q^(5*n+3))
and the g.f. of A340456,
Q(q) = Sum_{n>=0} q^n/(1 - q^(5*n+1)) - q^3*Sum_{n>=0} q^(4*n)/(1 - q^(5*n+4))
then
R(q)^3 = P(q)/Q(q) where
Q(q) = 1 + 2*q + 2*q^2 + q^3 + 2*q^4 + 2*q^5 + 2*q^6 + q^7 + 2*q^8 + 2*q^9 + 2*q^10 + 2*q^12 + 4*q^13 + 2*q^14 + q^16 + ...
R(q)^3 = 1 - 3*q + 6*q^2 - 7*q^3 + 3*q^4 + 6*q^5 - 17*q^6 + 24*q^7 - 21*q^8 + 6*q^9 + 21*q^10 - 54*q^11 + 77*q^12 - 72*q^13 + 24*q^14 + 64*q^15 + ...;
here, R(q) is the expansion of Ramanujan's continued fraction (A007325).

Cf. A007325, A055102, A340456, A340453, A340454.

{a(n) = my(A = prod(m=0,n\5+1, (1-x^(5*m+5) +x*O(x^n))^2 * (1-x^(5*m+1))*(1-x^(5*m+4)) / ( (1-x^(5*m+2))^2*(1-x^(5*m+3))^2 +x*O(x^n) ) ));polcoeff(A,n)}
for(n=0,100,print1(a(n),", "))

{S(j,k,n) = sum(m=0,n, x^(j*m)/(1 - x^(5*m+k) +x*O(x^n)) ) }
{a(n) = polcoeff( S(2,2,n) - x*S(3,3,n), n)}
for(n=0,100,print1(a(n),", "))

G.f.: Product_{n>=0} (1 - x^(n+1)) * (1 - x^(5*n+5)) / ( (1 - x^(5*n+2))^3 * (1 - x^(5*n+3))^3 ).
G.f.: Product_{n>=0} (1 - x^(5*n+5))^2 * (1 - x^(5*n+1))*(1 - x^(5*n+4)) / ( (1 - x^(5*n+2))^2*(1 - x^(5*n+3))^2 ).
G.f.: [ Sum_{n>=0} x^n/(1 - x^(5*n+3)) - x * Sum_{n>=0} x^(4*n)/(1 - x^(5*n+2)) ] * R(x), where R(q) is the expansion of Ramanujan's continued fraction (A007325).

A285443 Expansion of Product_{k>0} ((1-x^{5k-2}) * (1-x^{5k-3})/((1-x^{5k-1}) * (1-x^{5k-4})))^3 in powers of x.

Original entry on oeis.org

1, 3, 3, -2, -6, 0, 12, 9, -15, -28, 3, 48, 33, -48, -87, 7, 135, 90, -134, -234, 21, 356, 237, -330, -575, 42, 831, 540, -762, -1296, 107, 1848, 1191, -1633, -2769, 210, 3842, 2448, -3366, -5634, 444, 7722, 4889, -6624, -11028, 840, 14871, 9342, -12636, -20877

Offset: 0

Seiichi Manyama, Apr 19 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Prod_{k>0} ((1-x^{5k-1}) * (1-x^{5k-4})/((1-x^{5k-2}) * (1-x^{5k-3})))^m: A285444 (m=-4), this sequence (m=-3), A285442 (m=-2), A003823 (m=-1), A007325 (m=1), A055101 (m=2), A055102 (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.
Expansion of cube of continued fraction 1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...)))). - Ilya Gutkovskiy, Apr 19 2017
G.f.: ( Sum_{k in Z} x^k / (1 - x^(5*k+1)) ) / ( Sum_{k in Z} x^(2*k) / (1 - x^(5*k+2)) ). - Seiichi Manyama, Jul 29 2024

A340453 G.f.: Product_{n>=0} (1 - x^(5n+5))^2 / ( (1 - x^(5n+1))(1 - x^(5n+4)) ).

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 1, 0, 2, 1, 0, 2, 2, 0, 1, 1, 2, 1, 1, -1, 2, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 0, 2, 2, 1, 1, 2, -1, 1, 2, 2, 1, 0, 0, 3, 0, 1, 1, 2, 0, 1, 1, 2, 2, 1, 0, 2, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 2, 2, 1, 2, 0, 2, -1, 0, 1, 2

Offset: 0

Paul D. Hanna, Jan 16 2021

sign

			G.f.: P(q) = 1 + q + q^2 + q^3 + 2*q^4 + q^6 + q^7 + 2*q^8 + q^9 + q^10 + 2*q^12 + q^13 + 2*q^15 + 2*q^16 + q^18 + q^19 + 2*q^20 + ...
Given the g.f. of this sequence,
P(q) = Product_{n>=0} (1 - q^(5*n+5))^2 / ( (1 - q^(5*n+1))*(1 - q^(5*n+4)) ),
and the g.f. of A340454,
Q(q) = Product_{n>=0} (1 - q^(5*n+5))^2 / ( (1 - q^(5*n+2))*(1 - q^(5*n+3)) ),
then R(q) = P(q)/Q(q) where
Q(q) = 1 + q^2 + q^3 + q^4 - q^5 + 2*q^6 + q^8 + q^9 + q^10 + q^12 - q^13 + q^14 + 2*q^15 + q^16 + 2*q^18 - q^19 + q^20 + ...
and
R(q) = 1 + q - q^3 + q^5 + q^6 - q^7 - 2*q^8 + 2*q^10 + 2*q^11 - q^12 - 3*q^13 - q^14 + 3*q^15 + 3*q^16 - 2*q^17 - 5*q^18 - q^19 + 6*q^20 + ...;
here, R(q) is the expansion of Ramanujan's continued fraction (A007325).

Cf. A007325, A340454, A340455, A340456.

{a(n) = my(A = prod(m=0,n, (1 - x^(5*m+5))^2 / ( (1 - x^(5*m+1))*(1 - x^(5*m+4)) +x*O(x^n)) )); polcoeff(A,n)}
for(n=0,80,print1(a(n),", "))

{a(n) = my(A = sum(m=0,n, x^(1*m)/(1 - x^(5*m+2) +x*O(x^n)) ) - x^2 * sum(m=0,n, x^(3*m)/(1 - x^(5*m+4) +x*O(x^n)) )); polcoeff(A,n)}
for(n=0,80,print1(a(n),", "))

{a(n) = my(A = sum(m=0,n, x^(2*m)/(1 - x^(5*m+1) +x*O(x^n)) ) - x^2 * sum(m=0,n, x^(4*m)/(1 - x^(5*m+3) +x*O(x^n)) )); polcoeff(A,n)}
for(n=0,80,print1(a(n),", "))

G.f.: Sum_{n>=0} x^(1*n)/(1 - x^(5*n+2)) - x^2 * Sum_{n>=0} x^(3*n)/(1 - x^(5*n+4)).
G.f.: Sum_{n>=0} x^(2*n)/(1 - x^(5*n+1)) - x^2 * Sum_{n>=0} x^(4*n)/(1 - x^(5*n+3)).
G.f.: Sum_{n>=0} x^(1*n)/(1 - x^(5*n+2)) - x^2 * Sum_{n>=0} x^(4*n)/(1 - x^(5*n+3)).
G.f.: Sum_{n>=0} x^(2*n)/(1 - x^(5*n+1)) - x^2 * Sum_{n>=0} x^(3*n)/(1 - x^(5*n+4)).
G.f.: [ Sum_{n>=0} x^n/(1 - x^(5*n+1)) - x^3 * Sum_{n>=0} x^(4*n)/(1 - x^(5*n+4)) ] * R(x), where R(q) is the expansion of Ramanujan's continued fraction (A007325).

A285442 Expansion of Product_{k>0} ((1-x^{5k-2}) * (1-x^{5k-3})/((1-x^{5k-1}) * (1-x^{5k-4})))^2 in powers of x.

Original entry on oeis.org

1, 2, 1, -2, -2, 2, 5, 0, -8, -6, 7, 14, 1, -18, -15, 14, 30, 2, -40, -32, 32, 66, 6, -82, -65, 60, 125, 8, -157, -120, 117, 238, 19, -286, -222, 206, 419, 28, -507, -386, 366, 732, 55, -864, -659, 610, 1224, 86, -1442, -1090, 1016, 2024, 147, -2350, -1775, 1632

Offset: 0

Seiichi Manyama, Apr 19 2017

sign

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. 282-283.

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), this sequence (m=-2), A003823 (m=-1), A007325 (m=1), A055101 (m=2), A055102 (m=3), A055103 (m=4).

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

nmax = 60; CoefficientList[Series[Product[((1-x^(5k-2)) * (1-x^(5k-3)) / ((1-x^(5k-1)) * (1-x^(5k-4))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2017 *)

a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A109091(k)*a(n-k) for n > 0.
Expansion of square of continued fraction 1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...)))). - Ilya Gutkovskiy, Apr 19 2017
From Seiichi Manyama, Jul 29 2024: (Start)
G.f.: ( Sum_{k in Z} x^k / (1 - x^(5*k+1)) ) / ( Sum_{k in Z} x^(3*k) / (1 - x^(5*k+1)) ).
G.f.: ( Sum_{k in Z} x^k / (1 - x^(5*k+2)) ) / ( Sum_{k in Z} x^(2*k) / (1 - x^(5*k+2)) ). (End)

A340454 G.f.: Product_{n>=0} (1 - x^(5n+5))^2 / ( (1 - x^(5n+2))(1 - x^(5n+3)) ).

Original entry on oeis.org

1, 0, 1, 1, 1, -1, 2, 0, 1, 1, 1, 0, 1, -1, 1, 2, 1, 0, 2, -1, 1, 0, 1, 0, 2, 0, 0, 2, 2, -1, 1, 0, 1, 0, 1, 0, 2, -1, 1, 2, 0, 1, 2, 0, 1, 0, 1, -2, 1, 0, 2, 2, 1, -1, 2, 0, 1, 1, 1, 0, 2, -2, 1, 2, 0, 0, 1, 1, 0, 0, 1, 0, 3, 1, 1, 0, 1, -1, 2, 0, 2

Offset: 0

Paul D. Hanna, Jan 16 2021

sign

			G.f.: Q(q) = 1 + q^2 + q^3 + q^4 - q^5 + 2*q^6 + q^8 + q^9 + q^10 + q^12 - q^13 + q^14 + 2*q^15 + q^16 + 2*q^18 - q^19 + q^20 + ...
Given the g.f. of this sequence,
Q(q) = Product_{n>=0} (1 - q^(5*n+5))^2 / ( (1 - q^(5*n+2))*(1 - q^(5*n+3)) ),
and the g.f. of A340453,
P(q) = Product_{n>=0} (1 - q^(5*n+5))^2 / ( (1 - q^(5*n+1))*(1 - q^(5*n+4)) ),
then R(q) = P(q)/Q(q) where
P(q) = 1 + q + q^2 + q^3 + 2*q^4 + q^6 + q^7 + 2*q^8 + q^9 + q^10 + 2*q^12 + q^13 + 2*q^15 + 2*q^16 + q^18 + q^19 + 2*q^20 + ...
and
R(q) = 1 + q - q^3 + q^5 + q^6 - q^7 - 2*q^8 + 2*q^10 + 2*q^11 - q^12 - 3*q^13 - q^14 + 3*q^15 + 3*q^16 - 2*q^17 - 5*q^18 - q^19 + 6*q^20 + ...;
here, R(q) is the expansion of Ramanujan's continued fraction (A007325).

Cf. A007325, A340453, A340455, A340456.

{a(n) = my(A = prod(m=0,n, (1 - x^(5*m+5))^2 / ( (1 - x^(5*m+2))*(1 - x^(5*m+3)) +x*O(x^n) ) )); polcoeff(A,n)}
for(n=0,80,print1(a(n),", "))

{a(n) = my(A = sum(m=0,n, x^(1*m)/(1 - x^(5*m+3) +x*O(x^n)) ) - x * sum(m=0,n, x^(2*m)/(1 - x^(5*m+4) +x*O(x^n)) )); polcoeff(A,n)}
for(n=0,80,print1(a(n),", "))

{a(n) = my(A = sum(m=0,n, x^(3*m)/(1 - x^(5*m+1) +x*O(x^n)) ) - x * sum(m=0,n, x^(4*m)/(1 - x^(5*m+2) +x*O(x^n)) )); polcoeff(A,n)}
for(n=0,80,print1(a(n),", "))

G.f.: Sum_{n>=0} x^(1*n)/(1 - x^(5*n+3)) - x * Sum_{n>=0} x^(2*n)/(1 - x^(5*n+4)).
G.f.: Sum_{n>=0} x^(3*n)/(1 - x^(5*n+1)) - x * Sum_{n>=0} x^(4*n)/(1 - x^(5*n+2)).
G.f.: Sum_{n>=0} x^(1*n)/(1 - x^(5*n+3)) - x * Sum_{n>=0} x^(4*n)/(1 - x^(5*n+2)).
G.f.: Sum_{n>=0} x^(3*n)/(1 - x^(5*n+1)) - x * Sum_{n>=0} x^(2*n)/(1 - x^(5*n+4)).
G.f.: [ Sum_{n>=0} x^(2*n)/(1 - x^(5*n+2)) - x * Sum_{n>=0} x^(3*n)/(1 - x^(5*n+3)) ] / R(x), where R(q) is the expansion of Ramanujan's continued fraction (A007325).

A374900 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+1)).

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 1, 4, 2, 2, 2, 0, 2, 2, 3, 4, 2, 0, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 0, 2, 2, 0, 2, 2, 2, 3, 2, 0, 2, 2, 4, 0, 2, 2, 2, 2, 2, 3, 2, 0, 2, 4, 2, 2, 0, 0, 2, 2, 2, 4, 2, 0, 2, 2, 2, 2, 2, 0, 2, 4, 4, 2, 2, 0, 2, 2, 2, 2, 2

Offset: 0

Seiichi Manyama, Jul 31 2024

nonn

Cf. A008441, A033687, A097195, A340456.

my(N=110, x='x+O('x^N)); Vec(sum(k=-N, N, x^k/(1-x^(7*k+1))))

my(N=110, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^2*(1-x^(7*k-2))*(1-x^(7*k-5))/((1-x^(7*k-1))*(1-x^(7*k-6)))^2))

G.f.: Product_{k>0} (1-x^(7*k))^2 * (1-x^(7*k-2)) * (1-x^(7*k-5)) / ((1-x^(7*k-1)) * (1-x^(7*k-6)))^2.

A375062 Expansion of 1 / Sum_{k in Z} x^k / (1 - x^(5*k+1)).

Original entry on oeis.org

1, -2, 2, -1, -2, 6, -9, 9, -4, -7, 22, -34, 33, -13, -25, 71, -103, 97, -39, -69, 196, -282, 263, -102, -182, 499, -703, 645, -248, -433, 1181, -1650, 1499, -568, -988, 2652, -3660, 3294, -1240, -2129, 5681, -7790, 6960, -2595, -4438, 11732, -15959, 14161, -5252

Offset: 0

Seiichi Manyama, Jul 29 2024

sign

Convolution inverse of A340456.

Cf. A375061, A375063, A375064.

my(N=50, x='x+O('x^N)); Vec(1/sum(k=-N, N, x^k/(1-x^(5*k+1))))

my(N=50, x='x+O('x^N)); Vec(prod(k=1, N, ((1-x^(5*k-1))*(1-x^(5*k-4)))^3/((1-x^k)*(1-x^(5*k)))))

G.f.: Product_{k>0} ((1-x^(5*k-1)) * (1-x^(5*k-4)))^3 / ((1-x^k) * (1-x^(5*k))).

cp's OEIS Frontend

A097195 Expansion of s(12)^3*s(18)^2/(s(6)^2*s(36)), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815. Then replace q^6 with q.

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

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A340455 G.f.: Sum_{n>=0} x^(2*n)/(1 - x^(5*n+2)) - x*Sum_{n>=0} x^(3*n)/(1 - x^(5*n+3)).

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A285443 Expansion of Product_{k>0} ((1-x^{5k-2}) * (1-x^{5k-3})/((1-x^{5k-1}) * (1-x^{5k-4})))^3 in powers of x.

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A340453 G.f.: Product_{n>=0} (1 - x^(5*n+5))^2 / ( (1 - x^(5*n+1))*(1 - x^(5*n+4)) ).

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A285442 Expansion of Product_{k>0} ((1-x^{5k-2}) * (1-x^{5k-3})/((1-x^{5k-1}) * (1-x^{5k-4})))^2 in powers of x.

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A340454 G.f.: Product_{n>=0} (1 - x^(5*n+5))^2 / ( (1 - x^(5*n+2))*(1 - x^(5*n+3)) ).

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A097195 Expansion of s(12)^3s(18)^2/(s(6)^2s(36)), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815. Then replace q^6 with q.

A340455 G.f.: Sum_{n>=0} x^(2n)/(1 - x^(5n+2)) - xSum_{n>=0} x^(3n)/(1 - x^(5*n+3)).

A340453 G.f.: Product_{n>=0} (1 - x^(5n+5))^2 / ( (1 - x^(5n+1))(1 - x^(5n+4)) ).

A340454 G.f.: Product_{n>=0} (1 - x^(5n+5))^2 / ( (1 - x^(5n+2))(1 - x^(5n+3)) ).