A340453 -id:A340453 - cp's OEIS frontend

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

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

A340456 G.f.: Sum_{n>=0} x^n/(1 - x^(5n+1)) - x^3Sum_{n>=0} x^(4n)/(1 - x^(5n+4)).

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Jan 20 2021

sign

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

			G.f.: 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 + 2*q^17 + 2*q^18 + 2*q^19 + 2*q^20 + ...
Given the g.f. of this sequence
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))
and the g.f. of A340455,
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))
then
R(q)^3 = P(q)/Q(q) where
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 + ...
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, A340455, 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+2))*(1-x^(5*m+3)) / ( (1-x^(5*m+1))^2*(1-x^(5*m+4))^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(1,1,n) - x^3*S(4,4,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+1))^3 * (1 - x^(5*n+4))^3 ).
G.f.: Product_{n>=0} (1 - x^(5*n+5))^2 * (1 - x^(5*n+2))*(1 - x^(5*n+3)) / ( (1 - x^(5*n+1))^2*(1 - x^(5*n+4))^2 ).
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)) ] / R(x), where R(q) is the expansion of Ramanujan's continued fraction (A007325).

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).

A375148 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+2)).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Aug 01 2024

nonn

Cf. A374900, A375106, A375149, A375150.

Cf. A033687, A340453.

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

my(N=110, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^3*((1-x^(7*k-3))*(1-x^(7*k-4)))^2/(1-x^k)))

G.f.: Product_{k>0} (1-x^(7*k))^3 * ((1-x^(7*k-3)) * (1-x^(7*k-4)))^2 / (1-x^k).

G.f.: Sum_{k in Z} x^(2*k) / (1 - x^(7*k+1)).

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).

A375063 Expansion of 1 / Sum_{k in Z} x^k / (1 - x^(5*k+2)).

Original entry on oeis.org

1, -1, 0, 0, -1, 3, -3, 1, 0, -3, 9, -9, 3, 1, -9, 22, -22, 9, 2, -22, 51, -51, 22, 6, -51, 108, -108, 50, 13, -108, 221, -221, 105, 29, -220, 429, -429, 212, 57, -426, 810, -810, 407, 113, -801, 1479, -1478, 759, 208, -1457, 2640, -2637, 1371, 381, -2589, 4598, -4590, 2419, 669

Offset: 0

Seiichi Manyama, Jul 29 2024

sign

Convolution inverse of A340453.

Cf. A375061, A375062, A375064.

my(N=60, x='x+O('x^N)); Vec(1/sum(k=-N, N, x^k/(1-x^(5*k+2))))

my(N=60, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(5*k-1))*(1-x^(5*k-4))/(1-x^(5*k))^2))

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

cp's OEIS Frontend

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

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A340456 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)).

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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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A375148 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+2)).

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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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A375063 Expansion of 1 / Sum_{k in Z} x^k / (1 - x^(5*k+2)).

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A340456 G.f.: Sum_{n>=0} x^n/(1 - x^(5n+1)) - x^3Sum_{n>=0} x^(4n)/(1 - x^(5n+4)).

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

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