A007325 -id:A007325 - cp's OEIS frontend

A055103 Expansion of 4th power of continued fraction 1/ ( 1+q/ ( 1+q^2/ ( 1+q^3/ ( 1+q^4/... )))).

1, -4, 10, -16, 15, 0, -30, 64, -81, 60, 12, -128, 250, -312, 234, 32, -443, 848, -1014, 720, 109, -1312, 2448, -2880, 2033, 280, -3550, 6512, -7513, 5184, 744, -8832, 15980, -18252, 12492, 1712, -20745, 37168, -41942, 28352, 3918, -46288, 82146

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.

See A007325 for first power (which has an alternative g.f.), A055101 for square, A055102 for cube.

a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A109091(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 16 2017

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).

A285348 Expansion of r(q^2) / r(q)^2 in powers of q where r() is the Rogers-Ramanujan continued fraction.

Original entry on oeis.org

1, 2, 0, -4, -2, 6, 8, -4, -16, -6, 20, 24, -12, -44, -16, 52, 62, -28, -108, -40, 122, 144, -64, -244, -88, 266, 308, -136, -508, -180, 544, 624, -272, -1008, -356, 1060, 1206, -524, -1920, -672, 1988, 2244, -968, -3524, -1224, 3606, 4048, -1732, -6284

Offset: 0

Seiichi Manyama, Apr 17 2017

sign

Let k(q) = r(q) * r(q^2)^2.
G.f. satisfies: A(q) = (1 + k(q))/(1 - k(q)).
And r(q^2)^5 = k(q)^2 * A(q).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Rogers-Ramanujan continued fraction

r(q^k) / r(q)^k: this sequence (k=2), A285583 (k=3), A285584 (k=4), A285585 (k=5).

Cf. A007325, A078905 (r(q)^5), A112274 (k(q)), A112803 (1 + k(q)), A285349, A285355 (k(q)^2).

a(n) = A285349(n) - A138518(n) for n>0 (conjectured). - Thomas Baruchel, May 14 2018

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).

A285349 Expansion of r(q)^2 / r(q^2) in powers of q where r() is the Rogers-Ramanujan continued fraction.

Original entry on oeis.org

1, -2, 4, -4, 2, 2, -8, 12, -12, 6, 8, -24, 36, -36, 16, 20, -62, 92, -88, 40, 46, -144, 208, -196, 88, 102, -308, 440, -412, 180, 208, -624, 884, -816, 356, 404, -1206, 1692, -1552, 672, 760, -2244, 3128, -2852, 1224, 1378, -4048, 5612, -5084, 2174, 2428, -7104, 9796, -8836, 3760

Offset: 0

Seiichi Manyama, Apr 17 2017

sign

Let k(q) = r(q) * r(q^2)^2.
G.f. satisfies: A(q) = (1 - k(q))/(1 + k(q)).
And r(q)^5 = k(q) * A(q)^2.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Rogers-Ramanujan continued fraction

r(q)^k / r(q^k): this sequence (k=2), A285628 (k=3), A285629 (k=4), A285630 (k=5).

Cf. A007325, A078905 (r(q)^5), A112274 (k(q)), A285348.

a(n) = A138518(n) + A285348(n) for n>0 (conjectured). - Thomas Baruchel, May 14 2018

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

A285444 Expansion of Product_{k>0} ((1-x^{5k-2}) * (1-x^{5k-3})/((1-x^{5k-1}) * (1-x^{5k-4})))^4 in powers of x.

Original entry on oeis.org

1, 4, 6, 0, -11, -8, 18, 32, -10, -72, -42, 96, 153, -40, -288, -160, 344, 524, -146, -944, -501, 1080, 1602, -416, -2727, -1436, 2970, 4336, -1131, -7176, -3694, 7616, 10942, -2776, -17562, -8960, 18136, 25784, -6528, -40608, -20472, 41176, 57974, -14464

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: this sequence (m=-4), A285443 (m=-3), A285442 (m=-2), A003823 (m=-1), A007325 (m=1), A055101 (m=2), A055102 (m=3), A055103 (m=4).

a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A109091(k)*a(n-k) for n > 0.

Expansion of 4th power of continued fraction 1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...)))). - Ilya Gutkovskiy, Apr 19 2017

A286509 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...)))))).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, 0, 0, 1, -4, 6, -2, -1, 0, 1, -5, 10, -7, -1, 1, 0, 1, -6, 15, -16, 3, 4, -1, 0, 1, -7, 21, -30, 15, 6, -6, 1, 0, 1, -8, 28, -50, 40, 0, -17, 6, 0, 0, 1, -9, 36, -77, 84, -26, -30, 24, -3, -1, 0, 1, -10, 45, -112, 154, -90, -30, 64, -21, -2, 2, 0, 1, -11, 55, -156, 258, -217, 15, 125, -81, 6, 9, -3, 0

Offset: 0

Ilya Gutkovskiy, May 10 2017

			Square array begins:
1,  1,  1,  1,   1,   1,  ...
0, -1, -2, -3,  -4,  -5,  ...
0,  1,  3,  6,  10,  15,  ...
0,  0, -2, -7, -16, -30,  ...
0, -1, -1,  3,  15,  40,  ...
0,  1,  4,  6,   0, -26,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Columns k=0-5 give: A000007, A007325, A055101, A055102, A055103, A078905 (with offset 0).
Rows n=0-2 give: A000012, A001489, A000217.
Main diagonal gives A291651.
Antidiagonal sums give A302015.

Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[x^i, 1, {i, 1, n}])^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[(1 - x^(5 i - 1)) (1 - x^(5 i - 4))/((1 - x^(5 i - 2)) (1 - x^(5 i - 3))), {i, n}]^k, {x, 0, n}]][j - n], {j, 0, 12},{n, 0, j}] // Flatten

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

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)

cp's OEIS Frontend

A055103 Expansion of 4th power of continued fraction 1/ ( 1+q/ ( 1+q^2/ ( 1+q^3/ ( 1+q^4/... )))).

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

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)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula

A285348 Expansion of r(q^2) / r(q)^2 in powers of q where r() is the Rogers-Ramanujan continued fraction.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula

A285349 Expansion of r(q)^2 / r(q^2) in powers of q where r() is the Rogers-Ramanujan continued fraction.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Links

Crossrefs

Formula

A285444 Expansion of Product_{k>0} ((1-x^{5k-2}) * (1-x^{5k-3})/((1-x^{5k-1}) * (1-x^{5k-4})))^4 in powers of x.

Views

Author

Keywords

Links

Crossrefs

Formula

A286509 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...)))))).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

PARI

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)).

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