A285444 -id:A285444 - 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

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

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

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)

A285584 Expansion of r(q^4) / r(q)^4 in powers of q where r() is the Rogers-Ramanujan continued fraction.

Original entry on oeis.org

1, 4, 6, 0, -12, -12, 12, 32, 2, -60, -54, 64, 152, 24, -228, -224, 180, 488, 94, -688, -680, 528, 1448, 336, -1884, -1932, 1276, 3744, 944, -4680, -4828, 3088, 9154, 2464, -10980, -11520, 6744, 20792, 5832, -24304, -25618, 14584, 45424, 13184, -51696, -54972

Offset: 0

Seiichi Manyama, Apr 22 2017

sign

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

r(q^k) / r(q)^k: A285348 (k=2), A285583 (k=3), this sequence (k=4), A285585 (k=5).

Cf. A055103, A285444, A285629.

A291678 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 + 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, 0, 0, 1, 3, 1, -1, 0, 1, 4, 3, -2, 0, 0, 1, 5, 6, -2, -2, 1, 0, 1, 6, 10, 0, -6, 2, 1, 0, 1, 7, 15, 5, -11, 0, 5, -1, 0, 1, 8, 21, 14, -15, -8, 12, 0, -2, 0, 1, 9, 28, 28, -15, -24, 18, 9, -8, 0, 0, 1, 10, 36, 48, -7, -48, 15, 32, -15, -6, 2

Offset: 0

Seiichi Manyama, Aug 29 2017

			Square array begins:
   1,  1,  1,  1,   1, ...
   0,  1,  2,  3,   4, ...
   0,  0,  1,  3,   6, ...
   0, -1, -2, -2,   0, ...
   0,  0, -2, -6, -11, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..4 give A000007, A003823, A285442, A285443, A285444.
Rows n=0..1 give A000012, A001477.
Main diagonal gives A291679.
Antidiagonal sums give A302016.
Cf. A286509.

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

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

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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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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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A285584 Expansion of r(q^4) / r(q)^4 in powers of q where r() is the Rogers-Ramanujan continued fraction.

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A291678 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 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))).

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