A143064 -id:A143064 - cp's OEIS frontend

A185646 Square array A(n,m), n>=0, m>=0, read by antidiagonals, where column m is the expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))).

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, -1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 1, 2, 3, 3, 1, 0, 1, 1, 1, 2, 3, 4, 5, 1, -1, 1, 1, 1, 2, 3, 5, 7, 6, 1, 0, 1, 1, 1, 2, 3, 5, 8, 11, 10, 1, 0, 1, 1, 1, 2, 3, 5, 9, 13, 17, 14, 1, 0, 1, 1, 1, 2, 3, 5, 9, 14, 22, 28, 21, 1, 0

Offset: 0

Alois P. Heinz, Aug 29 2013

			Square array A(n,m) begins:
   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   0,  1,  1,  1,  1,  1,  1,  1,  1, ...
   0,  1,  2,  2,  2,  2,  2,  2,  2, ...
   0,  1,  2,  3,  3,  3,  3,  3,  3, ...
  -1,  1,  3,  4,  5,  5,  5,  5,  5, ...
   0,  1,  5,  7,  8,  9,  9,  9,  9, ...
   0,  1,  6, 11, 13, 14, 15, 15, 15, ...
  -1,  1, 10, 17, 22, 24, 25, 26, 26, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013

Columns m=0-10 give: A143064, A000012, A227360, A173173(n+1), A227374, A227375, A228646, A228644, A185648, A228645, A185649.

Diagonal gives: A005169.

nMax = 12; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A = Table[col[m][[1 ;; nMax + 1]], {m, 0, nMax}] // Transpose; a[n_ /; 0 <= n <= nMax, m_ /; 0 <= m <= nMax] := With[{n1 = n + 1, m1 = m + 1}, A[[n1, m1]]]; Table[a[n - m, m], {n, 0, nMax}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2016 *)

A228644 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=7.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 15, 26, 44, 76, 131, 225, 389, 670, 1156, 1994, 3439, 5934, 10236, 17661, 30470, 52569, 90699, 156483, 269985, 465811, 803677, 1386609, 2392357, 4127611, 7121498, 12286951, 21199078, 36575462, 63104849, 108876873, 187848862, 324101847

Offset: 0

Alois P. Heinz, Aug 28 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013
Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,0,-1,-1,-3,-2,-1,0,2,2,3,3,1,0,0,-2,-1,-1,-1).

Cf. A143064 (m=0), A227360 (m=2), A173173 (m=3), A227374 (m=4), A227375 (m=5), A228646(m=6), A228645 (m=9).

Column m=7 of A185646.

a:= n-> coeff(series(-(x^18 +x^17 +x^16 +2*x^15 +x^14 -2*x^11 -2*x^10 -2*x^9 -2*x^8 +x^5 +x^4 +x^3 +x^2-1) / ((x-1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)*(x^15 +x^14 +x^13 +2*x^12 -x^9 -2*x^8 -2*x^7 -x^6 +x^3 +x^2 +x-1)), x, n+1), x, n): seq(a(n), n=0..50);

nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A228644 = col[7][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)

G.f.: -(x^18 +x^17 +x^16 +2*x^15 +x^14 -2*x^11 -2*x^10 -2*x^9 -2*x^8 +x^5 +x^4 +x^3 +x^2-1) / ((x-1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)*(x^15 +x^14 +x^13 +2*x^12 -x^9 -2*x^8 -2*x^7 -x^6 +x^3 +x^2 +x-1)).

A228645 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=9.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 15, 26, 45, 78, 134, 232, 402, 695, 1205, 2086, 3613, 6259, 10841, 18780, 32531, 56354, 97621, 169111, 292954, 507488, 879136, 1522947, 2638242, 4570298, 7917253, 13715281, 23759370, 41159039, 71300984, 123516755, 213971647, 370669282

Offset: 0

Alois P. Heinz, Aug 28 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 0, 0, -1, -1, -2, -1, -3, -2, 0, 1, 3, 4, 4, 4, 4, 2, 0, -2, -3, -5, -4, -4, -3, -2, 0, 1, 1, 2, 2, 1, 1, 1).

Cf. A143064 (m=0), A227360 (m=2), A173173 (m=3), A227374 (m=4), A227375 (m=5), A228646 (m=6), A228644 (m=7).

Column m=9 of A185646.

a:= n-> coeff(series(-(x^30 +x^29 +x^28 +2*x^27 +2*x^26 +2*x^25 +x^24 +x^23 -x^22 -2*x^21 -2*x^20 -4*x^19 -4*x^18 -3*x^17 -2*x^16 -x^15 +2*x^13 +2*x^12 +3*x^11 +3*x^10 +x^9 +x^8 -x^5 -x^4 -x^3 -x^2+1) / ((x-1)*(x^2 +x+1)*(x^6 +x^3+1)*(x^26 +x^25 +x^24 +2*x^23 +2*x^22 +x^21 +x^20 -2*x^18 -2*x^17 -3*x^16 -3*x^15 -3*x^14 -x^13 -x^12 +x^11 +2*x^10 +2*x^9 +2*x^8 +x^7 +x^6 -x^3 -x^2 -x+1)), x, n+1), x, n): seq(a(n), n=0..50);

nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x] &; A228645 = col[9][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)

G.f.: -(x^30 +x^29 +x^28 +2*x^27 +2*x^26 +2*x^25 +x^24 +x^23 -x^22 -2*x^21 -2*x^20 -4*x^19 -4*x^18 -3*x^17 -2*x^16 -x^15 +2*x^13 +2*x^12 +3*x^11 +3*x^10 +x^9 +x^8 -x^5 -x^4 -x^3 -x^2+1) / ((x-1)*(x^2 +x+1)*(x^6 +x^3+1)*(x^26 +x^25 +x^24 +2*x^23 +2*x^22 +x^21 +x^20 -2*x^18 -2*x^17 -3*x^16 -3*x^15 -3*x^14 -x^13 -x^12 +x^11 +2*x^10 +2*x^9 +2*x^8 +x^7 +x^6 -x^3 -x^2 -x+1)).

A228646 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=6.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 15, 25, 43, 74, 126, 217, 372, 638, 1096, 1881, 3230, 5546, 9524, 16353, 28083, 48224, 82811, 142208, 244204, 419360, 720144, 1236670, 2123670, 3646879, 6262611, 10754485, 18468174, 31714525, 54461873, 93524824, 160605817, 275800867

Offset: 0

Alois P. Heinz, Aug 28 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013

Cf. A143064 (m=0), A227360 (m=2), A173173 (m=3), A227374 (m=4), A227375 (m=5), A228644 (m=7), A228645 (m=9).

Column m=6 of A185646.

nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A228646 = col[6][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)

A292420 Expansion of a q-series used by Ramanujan in his Lost Notebook.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 6, 8, 8, 11, 14, 16, 20, 24, 28, 34, 42, 48, 57, 68, 78, 94, 110, 126, 148, 172, 198, 230, 266, 304, 351, 404, 460, 526, 602, 684, 780, 888, 1004, 1140, 1290, 1456, 1646, 1856, 2088, 2351, 2644, 2964, 3326, 3728, 4168, 4664, 5212, 5812, 6484

Offset: 0

Michael Somos, Sep 16 2017

nonn

			G.f. = 1 + 2*x + 2*x^2 + 3*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=-1.

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A000009, A143064, A292445.

N:= 200: # to get a(0)..a(N)
g143064:= add(x^k/mul(1+x^(2*j+1),j=0..k),k=0..2*N):
g000009:= mul(1+x^(2*k),k=1..N):
S:= series(g143064*g000009,x,2*N+2):
seq(coeff(S,x,2*j),j=0..N); # Robert Israel, Sep 17 2017

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ QPochhammer[ x^2] / QPochhammer[ x] Sum[ (-1)^k x^(3 k^2 + 2 k) (1 + x^(2 k + 1)), {k, 0, Sqrt[n / 3]}], {x, 0, n}]];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A) * sum(k=0, sqrtint(n \ 3), (-1)^k * x^(3*k^2 + 2*k) * (1 + x^(2*k + 1)), A), n))};

G.f. is the product of the g.f. of A000009 and A143064.

Given g.f. A(x), then A(x^2) = 1 / (1+x) + x / (1+x^3) + x^2 * (1+x^2) / ((1+x^3) * (1+x^5)) + x^3 * (1+x^2) / ((1+x^5) * (1+x^7)) + x^4 * (1+x^2) * (1+x^4) / ((1+x^5) * (1+x^7) * (1+x^9)) + ...

A143065 Expansion of quotient of a Ramanujan false theta series by the theta series of triangular numbers in powers of x.

Original entry on oeis.org

1, 0, 0, -1, 1, -2, 2, -3, 4, -5, 6, -8, 11, -13, 16, -21, 27, -32, 39, -49, 61, -73, 87, -107, 131, -155, 184, -223, 267, -315, 372, -443, 526, -617, 722, -852, 1002, -1167, 1359, -1590, 1854, -2148, 2488, -2888, 3346, -3859, 4444, -5128, 5909, -6779, 7773

Offset: 0

Michael Somos, Jul 21 2008

sign

			G.f. = 1 - x^3 + x^4 - 2*x^5 + 2*x^6 - 3*x^7 + 4*x^8 - 5*x^9 + 6*x^10 + ...

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 13th equation.

Cf. A010054, A143064.

a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {x}, {x^2}, x^2, x^3], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=0, n, if( issquare( 3*k + 1, &m), (-1)^(m \ 3) * x^k ), A) / sum(k=0, (sqrtint(8*n + 1) - 1) \ 2, x^((k^2 + k) / 2), A), n))};

{a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n+1) - 1, (-1)^k * x^(k^2 + 2*k) * prod(j=1, k, (1 - x^(2*j - 1)) / (1 - x^(2*j))^2, 1 + O(x^(n + 1 - k^2 - 2*k)))), n))};

G.f.: ( 1 + x - x^5 - x^8 + x^16 + x^21 - ... ) / ( 1 + x + x^3 + x^6 + x^10 + x^15 + ... ). [Ramanujan]
G.f.: 1 - x^3 * (1 - x) / (1 - x^2)^2 + x^8 * (1 - x) * (1 - x^3) / ((1 - x^2)^2 * (1 - x^4)^2) - ... [Ramanujan]
Convolution with A010054 is A143064.

A291316 Expansion of x/(1-x) + x^4(1-x)/(1-x^3) + x^7(1-x)*(1-x^3)/(1-x^5) + ... in powers of x.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, -1, 1, 2, 0, 2, 1, 0, -1, 4, 2, -1, 2, -3, 4, 3, -1, 2, 0, 1, 1, 2, -2, 2, 5, 2, -3, 0, 1, -1, 6, 0, 4, -2, -1, 3, -1, 2, 0, 4, -2, 2, 4, -2, 1, 5, -2, -2, -2, 3, 6, 1, 3, -2, 4, -3, -1, -2, 3, 6, 2, 0, -4, 5, 1, 3, -1, 0, 0, 4, -1, -2, 4

Offset: 1

Michael Somos, Aug 22 2017

sign

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

George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See equation (1.5)

Cf. A008443, A143064, A290735, A290737.

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = sum(k=0, (n-1)\3, x^(3*k+1) * prod(i=1,k, 1 - x^(2*i-1), 1 + A) / (1 - x^(2*k+1)) ); polcoeff(A, n))};

a(n) = A008443(n) - A290735(n) = A290737(n) - A143064(n).

A308745 Expansion of 1/(1 - x(1 + x)/(1 - x^2(1 + x^2)/(1 - x^3(1 + x^3)/(1 - x^4(1 + x^4)/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 36, 76, 161, 342, 726, 1542, 3276, 6960, 14788, 31422, 66767, 141872, 301464, 640584, 1361188, 2892417, 6146164, 13060136, 27751818, 58970564, 125308114, 266270558, 565805452, 1202295228, 2554789536, 5428741218, 11535678790, 24512475453

Offset: 0

Ilya Gutkovskiy, Jun 21 2019

Cf. A005169, A053254, A092848, A143064.

nmax = 33; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^k (1 + x^k), 1, {k, 1, nmax}]), {x, 0, nmax}], x]

From Vaclav Kotesovec, Jun 25 2019: (Start)
a(n) ~ c * d^n, where
d = 2.124927028900893046638236231387101475346473032396641627320401...
c = 0.386397654364351443933577245182777062935616240164642598839093... (End)
From Peter Bala, Dec 18 2020: (Start)
Conjectural g.f.: 1/(2 - (1 + x)/(1 - x^2/(2 - (1 + x^3)/(1 - x^4/(2 - (1 + x^5)/(1 - x^6/(2 - ... ))))))).
More generally it appears that 1/(1 - t*x*(1 + u*x)/(1 - t*x^2*(1 + u*x^2)/(1 - t*x^3*(1 + u*x^3)/(1 - t*x^4*(1 + u*x^4)/(1 - ... ))))) = 1/(1 + u - (u + t*x)/(1 - t*x^2/(1 + u - (u + t*x^3)/(1 - t*x^4/(1 + u - (u + t*x^5)/(1 - ... )))))). (End)

cp's OEIS Frontend

A185646 Square array A(n,m), n>=0, m>=0, read by antidiagonals, where column m is the expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))).

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A228644 Expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))) for m=7.

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A228645 Expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))) for m=9.

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A228646 Expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))) for m=6.

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A292420 Expansion of a q-series used by Ramanujan in his Lost Notebook.

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A143065 Expansion of quotient of a Ramanujan false theta series by the theta series of triangular numbers in powers of x.

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A291316 Expansion of x/(1-x) + x^4*(1-x)/(1-x^3) + x^7*(1-x)*(1-x^3)/(1-x^5) + ... in powers of x.

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A308745 Expansion of 1/(1 - x*(1 + x)/(1 - x^2*(1 + x^2)/(1 - x^3*(1 + x^3)/(1 - x^4*(1 + x^4)/(1 - ...))))), a continued fraction.

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A185646 Square array A(n,m), n>=0, m>=0, read by antidiagonals, where column m is the expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))).

A228644 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=7.

A228645 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=9.

A228646 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=6.

A291316 Expansion of x/(1-x) + x^4(1-x)/(1-x^3) + x^7(1-x)*(1-x^3)/(1-x^5) + ... in powers of x.

A308745 Expansion of 1/(1 - x(1 + x)/(1 - x^2(1 + x^2)/(1 - x^3(1 + x^3)/(1 - x^4(1 + x^4)/(1 - ...))))), a continued fraction.