A286509 -id:A286509 - cp's OEIS frontend

A291651 a(n) = [x^n] (1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...)))))))^n, a continued fraction.

1, -1, 3, -7, 15, -26, 15, 153, -1049, 4790, -18522, 64481, -206181, 606384, -1615121, 3715993, -6289929, 550850, 61250694, -382787092, 1726745790, -6691501530, 23413714107, -75179994017, 221304346963, -586004040651, 1318720868416, -2044320913276, -1137686341077, 28530838758784, -165361803129585

Offset: 0

Ilya Gutkovskiy, Aug 28 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Main diagonal of A286509.

Cf. A291335.

Table[SeriesCoefficient[1/(1 + ContinuedFractionK[x^i, 1, {i, 1, n}])^n, {x, 0, n}], {n, 0, 30}]

a(n) = A286509(n,n).

A291652 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, 2, 0, 1, 4, 6, 6, 3, 0, 1, 5, 10, 13, 11, 5, 0, 1, 6, 15, 24, 27, 20, 9, 0, 1, 7, 21, 40, 55, 54, 38, 15, 0, 1, 8, 28, 62, 100, 120, 109, 70, 26, 0, 1, 9, 36, 91, 168, 236, 258, 216, 129, 45, 0, 1, 10, 45, 128, 266, 426, 540, 544, 423, 238, 78, 0, 1, 11, 55, 174, 402, 721, 1035, 1205, 1127, 824, 437, 135, 0

Offset: 0

Ilya Gutkovskiy, Aug 28 2017

			G.f. of column k: A_k(x) = 1 + k*x + k*(k + 1)*x^2/2 +  k*(k^2 + 3*k + 8)*x^3/6 + k*(k^3 + 6*k^2 + 35*k + 30)*x^4/24 + ...
Square array begins:
1,  1,   1,   1,    1,    1,  ...
0,  1,   2,   3,    4,    5,  ...
0,  1,   3,   6,   10,   15,  ...
0,  2,   6,  13,   24,   40,  ...
0,  3,  11,  27,   55,  100,  ...
0,  5,  20,  54,  120,  236,  ...

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

Columns k=0..1 give A000007, A005169.
Rows n=0..3 give A000012, A001477, A000217, A003600 (with a(0)=0).
Main diagonal gives A291653.
Cf. A286509, A286933.

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[((Sum[(-1)^i x^(i (i + 1))/Product[(1 - x^m), {m, 1, i}], {i, 0, n}])/(Sum[(-1)^i x^(i^2)/Product[(1 - x^m), {m, 1, i}], {i, 0, n}]))^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: (1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))))^k, a continued fraction.

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.

A286932 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 + kx/(1 + kx^2/(1 + kx^3/(1 + kx^4/(1 + k*x^5/(1 + ...)))))).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 4, 0, 0, 1, -4, 9, -4, -1, 0, 1, -5, 16, -18, 0, 1, 0, 1, -6, 25, -48, 27, 8, -1, 0, 1, -7, 36, -100, 128, -27, -24, 1, 0, 1, -8, 49, -180, 375, -320, -27, 48, 0, 0, 1, -9, 64, -294, 864, -1375, 704, 243, -64, -1, 0, 1, -10, 81, -448, 1715, -4104, 4875, -1280, -810, 48, 2, 0

Offset: 0

Ilya Gutkovskiy, May 16 2017

			G.f. of column k: A(x) = 1 - k*x + k^2*x^2 - (k - 1)*k^2*x^3 + (k - 2)*k^3*x^4 - k^3*(k^2 - 3*k + 1)*x^5 + ...
Square array begins:
  1,  1,  1,   1,    1,     1,  ...
  0, -1, -2,  -3,   -4,    -5,  ...
  0,  1,  4,   9,   16,    25,  ...
  0,  0, -4, -18,  -48,  -100,  ...
  0, -1,  0,  27,  128,   375,  ...
  0,  1,  8, -27, -320, -1375,  ...

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Columns k=0..1 give: A000007, A007325.
Rows n=0..3 give: A000012, A001489, A000290, A045991 (gives absolute value).
Main diagonal gives A291335.
Cf. A286509.

Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[k x^i, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: 1/(1 + k*x/(1 + k*x^2/(1 + k*x^3/(1 + k*x^4/(1 + k*x^5/(1 + ...)))))), a continued fraction.

G.f. of column k (for k > 0): (Sum_{j>=0} k^j*x^(j*(j+1))/Product_{i=1..j} (1 - x^i)) / (Sum_{j>=0} k^j*x^(j^2)/Product_{i=1..j} (1 - x^i)).

A302015 Expansion of 1/(1 - x/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, -1, 0, 1, 0, -1, 1, 1, -2, -1, 2, 0, -2, 2, 3, -3, -3, 4, 0, -7, 3, 9, -5, -7, 10, 4, -17, -1, 21, -7, -21, 21, 19, -36, -13, 47, -5, -56, 36, 64, -69, -54, 104, 15, -147, 41, 177, -115, -168, 221, 116, -344, -15, 442, -159, -481, 422, 443, -736, -280, 1034, -90, -1276, 681

Offset: 0

Ilya Gutkovskiy, Mar 30 2018

sign

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Antidiagonal sums of A286509.

Cf. A007325, A167750, A302016.

nmax = 68; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 68; CoefficientList[Series[1/(1 - x QPochhammer[x, x^5] QPochhammer[x^4, x^5]/(QPochhammer[x^2, x^5] QPochhammer[x^3, x^5])), {x, 0, nmax}], x]

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

a(0) = 1; a(n) = Sum_{k=1..n} A007325(k-1)*a(n-k).

A295703 Expansion of R(x*R(x)), where R(x) = 1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))), a continued fraction (g.f. for A007325).

Original entry on oeis.org

1, -1, 2, -3, 2, 4, -18, 43, -80, 123, -148, 78, 287, -1364, 3858, -8627, 15901, -23076, 20061, 18294, -140623, 420241, -930040, 1655753, -2293975, 1872682, 1835066, -12983537, 37871888, -83222132, 149287250, -212064236, 186932259, 131172644, -1139053896, 3449157957, -7710640256

Offset: 0

Ilya Gutkovskiy, Nov 29 2017

sign

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A003823, A007325, A055101, A055102, A055103, A127632, A286509, A291651.

nmax = 36; CoefficientList[Series[1/(1 + ContinuedFractionK[(x/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}]))^k, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
g[x_] := g[x] = QPochhammer[x, x^5] QPochhammer[x^4, x^5]/(QPochhammer[x^2, x^5] QPochhammer[x^3, x^5]); a[n_] := a[n] = SeriesCoefficient[g[x g[x]], {x, 0, n}];  Table[a[n], {n, 0, 36}]

G.f.: 1/(1 + x/(1 + x/(1 + x^2/(1 + x^3/(1 + ...))))/(1 + x^2/(1 + x/(1 + x^2/(1 + x^3/(1 + ...))))^2/(1 + x^3/(1 + x/(1 + x^2/(1 + x^3/(1 + ...))))^3/(1 + ...)))), a continued fraction.

cp's OEIS Frontend

A291651 a(n) = [x^n] (1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...)))))))^n, a continued fraction.

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A291652 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 - ...)))))).

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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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A286932 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 + k*x/(1 + k*x^2/(1 + k*x^3/(1 + k*x^4/(1 + k*x^5/(1 + ...)))))).

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A302015 Expansion of 1/(1 - x/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))))), a continued fraction.

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A295703 Expansion of R(x*R(x)), where R(x) = 1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))), a continued fraction (g.f. for A007325).

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Programs

Mathematica

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A286932 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 + kx/(1 + kx^2/(1 + kx^3/(1 + kx^4/(1 + k*x^5/(1 + ...)))))).