A005169 -id:A005169 - cp's OEIS frontend

A288267 Triangle read by rows: T(n,k) = T(n,k+1) + T(n-k,k-1) with T(0,0) = 1 and T(n,k) = 0 if k<0 or k > A003056(n).

1, 1, 1, 1, 1, 2, 2, 1, 3, 3, 1, 5, 5, 2, 9, 9, 4, 1, 15, 15, 6, 1, 26, 26, 11, 2, 45, 45, 19, 4, 78, 78, 33, 7, 1, 135, 135, 57, 12, 1, 234, 234, 99, 21, 2, 406, 406, 172, 37, 4, 704, 704, 298, 64, 7, 1222, 1222, 518, 112, 13, 1, 2120, 2120, 898, 194, 22, 1, 3679

Offset: 0

Seiichi Manyama, Sep 01 2017

			First few rows are:
   1;
   1,  1;
   1,  1;
   2,  2, 1;
   3,  3, 1;
   5,  5, 2;
   9,  9, 4, 1;
  15, 15, 6, 1.

Seiichi Manyama, Rows n = 0..481, flattened

Columns 0+1,2 give A005169, A289080 (for n>0).

Cf. A003056.

T:= proc(n,k) option remember; `if`(k<0 or k*(k+1)/2>n, 0,
      `if`(n=0, 1, T(n, k+1)+T(n-k, k-1)))
    end:
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20); # Alois P. Heinz, Sep 01 2017

T[n_, k_] := T[n, k] = If[k < 0 || k(k+1)/2 > n, 0, If[n == 0, 1, T[n, k+1] + T[n-k, k-1]]];
Table[T[n, k], {n, 0, 20}, {k, 0, Floor[(Sqrt[8n+1]-1)/2]}] // Flatten (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz *)

A290771 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the continued fraction 1/(1 - x/(1 - x^(2^k)/(1 - x^(3^k)/(1 - x^(4^k)/(1 - x^(5^k)/(1 - ...)))))).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 14, 1, 1, 1, 1, 3, 42, 1, 1, 1, 1, 1, 5, 132, 1, 1, 1, 1, 1, 2, 9, 429, 1, 1, 1, 1, 1, 1, 3, 15, 1430, 1, 1, 1, 1, 1, 1, 1, 4, 26, 4862, 1, 1, 1, 1, 1, 1, 1, 1, 5, 45, 16796, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 78, 58786, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 10, 135, 208012

Offset: 0

Ilya Gutkovskiy, Aug 10 2017

			Square array begins:
   1,  1,  1,  1,  1,  1, ...
   1,  1,  1,  1,  1,  1, ...
   2,  1,  1,  1,  1,  1, ...
   5,  2,  1,  1,  1,  1, ...
  14,  3,  1,  1,  1,  1, ...
  42,  5,  2,  1,  1,  1, ...

Columns k = 0..5 give A000108, A005169, A206739, A291146, A291149, A291168.

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

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

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.

A301627 G.f. A(x) satisfies: A(x) = 1/(1 - xA(x)/(1 - x^2A(x)^2/(1 - x^3A(x)^3/(1 - x^4A(x)^4/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 6, 20, 71, 265, 1024, 4059, 16414, 67451, 280856, 1182379, 5024361, 21522055, 92833874, 402879747, 1757852317, 7706728006, 33932931008, 149986338830, 665276977574, 2960306454110, 13210976195068, 59114318997648, 265166069469324, 1192145264317628, 5370983954821322

Offset: 0

Ilya Gutkovskiy, Mar 24 2018

nonn

			G.f. A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 71*x^5 + 265*x^6 + 1024*x^7 + 4059*x^8 + 16414*x^9 + 67451*x^10 + ...
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 55*x^4/4 + 236*x^5/5 + 1035*x^6/6 + 4593*x^7/7 + 20551*x^8/8 + ... + A291653(n)*x^n/n + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A005169, A192728, A192737, A291653, A301362, A301629.

a(n) ~ c * d^n / n^(3/2), where d = 4.760595370947474723688065553003203505424287110594102605580439495640678... and c = 0.395762805862214496152624315213041270339036... - Vaclav Kotesovec, Apr 08 2018

A301921 Expansion of e.g.f. 1/(1 - (exp(x) - 1)/(1 - (exp(x) - 1)^2/(1 - (exp(x) - 1)^3/(1 - ...)))), a continued fraction.

Original entry on oeis.org

1, 1, 3, 19, 159, 1651, 21303, 324619, 5653119, 110909251, 2424648903, 58430418619, 1537673312079, 43860906193651, 1347852526593303, 44392923532503019, 1560023977386027039, 58259266750803410851, 2303999137417453606503, 96188099015599819297819, 4227325636692027926037999

Offset: 0

Ilya Gutkovskiy, Jun 19 2018

nonn

From Peter Bala, Aug 19 2025: (Start)

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 9 we obtain the sequence [0, 1, 3, 1, 6, 4, 0, 7, 3, 1, 6, 4, 0, 7, 3, 1, 6, 4, 0, 7, ...] with an apparent period of 6 = phi(9) beginning at n = 2. Cf. A004123. (End)

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 159*x^4/4! + 1651*x^5/5! + 21303*x^6/6! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A005169, A019538, A301923.

nmax = 20; CoefficientList[Series[1/(1 + ContinuedFractionK[-(Exp[x] - 1)^k, 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
b[n_] := b[n] = SeriesCoefficient[1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, n}]), {x, 0, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k] k!, {k, 0, n}]; Table[a[n], {n, 0, 20}]

a(n) = Sum_{k=0..n} Stirling2(n,k)*A005169(k)*k!.

a(n) ~ c * d^n * n!, where d = 2.19787763261059933075080498218168228... and c = 0.250957960982243982921501085974065... - Vaclav Kotesovec, Dec 20 2018

A058300 Number of ways of piling up n wine bottles above a row of n+1 bottles at ground level.

Original entry on oeis.org

1, 1, 1, 3, 7, 16, 43, 115, 303, 813, 2203, 5991, 16371, 44917, 123598, 340988, 942930, 2612735, 7252407, 20163046, 56136326, 156488946, 436739752, 1220157514, 3412116339, 9550192161, 26751643663, 74991516850, 210364915858, 590490257667, 1658484275955

Offset: 0

Roland Bacher, Dec 08 2000

Related to the Catalan numbers (which count the ways of storing an arbitrary number of bottles above n bottles at ground level).

Related to fountains of n coins (A005169). [Joerg Arndt, Mar 18 2011]

			a(4) = 7: the seven possibilities are:
..............0.............0.........0...............0.........0............0
.0.0.0.0.....0.0.0.......0.0.0.......0.0...0.....0...0.0.......0.0.0......0.0.0
0.0.0.0.0.,.0.0.0.0.0.,.0.0.0.0.0.,.0.0.0.0.0.,.0.0.0.0.0.,.0.0.0.0.0,.0.0.0.0.0

R. P. Stanley, Enumerative Combinatorics (Volume 2); see Exercise 6.19(hhh).

Alois P. Heinz, Table of n, a(n) for n = 0..600
Andrew M. Odlyzko and Herbert S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.

Cf. A047998.

terms = 31; initialMax = 5; Clear[g]; g[max_] := g[max] = (Print["max = ", max]; f = 1/Fold[1 - y*x^#2/#1&, 1, Range[max] // Reverse]; b[n_, k_] := SeriesCoefficient[f, {x, 0, n}, {y, 0, k}]; b[0, 0] = 1; Clear[a]; a[n_] := a[n] = b[2n+1, n+1]; Array[a, terms, 0]); g[max = initialMax]; g[max = max+1]; While[g[max] != g[max-1], max = max+1]; A058300 = g[max] (* Jean-François Alcover, Oct 05 2017, after Alois P. Heinz's formula *)

Coefficient of w^(2*n+1)*z^(n+1) in the formal power series G(w, z) defined by G(w, z)=1+w*z*G(w, w*z).
a(n) = A047998(2n+1,n+1). - Alois P. Heinz, Jun 24 2015
a(n) ~ c * d^n / sqrt(n), where d = 2.8566122635122125634030051... and c = 0.19212135026441477122126... - Vaclav Kotesovec, Jul 17 2019

More terms from Alois P. Heinz, Jun 24 2015

A167751 Diagonal sums of A167749.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 7, 14, 27, 52, 102, 198, 387, 755, 1474, 2879, 5622, 10982, 21450, 41900, 81847, 159880, 312315, 610086, 1191768, 2328054, 4547732, 8883767, 17354001, 33900200, 66222412, 129362318, 252703135, 493643580, 964309346

Offset: 0

Paul Barry, Nov 10 2009

Hankel transform is A167753.

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 14*x^7 + 27*x^8 + 52*x^9 + ...

G.f.: 1/(1-x^2/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-...)))))) (continued fraction);

G.f.: 1/(1-x^2*f(x)), f(x) the g.f. of A005169.

A168445 Number of compositions a(1),...,a(k) of n, for some k, such that a(i+1) <= a(i) + 1 for 1 <= i < k and a(1) <= a(k) + 1.

Original entry on oeis.org

1, 2, 4, 6, 11, 18, 31, 52, 91, 155, 268, 464, 802, 1390, 2411, 4178, 7249, 12578, 21823, 37870, 65724, 114061, 197960, 343578, 596317, 1034983, 1796359, 3117837, 5411478, 9392460, 16302081, 28294850, 49110242, 85238716, 147945552, 256783448, 445689300

Offset: 1

Vladeta Jovovic, Nov 25 2009

nonn

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.

Alois P. Heinz, Table of n, a(n) for n = 1..400

Cf. A003116, A005169.

b:= proc(n,r,f) option remember; `if`(n=0, `if`(f-1<=r, 1, 0),
      add(b(n-i, i, f), i=1..min(r+1, n)))
    end:
a:= n-> add(b(n-i, i, i), i=1..n):
seq(a(n), n=1..40);  # Alois P. Heinz, Dec 15 2009

b[n_, r_, f_] := b[n, r, f] = If[n == 0, If[f - 1 <= r, 1, 0], Sum[b[n - i, i, f], {i, 1, Min [r + 1, n]}]];
a[n_] := Sum[b[n - i, i, i], {i, 1, n}];
Array[a, 40] (* Jean-François Alcover, Nov 10 2020, after Alois P. Heinz *)

a(n) ~ c / r^n, where r = A347901 = 0.576148769142756602297868573719938782354724663118974... is the lowest root of the equation Sum_{k>=0} (-1)^k * r^(k^2) / QPochhammer(r, r, k) = 0 and c = 0.6149126319329581124890112676009720339906790088212712130894... - Vaclav Kotesovec, May 01 2014, updated Sep 09 2020

More terms from Alois P. Heinz, Dec 15 2009

A187080 Triangle T(n,k) read by rows: fountains of n coins and height k.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 7, 1, 0, 0, 0, 0, 1, 12, 2, 0, 0, 0, 0, 0, 1, 20, 5, 0, 0, 0, 0, 0, 0, 1, 33, 11, 0, 0, 0, 0, 0, 0, 0, 1, 54, 22, 1, 0, 0, 0, 0, 0, 0, 0, 1, 88, 44, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 143, 85, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 232, 161, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 376, 302, 25, 0, 0, 0

Offset: 0

Joerg Arndt, Mar 08 2011

See A005169 for the definition of a "fountain of n coins". [John W. Layman, Mar 10 2011]

			Triangle begins:
1;
0,1;
0,1,0;
0,1,1,0;
0,1,2,0,0;
0,1,4,0,0,0;
0,1,7,1,0,0,0;
0,1,12,2,0,0,0,0;
0,1,20,5,0,0,0,0,0;
0,1,33,11,0,0,0,0,0,0;
0,1,54,22,1,0,0,0,0,0,0;
0,1,88,44,2,0,0,0,0,0,0,0;
0,1,143,85,5,0,0,0,0,0,0,0,0;
0,1,232,161,12,0,0,0,0,0,0,0,0,0;
0,1,376,302,25,0,0,0,0,0,0,0,0,0,0;
0,1,609,559,52,1,0,0,0,0,0,0,0,0,0,0;
0,1,986,1026,105,2,0,0,0,0,0,0,0,0,0,0,0;
0,1,1596,1870,207,5,0,0,0,0,0,0,0,0,0,0,0,0;
The 15 compositions corresponding to fountains of 7 coins are the following:
   #:    composition      height
   1:    [ 1 2 3 1 ]        3
   2:    [ 1 2 2 2 ]        2
   3:    [ 1 1 2 3 ]        3
   4:    [ 1 2 2 1 1 ]      2
   5:    [ 1 2 1 2 1 ]      2
   6:    [ 1 1 2 2 1 ]      2
   7:    [ 1 2 1 1 2 ]      2
   8:    [ 1 1 2 1 2 ]      2
   9:    [ 1 1 1 2 2 ]      2
  10:    [ 1 2 1 1 1 1 ]    2
  11:    [ 1 1 2 1 1 1 ]    2
  12:    [ 1 1 1 2 1 1 ]    2
  13:    [ 1 1 1 1 2 1 ]    2
  14:    [ 1 1 1 1 1 2 ]    2
  15:    [ 1 1 1 1 1 1 1 ]  1
  stats:  0 1 12 2 0 0 0 0

Seiichi Manyama, Rows n = 0..25, flattened

Row sums give A005169 (fountains of n coins).

Cf. A047998, A187081 (sandpiles by height).

b[n_, i_, h_] := b[n, i, h] = If[n == 0, x^h, Sum[b[n - j, j, Max[h, j]], {j, 1, Min[i + 1, n]}]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, 0, 0];
Table[T[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, May 31 2019, after Alois P. Heinz in A291878 *)

T(n,1) + T(n,2) = Fibonacci(n).

A206737 G.f.: 1/(1 - x/(1 - x^4/(1 - x^7/(1 - x^10/(1 - x^13/(1 - x^16/(1 -...- x^(3*n-2)/(1 -...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 20, 28, 39, 54, 76, 107, 150, 210, 294, 412, 578, 811, 1137, 1593, 2233, 3131, 4390, 6155, 8629, 12097, 16959, 23777, 33336, 46737, 65524, 91863, 128790, 180563, 253149, 354912, 497581, 697602, 978031, 1371190, 1922395

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

We have the simple continued fraction expansions (A(x) is the sequence o.g.f.): A(1/n) = [1; n - 2, 1, n^3 - 2, 1, n^4 - 2, 1, n^6 - 2, 1, n^7 - 2, 1, n^9 - 2, 1, n^10 - 2, 1, ...] for n >= 3 and A(-1/n) = [0; 1, n - 1, 1, n^3 - 1, n^4 - 1, 1, n^6 - 1, n^7 - 1, 1, n^9 - 1, n^10 - 1, 1, ...] for n >= 2. Cf. A005169, A111317 and A143951. - Peter Bala, Dec 15 2015

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + ...
Simple continued fraction expansions: A(1/10) = 1.11112345816325284441923227158 ... = [1, 8, 1, 998, 1, 9998, 1, 999998, 1, 9999998, 1, 999999998, 1, 9999999998, 1, ...]; A(-1/10) = 0.909082643877542661578687284018 ... = [0, 1, 9, 1, 999, 9999, 1, 999999, 9999999, 1, 999999999, 9999999999, 1, ...]. - _Peter Bala_, Dec 15 2015

Robert Israel, Table of n, a(n) for n = 0..2000
Peter Bala, Some simple continued fraction expansions associated with A206737

Cf. A206738, A143951, A005169, A111317, A143951, A224704.

N:= 100: # to get a(0) .. a(N)
C:= [0,[1,1],seq([-x^i,1],i=1..N,3)]:
S:= series(numtheory:-cfrac(C),x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Dec 28 2015

max = 15; CF = 1+x*O[x]^max; M = Sqrt[max+1]//Floor; For[k=0, k <= M, k++, CF = 1/(1-x^(3M-3k+1)*CF)]; CoefficientList[CF, x] (* Jean-François Alcover, Dec 29 2015, adapted from PARI *)
nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(3*Range[nmax + 1]-2)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)

{a(n)=local(CF=1+x*O(x^n),M=sqrtint(n+1)); for(k=0, M, CF=1/(1-x^(3*M-3*k+1)*CF)); polcoeff(CF, n, x)}
for(n=0,55,print1(a(n),", "))

a(n) ~ c * d^n, where d = 1.40198938377739909105003523518827... and c = 0.34165269320144328278000954698... - Vaclav Kotesovec, Aug 25 2017
From Peter Bala, Jul 03 2019: (Start)
O.g.f. as a ratio of q-series: N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*q^(3*n^2+n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))) and D(q) =  Sum_{n >= 0} (-1)^n*q^(3*n^2-2*n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))). Cf. A143951, A224704 and A206738.
D(q) has a simple real zero at x = 0.7132721628.... The constants c and d quoted in the above asymptotic approximation for a(n) are given by d = 1/x and c = - N(x)/(x*D'(x)), where the prime indicates differentiation w.r.t. q. (End)

cp's OEIS Frontend

A288267 Triangle read by rows: T(n,k) = T(n,k+1) + T(n-k,k-1) with T(0,0) = 1 and T(n,k) = 0 if k<0 or k > A003056(n).

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A290771 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the continued fraction 1/(1 - x/(1 - x^(2^k)/(1 - x^(3^k)/(1 - x^(4^k)/(1 - x^(5^k)/(1 - ...)))))).

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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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A301627 G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x)/(1 - x^2*A(x)^2/(1 - x^3*A(x)^3/(1 - x^4*A(x)^4/(1 - ...))))), a continued fraction.

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A301921 Expansion of e.g.f. 1/(1 - (exp(x) - 1)/(1 - (exp(x) - 1)^2/(1 - (exp(x) - 1)^3/(1 - ...)))), a continued fraction.

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A058300 Number of ways of piling up n wine bottles above a row of n+1 bottles at ground level.

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A167751 Diagonal sums of A167749.

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A168445 Number of compositions a(1),...,a(k) of n, for some k, such that a(i+1) <= a(i) + 1 for 1 <= i < k and a(1) <= a(k) + 1.

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A301627 G.f. A(x) satisfies: A(x) = 1/(1 - xA(x)/(1 - x^2A(x)^2/(1 - x^3A(x)^3/(1 - x^4A(x)^4/(1 - ...))))), a continued fraction.