A000932 -id:A000932 - cp's OEIS frontend

A180048 Coefficient triangle of the denominators of the (n-th convergents to) the continued fraction 1/(w+2/(w+3/(w+4/... . Conjectured to equal unsigned version of A137286.

1, 0, 1, 2, 0, 1, 0, 5, 0, 1, 8, 0, 9, 0, 1, 0, 33, 0, 14, 0, 1, 48, 0, 87, 0, 20, 0, 1, 0, 279, 0, 185, 0, 27, 0, 1, 384, 0, 975, 0, 345, 0, 35, 0, 1, 0, 2895, 0, 2640, 0, 588, 0, 44, 0, 1, 3840, 0, 12645, 0, 6090, 0, 938, 0, 54, 0, 1, 0, 35685, 0, 41685, 0, 12558, 0, 1422, 0, 65

Offset: 0

Wouter Meeussen, Aug 08 2010

Equivalence to the recurrence formula needs formal proof. This continued fraction converges to 0.525135276160981... for w=1. A conjecture by Ramanujan puts this equal to -1 + 1/(sqrt(e*Pi/2) - Sum_{k>=1} 1/(2k-1)!!).
From Alexander Kreinin, Oct 26 2015: (Start)
Let us denote the continued fraction by U(w).
Then it is easy to show that Mill's ratio, R(w) = (1 - Phi(w))/f(w), where Phi is the standard normal distribution function and f is the standard normal density function, satisfies R(w) = 1/(w + U(w)).
Indeed, R(w) = 1/(w+1/(w+2/(w+3/(w+... Then we find U(w) = 1/R(w) - w. It was proved in Alexander Kreinin (arXiv:1405.5852) that R(w+t) + Q(w, t) = exp(w*t + w^2/2)*R(t), where Q(w,t) = Sum_{k>=0} Sum_{m=0..k} q(k,m) * t^m * w^(k+1)/(k+1)!.
Substituting t=0, we obtain R(w) = exp(w^2/2)*sqrt(Pi/2) - Sum_{n>=0} w^(2n+1)/(2n+1)!!. If w=1 we obtain Ramanujan's formula. (End)

			The denominator of 1/(w+2/(w+3/(w+4/(w+5/(w+6/w))))) equals 48 + 87w^2 + 20w^4 + w^6.
From _Joerg Arndt_, Apr 20 2013: (Start)
Triangle begins
     1;
     0,     1;
     2,     0,     1;
     0,     5,     0,     1;
     8,     0,     9,     0,    1;
     0,    33,     0,    14,    0,   1;
    48,     0,    87,     0,   20,   0,   1;
     0,   279,     0,   185,    0,  27,   0,  1;
   384,     0,   975,     0,  345,   0,  35,  0,  1;
     0,  2895,     0,  2640,    0, 588,   0, 44,  0, 1;
  3840,     0, 12645,     0, 6090,   0, 938,  0, 54, 0, 1;
     0, 35685,     0, 41685,    0, ... (End)

G. C. Greubel, Table of n, a(n) for the first 75 rows, flattened
Authors?, Hungarian discussion forum
Alexander Kreinin, Combinatorial Properties of Mills' Ratio, arXiv:1405.5852 [math.CO], 2014. See Table 3.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, preprint, 2016.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences 19 (2016), Article 16.6.2.

Cf. A137286, A084950, A180047, A180049.

Cf. A111129.

Table[ CoefficientList[ Denominator[ Together[ Fold[ #2/(w+#1) &, Infinity, Reverse @ Table[ k, {k, 1, n} ] ] ] ], w ], {n, 16} ] (* or equivalently *) Clear[ p ];p[ 0 ]=1; p[ 1 ]=w; p[ n_ ]:=p[ n ]= w*p[ n-1 ] + n*p[ n-2 ]; Table[ CoefficientList[ p[ k ]//Expand, w ], {k,0,15} ]

p(0)=1; p(1)=w; p(n) = w*p(n-1) + n*p(n-2) (conjecture).
T(n,k) = T(n-1,k-1) + n*T(n-2,k), T(0,0) = 1, T(1,0) = 0, T(1,1) = 1. - Philippe Deléham, Oct 28 2013
sum_{k=0..n} T(n,k) = A000932(n). - Philippe Deléham, Oct 28 2013
T(2n,0) = A000165(n); T(2n+1,1) = A129890(n); T(2n+2,2) = A035101(n+2). - Philippe Deléham, Oct 28 2013

A225435 Numerators of convergents to the general continued fraction 1/(1 + 2/(1 + 3/(1 + 4/(1+ ...)))).

Original entry on oeis.org

1, 1, 2, 4, 7, 19, 68, 44, 416, 758, 6092, 24284, 10348, 110864, 997828, 4545476, 827252, 5166356, 255994804, 1289266004, 3332578444, 8757252244, 3766552348, 27079574548, 1434303566956, 4061479240156, 46849154788124, 54858398447372, 816458740546228, 189647639388428

Offset: 1

Eric W. Weisstein, May 07 2013

			1, 1/3, 2/3, 4/9, 7/12, 19/39, ... = A225435(n)/A225436(n).

Seiichi Manyama, Table of n, a(n) for n = 1..843
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Cf. A225436 (denominators).
Cf. A111129 (decimal digits of infinite c.f.).
Related to A000932.

Numerator[Table[ContinuedFractionK[k, 1, {k, 1, n}], {n, 30}]]

E.g.f.: (1/2)*(-2+e^((1/2)*z*(2+z))*(1+z)(2+sqrt(2*e*Pi)*erf(1/sqrt(2)))-e^((1/2)*(1+z)^2)*sqrt(2*Pi)*(1+z)*erf((1+z)/sqrt(2))).

Lim_{n->infinity} A225435(n)/A225436(n) = sqrt(2/(e*Pi))/erfc(1/sqrt(2))-1 = A111129.

A225436 Denominators of convergents to the general continued fraction 1/(1 + 2/(1 + 3/(1 + 4/(1+ ...)))).

Original entry on oeis.org

1, 3, 3, 9, 12, 39, 123, 87, 771, 1473, 11427, 46779, 19533, 212559, 1890093, 8691981, 1570137, 9863961, 486463449, 2459255649, 6337494039, 16694653089, 7166066763, 51605000913, 2729643372111, 7738039298811, 89176449644619, 104501330075607, 1554311845035993, 361227369257943

Offset: 1

Eric W. Weisstein, May 07 2013

			1, 1/3, 2/3, 4/9, 7/12, 19/39, ... = A225435(n)/A225436(n).

Seiichi Manyama, Table of n, a(n) for n = 1..843
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Cf. A225435 (numerators).
Cf. A111129 (decimal digits of infinite c.f.).
Related to A000932.

Denominator[Table[ContinuedFractionK[k, 1, {k, 1, n}], {n, 30}]]

E.g.f: (1/2)*(2+e^((1/2)*(1+z)^2)*sqrt(2*Pi)*(1+z)*(-erf(1/sqrt(2))+erf((1+z)/sqrt(2)))).

Limit_{n->oo} A225435(n)/a(n) = sqrt(2/(e*Pi))/erfc(1/sqrt(2))-1 = A111129.

A173895 E.g.f. satisfies: A'(x) = 1/(1 + x*A(x)) with A(0) = 1.

Original entry on oeis.org

1, 1, -1, 0, 9, -48, 15, 2448, -24927, 23424, 3091311, -47659200, 88056969, 10702667520, -225139993377, 679791291648, 78646340795265, -2128005345251328, 9456106738649631, 1053535684549174272

Offset: 0

Peter Bala, Nov 26 2010

Define a polynomial sequence P_n(x) recursively by
... P_0(x) = 1, and for n >= 1
... P_n(x) = (x-1)*P_(n-1)(x-1)-n*P_(n-1)(x+1).
The first few polynomials are
P_1(x) = x-2
P_2(x) = x^2-6*x+5
P_3(x) = x^3-12*x^2+32*x-12.
It appears that a(n+1) = P_n(1) (checked as far as a(19)).
Compare with A144010.

			E.g.f.: A(x) = 1 + x - x^2/2! + 9*x^4/4! - 48*x^5/5! + 15*x^6/6! + 2448*x^7/7! +...
where
1/(1 + x*A(x)) = 1 - x + 9*x^3/3! - 48*x^4/4! + 15*x^5/5! + 2448*x^6/6! +...
Also, A(G(x)) = 1 + x where
G(x) = x + x^2/2! + 3*x^3/3! + 6*x^4/4! + 18*x^5/5! + 48*x^6/6! + 156*x^7/7! + 492*x^8/8! +...+ A000932(n-1)*x^n/n! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..190

Cf. A144010, A000932.

m = 20; A[_] = 1;
Do[A[x_] = 1 + Integrate[1/(1+x*A[x])+O[x]^m, x]+O[x]^m // Normal, {m}];
CoefficientList[A[x], x] * Range[0, m-1]! (* Jean-François Alcover, Nov 02 2019 *)

{a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1/(1+x*A+x*O(x^n)) ));n!*polcoeff(A, n)}

E.g.f. satisfies: A(x) = 1 + Integral 1/(1 + x*A(x)) dx.

E.g.f. satisfies: A(G(x)) = 1 + x where G(x) is the e.g.f. of A000932 (offset 1). [Paul D. Hanna, Aug 23 2011]

A070895 Triangle read by rows where T(n+1,k)=T(n,k)+n*T(n-1,k) starting with T(n,n)=1 and T(n,k)=0 if n

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 6, 4, 1, 1, 26, 18, 8, 5, 1, 1, 76, 48, 28, 10, 6, 1, 1, 232, 156, 76, 40, 12, 7, 1, 1, 764, 492, 272, 110, 54, 14, 8, 1, 1, 2620, 1740, 880, 430, 150, 70, 16, 9, 1, 1, 9496, 6168, 3328, 1420, 636, 196, 88, 18, 10, 1, 1, 35696, 23568

Offset: 0

Henry Bottomley, May 23 2002

For n>k+1, T(n,k) is a multiple of k+2.

Eigentriangle of inverse of (-1)^(n-k)*A094587. Row sums are A187044. - Paul Barry, Mar 02 2011

			Rows start: 1; 1,1; 2,1,1; 4,3,1,1; 10,6,4,1,1; etc.
Triangle begins
1,
1, 1,
2, 1, 1,
4, 3, 1, 1,
10, 6, 4, 1, 1,
26, 18, 8, 5, 1, 1,
76, 48, 28, 10, 6, 1, 1,
232, 156, 76, 40, 12, 7, 1, 1
Production matrix begins
1, 1,
1, 0, 1,
1, 1, 0, 1,
2, 1, 1, 0, 1,
4, 3, 1, 1, 0, 1,
10, 6, 4, 1, 1, 0, 1,
26, 18, 8, 5, 1, 1, 0, 1,
76, 48, 28, 10, 6, 1, 1, 0, 1,
232, 156, 76, 40, 12, 7, 1, 1, 0, 1
Inverse begins
1,
-1, 1,
-1, -1, 1,
0, -2, -1, 1,
0, 0, -3, -1, 1,
0, 0, 0, -4, -1, 1,
0, 0, 0, 0, -5, -1, 1,
0, 0, 0, 0, 0, -6, -1, 1
- _Paul Barry_, Mar 02 2011

Columns include A000085, A000932, A059480. Right hand columns effectively include A000012 (twice), A000027, A005843, A028552. Cf. A062323 for a triangle with similar formulas.

T(n, k+1)=(T(n, k-1)-T(n-1, k))/k for 0

A289491 a(n) = denominator of 1/(1 + 1/(1 + 2/(1 + ... (1 + n)))).

Original entry on oeis.org

2, 4, 5, 13, 19, 58, 191, 131, 1187, 2231, 17519, 71063, 29881, 323423, 2887921, 13237457, 2397389, 15030317, 742458253, 3748521653, 9670072483, 25451905333, 10932619111, 78684575461, 4163946939067, 11799518538967, 136025604432743, 159359728522979

Offset: 1

Seiichi Manyama, Sep 02 2017

			1/2, 3/4, 3/5, 9/13, 12/19, 39/58, 123/191, 87/131, 771/1187, 1473/2231, 11427/17519, 46779/71063, 19533/29881, ... = A225436/A289491 -> A108088.
A225436(1)/a(1) = 1/2  = 1/(1 + 1)                         =  1/2,
A225436(2)/a(2) = 3/4  = 1/(1 + 1/(1 + 2))                 =  3/4,
A225436(3)/a(3) = 3/5  = 1/(1 + 1/(1 + 2/(1 + 3)))         =  6/10,
A225436(4)/a(4) = 9/13 = 1/(1 + 1/(1 + 2/(1 + 3/(1 + 4)))) = 18/26.

Seiichi Manyama, Table of n, a(n) for n = 1..843
OEIS Wiki, A remarkable formula of Ramanujan

Cf. A000085, A000932, A108088, A225435, A225436 (numerators).

p:= (i, n)-> `if`(i=n, (1+n), 1+i/p(i+1,n)):
a:= n-> denom(1/p(1,n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 02 2017

a(n) = A225435(n) + A225436(n).

A225436(n)/a(n) = 1/(1 + 1/(1 + 2/(1 + ... (1 + n)))) = A000932(n)/A000085(n+1).

A291856 a(0) = -1, a(1) = 1, a(n) = a(n-1) + (n-1)*a(n-2) for n > 1.

Original entry on oeis.org

-1, 1, 0, 2, 2, 10, 20, 80, 220, 860, 2840, 11440, 42680, 179960, 734800, 3254240, 14276240, 66344080, 309040160, 1503233600, 7374996640, 37439668640, 192314598080, 1015987308160, 5439223064000, 29822918459840, 165803495059840, 941199375015680

Offset: 0

Seiichi Manyama, Sep 04 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..800

Cf. A000085, A000932, A249059.

a:=[-1,1];; for n in [3..10^2] do a[n]:=a[n-1]+(n-2)*a[n-2]; od; a;  # Muniru A Asiru, Sep 07 2017

RecurrenceTable[{a[0] == -1, a[1] == 1, a[n] == a[n-1] + (n-1)*a[n-2]}, a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 04 2017 *)
CoefficientList[Series[E^(x*(2 + x)/2) * (E^(1/2)*Sqrt[2*Pi]*(Erf[(1 + x)/Sqrt[2]] - Erf[1/Sqrt[2]]) - 1), {x, 0, 20}], x]*Range[0, 20]! (* Vaclav Kotesovec, Sep 05 2017 *)

a(n+4) = 2*A249059(n) for n >= 0.
E.g.f.: exp(x*(2+x)/2) * (exp(1/2) * sqrt(2*Pi) * (erf((1+x)/sqrt(2)) - erf(1/sqrt(2))) - 1). - Vaclav Kotesovec, Sep 05 2017
a(n) ~ (sqrt(Pi) * exp(1/2) * (1 - erf(1/sqrt(2))) - sqrt(2)/2) * n^(n/2) * exp(sqrt(n) - n/2 - 1/4). - Vaclav Kotesovec, Sep 05 2017

A346943 a(n) = a(n-1) + n(n+1)a(n-2) with a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 7, 19, 159, 729, 7407, 48231, 581535, 4922325, 68891175, 718638075, 11465661375, 142257791025, 2550046679775, 36691916525775, 730304613424575, 11958031070311725, 261722208861516375, 4805774015579971875, 114729101737416849375, 2334996696935363855625

Offset: 0

Vaclav Kotesovec, Aug 08 2021, following a suggestion from John M. Campbell

From Peter Bala, Dec 09 2024: (Start)

b(n) := A000246(n+2) = (n+2)!/2^(n+1) * binomial(n+1, floor((n+1)/2)) satisfies the same second-order recurrence as a(n) with the initial conditions b(0) = 1 and b(1) = 3. This leads to the finite continued fraction a(n)/b(n) = 1/(1 + 2/(1 + 6/(1 + ... + n*(n+1)/1). Letting n tend to infinity gives the continued fraction representation 1/(1 + 2/(1 + 6/(1 + ... + n*(n+1)/(1 + ...) = Pi/2 - 1, due to Euler - see paragraph 31, p. 48. (End)

Leonhard Euler, E123: De fractionibus continuis observationes, originally published in Commentarii academiae scientiarum Petropolitanae 11, 1750, pp. 32-81, reprinted in Opera Omnia: Series 1, Volume 14, 291-349.

Cf. A000246, A000932, A024167.

RecurrenceTable[{a[n] == a[n-1] + n*(n+1)*a[n-2], a[0]==1, a[1]==1}, a, {n,0,20}]
nmax = 20; CoefficientList[Series[(-2 + Pi + 2*Pi*x + 4*Sqrt[1 - x^2] + 2*x*(-2 + Sqrt[1 - x^2]) - 4*(1 + 2*x) * ArcSin[Sqrt[1 - x]/Sqrt[2]]) / (2*(1 - x)^(5/2) * (1 + x)^(3/2)), {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ n! * (Pi - 2) * n^(3/2) / sqrt(2*Pi).
a(n) ~ (Pi - 2) * n^(n+2) / exp(n).
E.g.f. A(x) satisfies the differential equation -6*A(x) - (6*x + 1)*A'(x) + (1 - x^2)*A''(x) = 0, A(0)=1, A'(0)=1.
E.g.f.: (-2 + Pi + 2*Pi*x + 4*sqrt(1-x^2) + 2*x*(-2+sqrt(1-x^2)) - 4*(1+2*x) * arcsin(sqrt(1-x)/sqrt(2))) / (2*(1-x)^(5/2) * (1+x)^(3/2)).

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cp's OEIS Frontend

A180048 Coefficient triangle of the denominators of the (n-th convergents to) the continued fraction 1/(w+2/(w+3/(w+4/... . Conjectured to equal unsigned version of A137286.

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A225435 Numerators of convergents to the general continued fraction 1/(1 + 2/(1 + 3/(1 + 4/(1+ ...)))).

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A225436 Denominators of convergents to the general continued fraction 1/(1 + 2/(1 + 3/(1 + 4/(1+ ...)))).

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A173895 E.g.f. satisfies: A'(x) = 1/(1 + x*A(x)) with A(0) = 1.

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PARI

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A070895 Triangle read by rows where T(n+1,k)=T(n,k)+n*T(n-1,k) starting with T(n,n)=1 and T(n,k)=0 if n

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A289491 a(n) = denominator of 1/(1 + 1/(1 + 2/(1 + ... (1 + n)))).

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A291856 a(0) = -1, a(1) = 1, a(n) = a(n-1) + (n-1)*a(n-2) for n > 1.

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A346943 a(n) = a(n-1) + n*(n+1)*a(n-2) with a(0)=1, a(1)=1.

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A346943 a(n) = a(n-1) + n(n+1)a(n-2) with a(0)=1, a(1)=1.