A216966 -id:A216966 - cp's OEIS frontend

A227887 O.g.f.: 1/(1 - x/(1 - 2^4x/(1 - 3^4x/(1 - 4^4x/(1 - 5^4x/(1 - 6^4*x/(1 -...))))))), a continued fraction.

1, 1, 17, 1585, 485729, 372281761, 601378506737, 1820943071778385, 9489456505643743169, 79759396929125826861121, 1027412704023984825792488657, 19464301715272748317827942755185, 524230105465412991467916306841439009, 19509134827116013764271741468197795034081

Offset: 0

Paul D. Hanna, Oct 26 2013

nonn

Compare to the continued fraction for the Euler numbers (A000364):
1/(1-x/(1-2^2*x/(1-3^2*x/(1-4^2*x/(1-5^2*x/(1-6^2*x/(1-...))))))).
From Vaclav Kotesovec, Sep 24 2020: (Start)
In general, if s>0 and g.f. = 1/(1 - x/(1 - 2^s*x/(1 - 3^s*x/(1 - 4^s*x/(1 - 5^s*x/(1 - 6^s*x/(1 -...))))))), a continued fraction, then
a(n,s) ~ c(s) * d(s)^n * (n!)^s / sqrt(n), where
d(s) = (2*s*Gamma(2/s) / Gamma(1/s)^2)^s
c(s) = sqrt(s*d(s)/(2*Pi)). (End)

			G.f.: A(x) = 1 + x + 17*x^2 + 1585*x^3 + 485729*x^4 + 372281761*x^5 +...

Cf. A000364, A216966, A337807, A337808, A337809.

T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else (k - n - 1)^4 * T(n, k - 1) + T(n - 1, k) fi fi end:
a := n -> T(n, n): seq(a(n), n = 0..13);  # Peter Luschny, Oct 02 2023

nmax = 20; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[Range[nmax + 1]^4*x]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)

{a(n)=local(CF=1+x*O(x^n)); for(k=1, n, CF=1/(1-(n-k+1)^4*x*CF)); polcoeff(CF, n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * d^n * (n!)^4 / sqrt(n), where d = 4096 * Pi^2 / Gamma(1/4)^8 = 1.353976395034780345656335026823167975194... and c = sqrt(2*d/Pi) = 64 * sqrt(2*Pi) / Gamma(1/4)^4 = 0.9284223954634658948993105287957575... - Vaclav Kotesovec, Aug 25 2017, updated Sep 23 2020

A343439 G.f.: 1 + 2^0x/(1 + 2^1x/(1 + 2^2x/(1 + 2^3x/(1 + 2^4*x/(1 + ...))))).

Original entry on oeis.org

1, 1, -2, 12, -136, 2736, -99616, 6810816, -900563072, 234247256832, -120883821425152, 124271556482829312, -255006726559759042560, 1045529090595650037657600, -8569159507007490469146992640, 140431398588497630920722150113280, -4602217897540461023955069241211781120

Offset: 0

Seiichi Manyama, Apr 15 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..81

Cf. A015083, A168491, A216966.

a(n) = my(A=1+O(x)); for(i=1, n, A=1+2^(n-i)*x/A); polcoef(A, n);

a(n) = if(n<2, 1, -sum(k=1, n-1, 2^k*a(k)*a(n-k)));

G.f. satisfies: A(x) = 1 + x/A(2*x).
G.f.: 1/(Sum_{k>=0} A015083(k) * (-x)^k).
a(0) = a(1) = 1 and a(n) = -Sum_{k=1..n-1} 2^k*a(k)*a(n-k) for n > 1.
a(n) = (-2)^(n-1) * A015083(n-1) for n > 0.

A290569 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - x/(1 - 2^kx/(1 - 3^kx/(1 - 4^kx/(1 - 5^kx/(1 - ...)))))).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 5, 15, 14, 1, 1, 9, 61, 105, 42, 1, 1, 17, 297, 1385, 945, 132, 1, 1, 33, 1585, 24273, 50521, 10395, 429, 1, 1, 65, 8865, 485729, 3976209, 2702765, 135135, 1430, 1, 1, 129, 50881, 10401345, 372281761, 1145032281, 199360981, 2027025, 4862

Offset: 0

Ilya Gutkovskiy, Aug 08 2017

			G.f. of column k: A_k(x) = 1 + x + (2^k + 1)*x^2 + (2^(k+1) + 4^k + 6^k + 1)*x^3 + ...
Square array begins:
:  1,    1,      1,        1,          1,            1,  ...
:  1,    1,      1,        1,          1,            1,  ...
:  2,    3,      5,        9,         17,           33,  ...
:  5,   15,     61,      297,       1585,         8865,  ...
: 14,  105,   1385,    24273,     485729,     10401345,  ...
: 42,  945,  50521,  3976209,  372281761,  38103228225,  ...

Columns k=0-4 give: A000108, A001147, A000364, A216966, A227887.
Main diagonal gives A291333.
Cf. A000051 (row 2).

Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-i^k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 9}, {n, 0, j}] // Flatten

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

A337807 O.g.f.: 1/(1 - x/(1 - 2^5x/(1 - 3^5x/(1 - 4^5x/(1 - 5^5x/(1 - 6^5*x/(1 -...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 33, 8865, 10401345, 38103228225, 352780110115425, 7139708074971014625, 284135772494258636522625, 20513418606891012201924650625, 2521576999908767233729260158270625, 501403316300941434382591838239147790625, 154613553816538472779474765739707728587090625

Offset: 0

Vaclav Kotesovec, Sep 23 2020

nonn

Cf. A001147, A000364, A216966, A227887, A337808, A337809.

nmax = 15; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[Range[nmax + 1]^5*x]], {x, 0, nmax}], x]

a(n) ~ c * d^n * (n!)^5 / sqrt(n), where
d = 10^5 * Gamma(2/5)^5 / Gamma(1/5)^10 = 1.29133469292029895399895579276779248119508048136258551947012306768...
c = sqrt(5*d/(2*Pi)) = 1.0137117429632556021458475899461841562723826775113969124...

A337808 O.g.f.: 1/(1 - x/(1 - 2^6x/(1 - 3^6x/(1 - 4^6x/(1 - 5^6x/(1 - 6^6*x/(1 -...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 65, 50881, 231455105, 4104215813761, 220579355255364545, 30221200809380332664641, 9302731197994837387680880385, 5843241203886533657008940262539521, 6942621504765123845961888310824174754625, 14676663615276526648053662674115841827580734401

Offset: 0

Vaclav Kotesovec, Sep 23 2020

nonn

Cf. A001147, A000364, A216966, A227887, A337807, A337809.

nmax = 15; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[Range[nmax + 1]^6*x]], {x, 0, nmax}], x]

a(n) ~ c * d^n * (n!)^6 / sqrt(n), where
d = 12^6 * Gamma(1/3)^6 / Gamma(1/6)^12 = 1.247526211138803516951432724912344768287048859917010218640563910368330569...
c = sqrt(3*d/Pi) = 1.091466801527496975151641909038030945303075901009696629477...

A337809 O.g.f.: 1/(1 - x/(1 - 2^7x/(1 - 3^7x/(1 - 4^7x/(1 - 5^7x/(1 - 6^7*x/(1 -...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 129, 296577, 5273061633, 456296857756929, 143521873041157216641, 134210828762693919568092033, 322179101908965036802512977670657, 1775143826590061506939568896182460951041, 20554318541749460884980441781629250054049026689

Offset: 0

Vaclav Kotesovec, Sep 23 2020

nonn

In general, if s>0 and g.f. = 1/(1 - x/(1 - 2^s*x/(1 - 3^s*x/(1 - 4^s*x/(1 - 5^s*x/(1 - 6^s*x/(1 -...))))))), a continued fraction, then a(n,s) ~ c(s) * d(s)^n * (n!)^s / sqrt(n), where d(s) = (2*s*Gamma(2/s) / Gamma(1/s)^2)^s and c(s) = sqrt(s*d(s)/(2*Pi)). - Vaclav Kotesovec, Sep 24 2020

Cf. A001147, A000364, A216966, A227887, A337807, A337808.

nmax = 15; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[Range[nmax + 1]^7*x]], {x, 0, nmax}], x]

a(n) ~ c * d^n * (n!)^7 / sqrt(n), where
d = 14^7 * Gamma(2/7)^7 / Gamma(1/7)^14 = 1.2151675804792498774003050188354949771364793751019885755525736...
c = sqrt(7*d/(2*Pi)) = 1.1635288951410008357326423559026931516828251494058147648...

A338633 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^3x/(A(x) - 3^3x/(A(x) - 4^3x/(A(x) - 5^3x/(A(x) - 6^3*x/(A(x) - ...)))))), a continued fraction relation.

Original entry on oeis.org

1, 1, 7, 250, 21867, 3725702, 1096355494, 513875333940, 361121449989171, 362961084011245198, 502496711191618404882, 929337000359116522329132, 2238572532534241145084855934, 6875030222633195280825967544508, 26436454884630260855874989243890732

Offset: 0

Paul D. Hanna, Nov 04 2020

nonn

Compare to the continued fraction relation for the g.f. of A158119 and A338634.

			G.f.: A(x) = 1 + x + 7*x^2 + 250*x^3 + 21867*x^4 + 3725702*x^5 + 1096355494*x^6 + 513875333940*x^7 + 361121449989171*x^8 + 362961084011245198*x^9 + ...
where
1 = A(x) - x/(A(x) - 2^3*x/(A(x) - 3^3*x/(A(x) - 4^3*x/(A(x) - 5^3*x/(A(x) - 6^3*x/(A(x) - 7^3*x/(A(x) - 8^3*x/(A(x) - 9^3*x/(A(x) - ...))))))))), a continued fraction relation.

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A000699, A158119, A216966, A338634.

{a(n) = my(A=[1],CF=1); for(i=1,n, A=concat(A,0); for(i=1,#A, CF = Ser(A) - (#A-i+1)^3*x/CF ); A[#A] = -polcoeff(CF,#A-1) );A[n+1] }
for(n=0,20,print1(a(n),", "))

For n > 0, a(n) is odd iff n is a power of 2 (conjecture).
From Vaclav Kotesovec, Nov 12 2020: (Start)
a(n) ~ sqrt(3/(2*Pi)) * (6*Gamma(2/3)/Gamma(1/3)^2)^(3*n + 3/2) * (n!)^3 / sqrt(n).
a(n) ~ 2^(6*n + 4) * 3^(3*n/2 + 5/4) * Pi^(3*n + 5/2) * n^(3*n + 1) / Gamma(1/3)^(9*(n + 1/2)) / exp(3*n). (End)

A372001 Array read by descending antidiagonals: A family of generalized Catalan numbers generated by a generalization of Deléham's Delta operator.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 14, 15, 5, 1, 1, 42, 105, 61, 9, 1, 1, 132, 945, 1385, 297, 17, 1, 1, 429, 10395, 50521, 24273, 1585, 33, 1, 1, 1430, 135135, 2702765, 3976209, 485729, 8865, 65, 1, 1, 4862, 2027025, 199360981, 1145032281, 372281761, 10401345, 50881, 129, 1, 1

Offset: 1

Peter Luschny, Apr 21 2024

Deléham's Delta operator is defined in A084938. It maps two sequences (a, b) to a triangle T. The given sequences are the coefficients of the linear function p = a + x*b which is the starting point of a recurrence described in A084938 and implemented in A371637. The generalization given here extends the number of input sequences to any number, mapping (a, b, c, ...) to p = a + x*b + x^2*c ... but leaves the recurrence unchanged.
The result, as said, is a triangle that we can evaluate in two ways: Firstly, we only return the main diagonal. In this case, we created a new sequence from n given sequences. This case is implemented by the function A(n, dim) below.
Alternatively, we return the entire triangle. But since the triangle is irregular, we convert it into a regular one by taking only every n-th term of a row. This case is handled by the function T(n, dim). For the first few triangles generated this way, see the link section.

			Array starts:
  [0] 1, 1,  2,     5,        14,            42,                132, ...
  [1] 1, 1,  3,    15,       105,           945,              10395, ...
  [2] 1, 1,  5,    61,      1385,         50521,            2702765, ...
  [3] 1, 1,  9,   297,     24273,       3976209,         1145032281, ...
  [4] 1, 1, 17,  1585,    485729,     372281761,       601378506737, ...
  [5] 1, 1, 33,  8865,  10401345,   38103228225,    352780110115425, ...
  [6] 1, 1, 65, 50881, 231455105, 4104215813761, 220579355255364545, ...
.
Seen as a triangle T(n, k) = A(k, n - k):
  [0] [  1]
  [1] [  1,     1]
  [2] [  2,     1,     1]
  [3] [  5,     3,     1,     1]
  [4] [ 14,    15,     5,     1,    1]
  [5] [ 42,   105,    61,     9,    1,  1]
  [6] [132,   945,  1385,   297,   17,  1, 1]
  [7] [429, 10395, 50521, 24273, 1585, 33, 1, 1]

Peter Luschny, First few triangles generated by A372001.

By ascending antidiagonals: A290569.
Family: A000108 (n=0), A001147 (n=1), A000364 (n=2), A216966 (n=3), A227887 (n=4), A337807 (n=5), A337808 (n=6), A337809 (n=7).
Cf. A291333 (main diagonal), A371999 (row sums of triangle).
Cf. A084938, A131427, A371994, A371637.

def GeneralizedDelehamDelta(F, dim, seq=True):  # The algorithm.
    ring = PolynomialRing(ZZ, 'x')
    x = ring.gen()
    A = [sum(F[j](k) * x^j for j in range(len(F))) for k in range(dim)]
    C = [ring(0)] + [ring(1) for i in range(dim)]
    for k in range(dim):
        for n in range(k, 0, -1):
            C[n] = C[n-1] + C[n+1] * A[n-1]
        yield list(C[1])[-1] if seq else list(C[1])
def F(n):  # Define the input functions.
    def p0(): return lambda n: pow(n, n^0)
    def p(k): return lambda n: pow(n + 1, k)
    return [p0()] + [p(k) for k in range(n + 1)]
def A(n, dim): # Return only the main diagonal of the triangle.
    return [r for r in GeneralizedDelehamDelta(F(n), dim)]
for n in range(7): print(A(n, 7))
def T(n, dim): # Return the regularized triangle.
    R = GeneralizedDelehamDelta(F(n), dim, False)
    return [[r[k] for k in range(0, len(r), n + 1)] for r in R]
for n in range(0, 4):
    for row in T(n, 6): print(row)

A = DELTA([x -> (x + 1)^k : 0 <= k <= n]), i.e. here the input functions of the generalized Delta operator are the (shifted) power functions. The returned sequence is the main diagonal of the generated triangle.

A218221 G.f.: 1/(1-x/(1-4x/(1-10x/(1-20x/(1-35x/(1-56*x/(1-...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 5, 65, 1725, 81225, 6181125, 710984625, 117537778125, 26848583825625, 8210318193703125, 3275053250628290625, 1667519951972905828125, 1063947235962694359515625, 837322677987349287566953125, 801714108831393845941434140625

Offset: 0

Paul D. Hanna, Oct 23 2012

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 65*x^3 + 1725*x^4 + 81225*x^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..198

Cf. A000292, A002105, A216966.

nmax = 20; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[Binomial[Range[nmax + 1]+2,3]*x]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)
eulerCF[f_, len_] := Module[{g}, g[len - 1] = 1;
g[k_] := g[k] = 1 - f[k]/(f[k] - 2/g[k + 1]); CoefficientList[g[0] + O[x]^len, x]];
A218221List[len_] := eulerCF[(1/3) x (# + 1) (# + 2) (# + 3) &, len];
A218221List[16] (* Peter Luschny, Aug 09 2019 *)

{a(n)=local(CF=1+x*O(x^n)); for(k=1, n, CF=1/(1-(n-k+1)*(n-k+2)*(n-k+3)/6*x*CF)); polcoeff(CF, n)}
for(n=0,20,print1(a(n),", "))

a(n) == 0 (mod 5) for n>1.
a(n) == 0 (mod 3) for n>3.
G.f.: Q(0), where Q(k) = 1 - x*(k+1)*(k+2)*(k+3)/(x*(k+1)*(k+2)*(k+3) - 6/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 19 2013
a(n) ~ c * d^n * (n!)^3 * n^(5/2), where d = 32*sqrt(3)*Pi^3 / Gamma(1/3)^9 = 0.241821816937867322064737317131437... and c = 2^(15/2) * 3^(11/4) * Pi^4 / Gamma(1/3)^(27/2) = 0.60382458700655692263505976... - Vaclav Kotesovec, Aug 25 2017, updated Mar 16 2024

A371996 Triangle read by rows: T(m, n, k) = 1 if k = 0 and T(m, n, k - 1) if k = n; otherwise (-1)^m(k - n - 1)^m T(m, n, k - 1) + T(m, n - 1, k) where m = 3.

Original entry on oeis.org

1, 1, 1, 1, 9, 9, 1, 36, 297, 297, 1, 100, 2997, 24273, 24273, 1, 225, 17397, 493992, 3976209, 3976209, 1, 441, 72522, 5135400, 142632009, 1145032281, 1145032281, 1, 784, 241866, 35368650, 2406225609, 66113123724, 530050022073, 530050022073

Offset: 0

Peter Luschny, Apr 24 2024

			Triangle read by rows:
[0] 1;
[1] 1,   1;
[2] 1,   9,     9;
[3] 1,  36,   297,     297;
[4] 1, 100,  2997,   24273,     24273;
[5] 1, 225, 17397,  493992,   3976209,    3976209;
[6] 1, 441, 72522, 5135400, 142632009, 1145032281, 1145032281;

Family: A009766 (Catalan's triangle, m=0), A001498 (m=1), A060058 (m=2), this triangle (m=3), A371997 (m=4).

Cf. A216966 (main diagonal).

T := proc(m, n, k) option remember; if k = 0 then 1 elif k = n then T(m, n, k-1) else (-1)^m*(k - n - 1)^m * T(m, n, k - 1) + T(m, n - 1, k) fi end:
seq(seq(T(3, n, k), k = 0..n), n = 0..8));

cp's OEIS Frontend

A227887 O.g.f.: 1/(1 - x/(1 - 2^4*x/(1 - 3^4*x/(1 - 4^4*x/(1 - 5^4*x/(1 - 6^4*x/(1 -...))))))), a continued fraction.

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A343439 G.f.: 1 + 2^0*x/(1 + 2^1*x/(1 + 2^2*x/(1 + 2^3*x/(1 + 2^4*x/(1 + ...))))).

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A290569 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - x/(1 - 2^k*x/(1 - 3^k*x/(1 - 4^k*x/(1 - 5^k*x/(1 - ...)))))).

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A337807 O.g.f.: 1/(1 - x/(1 - 2^5*x/(1 - 3^5*x/(1 - 4^5*x/(1 - 5^5*x/(1 - 6^5*x/(1 -...))))))), a continued fraction.

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A337808 O.g.f.: 1/(1 - x/(1 - 2^6*x/(1 - 3^6*x/(1 - 4^6*x/(1 - 5^6*x/(1 - 6^6*x/(1 -...))))))), a continued fraction.

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A337809 O.g.f.: 1/(1 - x/(1 - 2^7*x/(1 - 3^7*x/(1 - 4^7*x/(1 - 5^7*x/(1 - 6^7*x/(1 -...))))))), a continued fraction.

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A338633 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^3*x/(A(x) - 3^3*x/(A(x) - 4^3*x/(A(x) - 5^3*x/(A(x) - 6^3*x/(A(x) - ...)))))), a continued fraction relation.

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A372001 Array read by descending antidiagonals: A family of generalized Catalan numbers generated by a generalization of Deléham's Delta operator.

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A218221 G.f.: 1/(1-x/(1-4*x/(1-10*x/(1-20*x/(1-35*x/(1-56*x/(1-...))))))), a continued fraction.

A227887 O.g.f.: 1/(1 - x/(1 - 2^4x/(1 - 3^4x/(1 - 4^4x/(1 - 5^4x/(1 - 6^4*x/(1 -...))))))), a continued fraction.

A343439 G.f.: 1 + 2^0x/(1 + 2^1x/(1 + 2^2x/(1 + 2^3x/(1 + 2^4*x/(1 + ...))))).

A290569 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - x/(1 - 2^kx/(1 - 3^kx/(1 - 4^kx/(1 - 5^kx/(1 - ...)))))).

A337807 O.g.f.: 1/(1 - x/(1 - 2^5x/(1 - 3^5x/(1 - 4^5x/(1 - 5^5x/(1 - 6^5*x/(1 -...))))))), a continued fraction.

A337808 O.g.f.: 1/(1 - x/(1 - 2^6x/(1 - 3^6x/(1 - 4^6x/(1 - 5^6x/(1 - 6^6*x/(1 -...))))))), a continued fraction.

A337809 O.g.f.: 1/(1 - x/(1 - 2^7x/(1 - 3^7x/(1 - 4^7x/(1 - 5^7x/(1 - 6^7*x/(1 -...))))))), a continued fraction.

A338633 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^3x/(A(x) - 3^3x/(A(x) - 4^3x/(A(x) - 5^3x/(A(x) - 6^3*x/(A(x) - ...)))))), a continued fraction relation.

A218221 G.f.: 1/(1-x/(1-4x/(1-10x/(1-20x/(1-35x/(1-56*x/(1-...))))))), a continued fraction.

A371996 Triangle read by rows: T(m, n, k) = 1 if k = 0 and T(m, n, k - 1) if k = n; otherwise (-1)^m(k - n - 1)^m T(m, n, k - 1) + T(m, n - 1, k) where m = 3.