cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A216966 O.g.f.: 1/(1 - x/(1 - 2^3*x/(1 - 3^3*x/(1 - 4^3*x/(1 - 5^3*x/(1 - 6^3*x/(1 -...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 9, 297, 24273, 3976209, 1145032281, 530050022073, 369626762653857, 369614778179835681, 509880429246329788329, 940535818601273787325257, 2261104378216803649437779313, 6933711495845384616312688513329, 26630255658298074277771723491847161
Offset: 0

Views

Author

Paul D. Hanna, Sep 20 2012

Keywords

Comments

Compare to the continued fraction o.g.f. 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)

Examples

			G.f.: A(x) = 1 + x + 9*x^2 + 297*x^3 + 24273*x^4 + 3976209*x^5 +...

Crossrefs

Cf. A000364, A227887, A337807, A337808, A337809.

Programs

  • Maple
    T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else -(k - n - 1)^3 * T(n, k - 1) + T(n - 1, k) fi fi end:
a := n -> T(n, n): seq(a(n), n = 0..14);  # Peter Luschny, Oct 02 2023
  • Mathematica
    nmax = 20; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[Range[nmax + 1]^3*x]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)
  • PARI
    {a(n)=local(CF=1+x*O(x^n)); for(k=1, n, CF=1/(1-(n-k+1)^3*x*CF)); polcoeff(CF,n)}
for(n=0,20,print1(a(n),", "))

Formula

G.f.: T(0), where T(k) = 1 - x*(k+1)^3/(x*(k+1)^3 -1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 12 2013
a(n) ~ c * d^n * (n!)^3 / sqrt(n), where d = 192 * sqrt(3) * Pi^3 / Gamma(1/3)^9 = 1.450930901627203932388423902788627... and c = 12 * sqrt(2) * 3^(1/4) * Pi / Gamma(1/3)^(9/2) = sqrt(3*d/(2*Pi)) = 0.8323271443586650769764930497... - Vaclav Kotesovec, Aug 25 2017, updated Sep 23 2020

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.

Original entry on oeis.org

1, 1, 17, 1585, 485729, 372281761, 601378506737, 1820943071778385, 9489456505643743169, 79759396929125826861121, 1027412704023984825792488657, 19464301715272748317827942755185, 524230105465412991467916306841439009, 19509134827116013764271741468197795034081
Offset: 0

Views

Author

Paul D. Hanna, Oct 26 2013

Keywords

Comments

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)

Examples

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

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    {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), ", "))

Formula

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

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.

Original entry on oeis.org

1, 1, 33, 8865, 10401345, 38103228225, 352780110115425, 7139708074971014625, 284135772494258636522625, 20513418606891012201924650625, 2521576999908767233729260158270625, 501403316300941434382591838239147790625, 154613553816538472779474765739707728587090625
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 23 2020

Keywords

Crossrefs

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

Programs

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

Formula

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...

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.

Original entry on oeis.org

1, 1, 129, 296577, 5273061633, 456296857756929, 143521873041157216641, 134210828762693919568092033, 322179101908965036802512977670657, 1775143826590061506939568896182460951041, 20554318541749460884980441781629250054049026689
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 23 2020

Keywords

Comments

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

Crossrefs

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

Programs

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

Formula

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...

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

Views

Author

Peter Luschny, Apr 21 2024

Keywords

Comments

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.

Examples

			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]

Links

Crossrefs

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.

Programs

  • SageMath
    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)

Formula

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.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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