A227887 -id:A227887 - cp's OEIS frontend

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

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

Offset: 0

Paul D. Hanna, Sep 20 2012

nonn

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)

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

Cf. A000364, A227887, 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)^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

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 *)

{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),", "))

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

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...

A338634 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^4x/(A(x) - 3^4x/(A(x) - 4^4x/(A(x) - 5^4x/(A(x) - 6^4*x/(A(x) - ... )))))), a continued fraction relation.

Original entry on oeis.org

1, 1, 15, 1490, 472475, 367254494, 596838469302, 1812465211795364, 9460229930323620755, 79588323526110945959270, 1025816228173896271039326050, 19441688693651416990291991566332, 523762848713992063145153491388390686, 19495503038639783268900576813041922912172

Offset: 0

Paul D. Hanna, Nov 04 2020

nonn

			G.f.: A(x) = 1 + x + 15*x^2 + 1490*x^3 + 472475*x^4 + 367254494*x^5 + 596838469302*x^6 + 1812465211795364*x^7 + 9460229930323620755*x^8 + ...
where
1 = A(x) - x/(A(x) - 2^4*x/(A(x) - 3^4*x/(A(x) - 4^4*x/(A(x) - 5^4*x/(A(x) - 6^4*x/(A(x) - 7^4*x/(A(x) - 8^4*x/(A(x) - 9^4*x/(A(x) - 10^4*x/(A(x) - ... )))))))))), a continued fraction relation.

Paul D. Hanna, Table of n, a(n) for n = 0..130

Cf. A000699, A158119, A227887, A338633, A338635.

{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)^4*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(2/Pi) * (8*sqrt(Pi) / Gamma(1/4)^2)^(4*n + 2) * (n!)^4 / sqrt(n).
a(n) ~ 2^(12*n + 17/2) * Pi^(2*n + 5/2) * n^(4*n + 3/2) / (Gamma(1/4)^(8*n + 4) * exp(4*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.

A371997 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 = 4.

Original entry on oeis.org

1, 1, 1, 1, 17, 17, 1, 98, 1585, 1585, 1, 354, 30259, 485729, 485729, 1, 979, 280883, 23237252, 372281761, 372281761, 1, 2275, 1702758, 459143300, 37562889061, 601378506737, 601378506737, 1, 4676, 7762854, 5310927050, 1397160213861, 113771355829478, 1820943071778385, 1820943071778385

Offset: 0

Peter Luschny, Apr 24 2024

			Triangle begins:
[0] 1;
[1] 1,    1;
[2] 1,   17,      17;
[3] 1,   98,    1585,      1585;
[4] 1,  354,   30259,    485729,      485729;
[5] 1,  979,  280883,  23237252,   372281761,    372281761;
[6] 1, 2275, 1702758, 459143300, 37562889061, 601378506737, 601378506737;

Family: A009766 (Catalan's triangle, m=0), A001498 (m=1), A060058 (m=2), A371996 (m=3), this triangle (m=4).

Cf. A227887 (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(4, n, k), k = 0..n), n = 0..8));

cp's OEIS Frontend

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.

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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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A338634 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^4*x/(A(x) - 3^4*x/(A(x) - 4^4*x/(A(x) - 5^4*x/(A(x) - 6^4*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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A371997 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 = 4.

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

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.

A338634 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^4x/(A(x) - 3^4x/(A(x) - 4^4x/(A(x) - 5^4x/(A(x) - 6^4*x/(A(x) - ... )))))), a continued fraction relation.

A371997 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 = 4.