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.

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 - ...)))))).

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

Views

Author

Ilya Gutkovskiy, Aug 08 2017

Keywords

Examples

			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,  ...

Crossrefs

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

Programs

  • Mathematica
    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

Formula

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.

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.

A291331 a(n) = [x^n] 1/(1 - 2^n*x/(1 - 4^n*x/(1 - 6^n*x/(1 - 8^n*x/(1 - 10^n*x/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 2, 80, 152064, 31832735744, 1278532180456243200, 15158097871912903189326725120, 75553979800594222861911290918096439607296, 213679399657239557797941463213636090471439135194537263104
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 22 2017

Keywords

Crossrefs

Main diagonal of A291260.
Cf. A291333.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-(2 i)^n x, 1, {i, 1, n}]), {x, 0, n}], {n, 0, 8}]

Formula

a(n) = A291260(n,n).
a(n) ~ c * 2^(n^2) * (n!)^n ~ c * Pi^(n/2) * (2*n)^(n^2 + n/2) / exp(n^2 - 1/12), where c = 1/QPochhammer(exp(-1)) = 1.982440907412873703685682465561312... - Vaclav Kotesovec, Jun 08 2019

A292920 a(n) = [x^n] 1/(1 + x/(1 - 2^n*x/(1 + 3^n*x/(1 - 4^n*x/(1 + 5^n*x/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, -1, -3, 167, 262305, -19802585281, -111307539961183167, 66192452118355875767376767, 5609362049224731266886822845131449345, -87773600779729250264394909974880988558750376171521, -318955450227538853365057634352430260175496937997604285602324132863
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 26 2017

Keywords

Crossrefs

Main diagonal of A291207.
Cf. A291333.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-(-1)^k k^n x, 1, {k, 1, n}]), {x, 0, n}], {n, 0, 10}]

Formula

a(n) = A291207(n,n).

A317103 Expansion of e.g.f. -LambertW(-x) * Product_{k>=1} 1/(1-x^k).

Original entry on oeis.org

0, 1, 4, 27, 220, 2265, 27246, 393421, 6548536, 126257697, 2767122010, 68387691141, 1882488882660, 57198150690577, 1900138953826582, 68502961685976525, 2662089147552365296, 110887849449189768513, 4926985461324765096498, 232544882903837769171829
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 21 2018

Keywords

Links

Crossrefs

Cf. A000041, A000169, A252761, A291332, A291333, A317104.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[-LambertW[-x]*Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
Table[n!*Sum[PartitionsP[n-k]*k^(k-1)/k!, {k, 1, n}], {n, 0, 20}]

Formula

a(n) ~ c * n^(n-1), where c = 1/QPochhammer(exp(-1)) = 1.98244090741287370368568246556131... - Vaclav Kotesovec, Jul 21 2018
Showing 1-5 of 5 results.

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