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 1-4 of 4 results.

A346208 Expansion of e.g.f.: exp(-3*x) / (2 - exp(x)).

Original entry on oeis.org

1, -2, 6, -14, 54, -62, 966, 4786, 71574, 875938, 12810726, 202739986, 3511712694, 65856494338, 1330170266886, 28785391689586, 664456856787414, 16296345814039138, 423191833100881446, 11600198414334789586, 334710974532291679734, 10140603124807778534338
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

Links

Crossrefs

Cf. A000670, A002620, A052841, A259533, A330603, A344037.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-3*x)/(2-Exp(x)) ))); // G. C. Greubel, Jun 11 2024
  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[-3 x]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[HurwitzLerchPhi[1/2, -n, -3]/2, {n, 0, 21}]
a[n_] := a[n] = (-3)^n + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
  • SageMath
    def A346208_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-3*x)/(2-exp(x)) ).egf_to_ogf().list()
A346208_list(40) # G. C. Greubel, Jun 11 2024

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * A000670(k).
a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A002620(k+2).
a(n) = Sum_{k>=0} (k - 3)^n / 2^(k+1).
a(n) = (-3)^n + Sum_{k=0..n-1} binomial(n,k) * a(k).
a(n) ~ n! / (16 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 15 2021

A355220 a(n) = Sum_{k>=1} (4*k - 1)^n / 2^k.

Original entry on oeis.org

1, 7, 81, 1399, 32289, 931687, 32259441, 1303134679, 60160827969, 3124574220487, 180312309395601, 11445969681199159, 792626097462398049, 59462922484586318887, 4804064349575887075761, 415847988794676360818839, 38396277196654611908582529, 3766800071614388562865514887
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 24 2022

Keywords

Crossrefs

Cf. A000629, A000670, A080253, A259533, A285067, A328183, A355218, A355219.

Programs

  • Mathematica
    nmax = 17; CoefficientList[Series[Exp[3 x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3^n + Sum[Binomial[n, k] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

Formula

E.g.f.: exp(3*x) / (2 - exp(4*x)).
a(0) = 1; a(n) = 3^n + Sum_{k=1..n} binomial(n,k) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * 4^k * A000670(k).
a(n) ~ n! * 2^(2*n - 1/4) / log(2)^(n+1). - Vaclav Kotesovec, Jun 24 2022

A292915 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(2 - exp(x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 11, 26, 75, 1, 5, 18, 51, 150, 541, 1, 6, 27, 94, 299, 1082, 4683, 1, 7, 38, 161, 582, 2163, 9366, 47293, 1, 8, 51, 258, 1083, 4294, 18731, 94586, 545835, 1, 9, 66, 391, 1910, 8345, 37398, 189171, 1091670, 7087261, 1, 10, 83, 566, 3195, 15666, 74067, 378214, 2183339, 14174522, 102247563
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 26 2017

Keywords

Comments

A(n,k) is the k-th binomial transform of A000670 evaluated at n.

Examples

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 3)*x^2/2! + (k^3 + 3*k^2 + 9*k + 13)*x^3/3! +  (k^4 + 4*k^3 + 18*k^2 + 52*k + 75) x^4/4! + ...
Square array begins:
    1,     1,     1,     1,     1,      1,  ...
    1,     2,     3,     4,     5,      6,  ...
    3,     6,    11,    18,    27,     38,  ...
   13,    26,    51,    94,   161,    258,  ...
   75,   150,   299,   582,  1083,   1910,  ...
  541,  1082,  2163,  4294,  8345,  15666,  ...

Links

Crossrefs

Columns k=0..4 give A000670, A000629, A007047, A259533, A368317.
Rows n=0..2 give A000012, A000027, A102305.
Main diagonal gives A292916.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 50);
T:= func< n,k | Coefficient(R!(Laplace( Exp(k*x)/(2-Exp(x)) )), n) >;
A292915:= func< n,k | T(k,n-k) >;
[A292915(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 12 2024
  • Maple
    A:= proc(n, k) option remember; k^n +add(
       binomial(n, j)*A(j, k), j=0..n-1)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 27 2017
  • Mathematica
    Table[Function[k, n! SeriesCoefficient[Exp[k x]/(2 - Exp[x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, HurwitzLerchPhi[1/2, -n, k]/2][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
  • PARI
    a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
A(n, k) = 2^k*a000670(n)-sum(j=0, k-1, 2^j*(k-1-j)^n); \\ Seiichi Manyama, Dec 25 2023
  • SageMath
    def T(n,k): return factorial(n)*( exp(k*x)/(2-exp(x)) ).series(x, n+1).list()[n]
def A292915(n,k): return T(k,n-k)
flatten([[A292915(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 12 2024

Formula

E.g.f. of column k: exp(k*x)/(2 - exp(x)).
A(n,k) = 2^k*A000670(n) - Sum_{j=0..k-1} 2^j*(k-1-j)^n. - Seiichi Manyama, Dec 25 2023

A368317 Expansion of e.g.f. exp(4*x) / (2 - exp(x)).

Original entry on oeis.org

1, 5, 27, 161, 1083, 8345, 74067, 754241, 8726283, 113375465, 1635899907, 25961939921, 449464541883, 8429731963385, 170261482711347, 3684531041231201, 85050474868523883, 2085932272336772105, 54168554611721580387, 1484825397108091268081
Offset: 0

Views

Author

Seiichi Manyama, Dec 21 2023

Keywords

Crossrefs

Cf. A000670, A000629, A007047, A259533.

Programs

  • PARI
    b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=4, t=1) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

Formula

a(n) = 4^n + Sum_{k=1..n} binomial(n,k) * a(n-k).
a(n) = 16*A000670(n) - (3^n + 2*2^n + 4 + 8*0^n).
Showing 1-4 of 4 results.

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