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.

A369795 Binomial transform of A355408.

Original entry on oeis.org

1, 3, 21, 225, 3201, 56913, 1214361, 30229545, 860016801, 27525472353, 978858962601, 38291126920665, 1634047719138801, 75542860973042193, 3761030066169432441, 200624240375801784585, 11415336789685550907201, 690117422445926970890433, 44175435307592982599575881
Offset: 0

Views

Author

Prabha Sivaramannair, Feb 01 2024

Keywords

Crossrefs

Cf. A355408.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[E^x/(1 + E^x - E^(3*x)), {x, 0, nmax}], x]*
Range[0, nmax]! (* Vaclav Kotesovec, Feb 01 2024*)
  • SageMath
    def a(m):
    if m==0:
        return 1
    else:
        return 1+sum([(3^j-1)*binomial(m,j)*a(m-j) for j in [1,..,m]])
list(a(m) for m in [1,..,50])

Formula

a(n) = 1 + Sum_{k=1..n} (3^k - 1) * binomial(n,k) * a(n-k) for n > 0.
E.g.f.: exp(x)/(1 + exp(x) - exp(3*x)). - Vaclav Kotesovec, Feb 01 2024

A355409 Expansion of e.g.f. 1/(1 + exp(2*x) - exp(3*x)).

Original entry on oeis.org

1, 1, 7, 55, 571, 7471, 117307, 2148175, 44958571, 1058555791, 27693129307, 796934764495, 25018548004171, 850870651904911, 31163746960955707, 1222922731101304015, 51189052318085027371, 2276586205163067346831, 107204914362429152404507
Offset: 0

Views

Author

Seiichi Manyama, Jul 01 2022

Keywords

Comments

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with period dividing phi(k) = A000010(k). For example, modulo 9 we obtain the sequence [1, 1, 7, 1, 4, 1, 1, 1, 7, 1, 4, 1, 1, 1, 7, 1, 4, 1, 1, ...] with an apparent period of 6 = phi(9) beginning at a(1). Cf. A354242. - Peter Bala, Apr 16 2024

Crossrefs

Cf. A371460 (binomial transform).
Cf. A000010, A354242, A355381, A355408, A370092.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+exp(2*x)-exp(3*x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (3^j-2^j)*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 2^k) * binomial(n,k) * a(n-k).
a(n) ~ n! / ((3 + r^2) * log(r)^(n+1)), where r = (1 + 2*cosh(log((29 + 3*sqrt(93))/2)/3))/3. - Vaclav Kotesovec, Jul 01 2022

A355410 Expansion of e.g.f. 1/(3 - exp(x) - exp(3*x)).

Original entry on oeis.org

1, 4, 42, 652, 13482, 348484, 10809282, 391162972, 16177467642, 752689508404, 38911563009522, 2212759299753292, 137270821971529002, 9225382887659221924, 667690580181890112162, 51776098497454677943612, 4282645413209764715753562
Offset: 0

Views

Author

Seiichi Manyama, Jul 01 2022

Keywords

Crossrefs

Cf. A004700, A355408.

Programs

  • Maple
    A355410 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        add((3^k + 1) * binomial(n,k) * procname(n-k),k=1..n) ;
    end if;
end proc:
seq(A355410(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(3-exp(x)-exp(3*x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (3^j+1)*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (3^k + 1) * binomial(n,k) * a(n-k).
a(n) ~ n! / ((9 - 2*r) * log(r)^(n+1)), where r = -2*sinh(log((-9*sqrt(3) + sqrt(247))/2)/3)/sqrt(3). - Vaclav Kotesovec, Jul 01 2022

A368014 Expansion of e.g.f. 1/(1 - exp(x) + exp(3*x)).

Original entry on oeis.org

1, -2, 0, 22, -48, -722, 5160, 39142, -731808, -1875362, 138079320, -428616938, -32571174768, 360902614798, 8748630825480, -218739750924218, -2165282730308928, 135578968355309758, -45942716160492360, -89526354705694434698, 1011199293695966345712
Offset: 0

Views

Author

Seiichi Manyama, Dec 08 2023

Keywords

Crossrefs

Cf. A355408, A355410.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (1-3^j)*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (1 - 3^k) * binomial(n,k) * a(n-k).

A372738 Binomial transform of A369795.

Original entry on oeis.org

1, 4, 28, 298, 4240, 75394, 1608688, 40045618, 1139279680, 36463487554, 1296712045648, 50724943433938, 2164652356532320, 100072984472662114, 4982304066392196208, 265770533884409878258, 15122101633293034668160, 914210942121577873619074, 58519992421072004957876368, 3954059527570115477197922578
Offset: 0

Views

Author

Prabha Sivaramannair, May 11 2024

Keywords

Crossrefs

Cf. A369795, A355408.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[E^(2*x)/(1 + E^x - E^(3*x)), {x, 0, nmax}], x]*Range[0, nmax]! (* Vaclav Kotesovec, Jun 01 2024 *)
  • SageMath
    def a(n):
    if n==0:
        return 1
    else:
        return sum([(1-(-1)^j-(-2)^j)*binomial(n,j)*a(n-j) for j in [1,..,n]])
list(a(n) for n in [0,..,20])

Formula

a(n) = Sum_{j=1..n} (1-(-1)^j-(-2)^j)*binomial(n,j)*a(n-j) for n > 0.
a(n) = 2^n + Sum_{j=1..n} (3^j-1)*binomial(n,j)*a(n-j).
a(n) = 1 + Sum_{j=1..n} (2^j-(-1)^j)*binomial(n,j)*a(n-j).
E.g.f.: exp(2*x)/(1 + exp(x) - exp(3*x)). - Vaclav Kotesovec, Jun 01 2024
Showing 1-5 of 5 results.

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