A158757 -id:A158757 - cp's OEIS frontend

A330044 Expansion of e.g.f. exp(x) / (1 - x^3).

1, 1, 1, 7, 25, 61, 841, 5251, 20497, 423865, 3780721, 20292031, 559501801, 6487717237, 44317795705, 1527439916731, 21798729916321, 180816606476401, 7478345832314977, 126737815733490295, 1236785588298582841, 59677199741873516461, 1171057417377450325801

Offset: 0

Ilya Gutkovskiy, Nov 28 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000522, A087208, A100732, A158757, A330045.

[n le 3 select 1 else 1 + 6*Binomial(n-1,3)*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 05 2021

nmax = 22; CoefficientList[Series[Exp[x]/(1 - x^3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[n!/(n - 3 k)!, {k, 0, Floor[n/3]}], {n, 0, 22}]

[sum(factorial(3*k)*binomial(n, 3*k) for k in (0..n//3)) for n in (0..40)] # G. C. Greubel, Dec 05 2021

G.f.: Sum_{k>=0} (3*k)! * x^(3*k) / (1 - x)^(3*k + 1).
a(0) = a(1) = a(2) = 1; a(n) = n * (n - 1) * (n - 2) * a(n - 3) + 1.
a(n) = Sum_{k=0..floor(n/3)} n! / (n - 3*k)!.
a(n) ~ n! * (exp(1)/3 + 2*cos(sqrt(3)/2 - 2*Pi*n/3) / (3*exp(1/2))). - Vaclav Kotesovec, Apr 18 2020
a(n) = A158757(n, 2*n). - G. C. Greubel, Dec 05 2021

A158785 Expansion of e.g.f.: exp(tx)/(1 - x^2/t - t^3x^3).

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 1, 0, 6, 0, 0, 7, 24, 0, 0, 12, 0, 0, 25, 0, 120, 0, 0, 260, 0, 0, 61, 720, 0, 0, 360, 0, 0, 1470, 0, 0, 841, 0, 5040, 0, 0, 15960, 0, 0, 5082, 0, 0, 5251, 40320, 0, 0, 20160, 0, 0, 122640, 0, 0, 134456, 0, 0, 20497, 0, 362880, 0, 0, 1512000

Offset: 0

Roger L. Bagula, Mar 26 2009

			Irregular triangle begins as:
      1;
      0,    1;
      2,    0, 0,     1;
      0,    6, 0,     0,     7;
     24,    0, 0,    12,     0, 0,     25;
      0,  120, 0,     0,   260, 0,      0,   61;
    720,    0, 0,   360,     0, 0,   1470,    0, 0,    841;
      0, 5040, 0,     0, 15960, 0,      0, 5082, 0,      0, 5251;
  40320,    0, 0, 20160,     0, 0, 122640,    0, 0, 134456,    0, 0, 20497;

G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

Cf. A005212, A005359, A158706, A158757, A330044.

Table[CoefficientList[Expand[t^Floor[n/2]*n!*SeriesCoefficient[Series[Exp[t*x]/(1 - x^2/t - t^3*x^3), {x, 0, 20}], n]], t], {n, 0, 10}]//Flatten

f(x, t) = exp(t*x)/(1 - x^2/t - t^3*x^3)
def A158785(n, k): return ( factorial(n)*t^(n//2)*( f(x, t) ).series(x, 20).list()[n] ).series(t, 2*n+1).list()[k]
flatten([[A158785(n, k) for k in (0..n+(n//2))] for n in (0..10)]) # G. C. Greubel, Dec 05 2021

T(n, k) = coefficients of e.g.f.: t^floor(n/2)*exp(t*x)/(1 - x^2/t - t^3*x^3).
From G. C. Greubel, Dec 05 2021: (Start)
T(n, floor(n/2) + n) = A330044(n).
T(n, 0) = A005359(n).
T(n, 1) = A005212(n). (End)

Edited by G. C. Greubel, Dec 05 2021

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A330044 Expansion of e.g.f. exp(x) / (1 - x^3).

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A158785 Expansion of e.g.f.: exp(tx)/(1 - x^2/t - t^3x^3).

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cp's OEIS Frontend

A330044 Expansion of e.g.f. exp(x) / (1 - x^3).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A158785 Expansion of e.g.f.: exp(t*x)/(1 - x^2/t - t^3*x^3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A158785 Expansion of e.g.f.: exp(tx)/(1 - x^2/t - t^3x^3).