A330044 -id:A330044 - cp's OEIS frontend

A330045 Expansion of e.g.f. exp(x) / (1 - x^4).

1, 1, 1, 1, 25, 121, 361, 841, 42001, 365905, 1819441, 6660721, 498971881, 6278929801, 43710250585, 218205219961, 21795091762081, 358652470233121, 3210080802962401, 20298322381652065, 2534333270094778681, 51516840824285500441, 563561785768079119561

Offset: 0

Ilya Gutkovskiy, Nov 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000522, A087208, A100733, A330044, A334157.

Outer diagonal of A158777.

nmax = 22; CoefficientList[Series[Exp[x]/(1 - x^4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[n!/(n - 4 k)!, {k, 0, Floor[n/4]}], {n, 0, 22}]

G.f.: Sum_{k>=0} (4*k)! * x^(4*k) / (1 - x)^(4*k + 1).
a(0) = a(1) = a(2) = a(3) = 1; a(n) = n*(n - 1)*(n - 2)*(n - 3)*a(n - 4) + 1.
a(n) = Sum_{k=0..floor(n/4)} n! / (n - 4*k)!.
a(n) ~ n! * (2*cos(Pi*n/2 - 1) + exp(1) + (-1)^n*exp(-1))/4. - Vaclav Kotesovec, Apr 18 2020

A337725 a(n) = (3n+1)! Sum_{k=0..n} 1 / (3*k+1)!.

Original entry on oeis.org

1, 25, 5251, 3780721, 6487717237, 21798729916321, 126737815733490295, 1171057417377450325801, 16160592359808814496053801, 317652603424402057734433512457, 8567090714356123497097671830965291, 307592825008242258039794809418977808065

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100089, A143820, A330044, A337726, A337727, A337728, A337729, A337730.

Table[(3 n + 1)! Sum[1/(3 k + 1)!, {k, 0, n}], {n, 0, 11}]
Table[(3 n + 1)! SeriesCoefficient[(Exp[3 x/2] - 2 Sin[Pi/6 - Sqrt[3] x/2])/(3 Exp[x/2] (1 - x^3)), {x, 0, 3 n + 1}], {n, 0, 11}]
Table[Floor[(Exp[3/2] + 2 Sin[(3 Sqrt[3] - Pi)/6])/(3 Sqrt[Exp[1]]) (3 n + 1)!], {n, 0, 11}]

a(n) = (3*n+1)!*sum(k=0, n, 1/(3*k+1)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (exp(3*x/2) - 2 * sin(Pi/6 - sqrt(3)*x/2)) / (3*exp(x/2) * (1 - x^3)) = x + 25*x^4/4! + 5251*x^7/7! + 3780721*x^10/10! + ...

a(n) = floor(c * (3*n+1)!), where c = (exp(3/2) + 2 * sin((3 * sqrt(3) - Pi) / 6))/(3 * sqrt(exp(1))) = A143820.

A337726 a(n) = (3n+2)! Sum_{k=0..n} 1 / (3*k+2)!.

Original entry on oeis.org

1, 61, 20497, 20292031, 44317795705, 180816606476401, 1236785588298582841, 13142083661260741268467, 205016505115667563788085201, 4494781858155895668489979946725, 133764708098719455094261803214536001, 5252940087036713001551661012234828759271

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100043, A143821, A330044, A337725, A337727, A337728, A337729, A337730.

Table[(3 n + 2)! Sum[1/(3 k + 2)!, {k, 0, n}], {n, 0, 11}]
Table[(3 n + 2)! SeriesCoefficient[(Exp[3 x/2] - 2 Sin[Sqrt[3] x/2 + Pi/6])/(3 Exp[x/2] (1 - x^3)), {x, 0, 3 n + 2}], {n, 0, 11}]
Table[Floor[(Exp[3/2] - 2 Sin[(3 Sqrt[3] + Pi)/6])/(3 Sqrt[Exp[1]]) (3 n + 2)!], {n, 0, 11}]

a(n) = (3*n+2)!*sum(k=0, n, 1/(3*k+2)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (exp(3*x/2) - 2 * sin(sqrt(3)*x/2 + Pi/6)) / (3*exp(x/2) * (1 - x^3)) = x^2/2! + 61*x^5/5! + 20497*x^8/8! + 20292031*x^11/11! + ...

a(n) = floor(c * (3*n+2)!), where c = (exp(3/2) - 2 * sin((3 * sqrt(3) + Pi) / 6))/(3 * sqrt(exp(1))) = A143821.

A158757 Expansion of e.g.f. exp(tx)/(1 - x^2/t^2 - t^3 x^3).

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, Mar 25 2009

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

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, page 221.

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

Cf. A005212, A005359, A158706, A330044.

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

f(x,t) = exp(t*x)/(1 - x^2/t^2 - t^3*x^3)
def A158757(n,k): return ( factorial(n)*t^n*( f(x,t) ).series(x, 20).list()[n] ).series(t,2*n+1).list()[k]
flatten([[A158757(n,k) for k in (0..2*n)] for n in (0..10)]) # G. C. Greubel, Dec 05 2021

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

Edited by G. C. Greubel, Dec 01 2021

A337750 a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k / (n-3*k)!.

Original entry on oeis.org

1, 1, 1, -5, -23, -59, 601, 4831, 19825, -302903, -3478319, -19626749, 399831961, 5968795405, 42864819817, -1091541253529, -20055152560799, -174888464853359, 5344185977277985, 116600656988485387, 1196237099596975561, -42646604098678320299, -1077390070573604975879

Offset: 0

Ilya Gutkovskiy, Sep 18 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A182386, A330044, A337749, A337751.

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

a(n) = n!*sum(k=0, n\3, (-1)^k / (n-3*k)!); \\ Michel Marcus, Sep 18 2020

G.f.: Sum_{k>=0} (-1)^k * (3*k)! * x^(3*k) / (1 - x)^(3*k+1).
E.g.f.: exp(x) / (1 + x^3).
a(0) = a(1) = a(2) = 1; a(n) = 1 - n * (n-1) * (n-2) * a(n-3).

A334157 Row sums of array A158777.

Original entry on oeis.org

1, 2, 5, 16, 89, 686, 5917, 54860, 588401, 7370074, 103522421, 1573237832, 25869057865, 462768222086, 8965777751309, 186025937645956, 4106375449878497, 96241703493486770, 2390797380938894821, 62730027061416412544

Offset: 0

Petros Hadjicostas, Apr 16 2020

nonn

Cf. A003269, A087208, A158777, A330044, A330045.

W := proc(n, m) local v, s, h; v := 0;
for s from 0 to m do
if 0 = (m - s) mod 4 then
h := (m - s)/4;
v := v + binomial(n - s - 3*h, h)/s!;
end if; end do; n!*v; end proc;
seq(add(W(n1, m1), m1 = 0 .. n1), n1 = 0 .. 35);

Table[Apply[Plus, CoefficientList[Expand[t^n*n!*SeriesCoefficient[Series[Exp[t*x]/( 1 - x/t - t^4*x^4), {x, 0, 50}], n]], t]], {n, 0, 40}]; (* Program due to Roger L. Bagula from A158777 *)

a(n) = n!*Sum_{k=0..n} A003269(k+1)/(n-k)!.
a(n) = n!*Sum_{k=0..n} Sum_{s=0..floor(k/3)} binomial(k-3*s, s)/(n-k)!.
E.g.f.: exp(x)/(1 - x - x^4).

A358547 a(n) = Sum_{k=0..floor(n/3)} (n-k)!/(n-3*k)!.

Original entry on oeis.org

1, 1, 1, 3, 7, 13, 45, 151, 403, 1617, 6793, 23275, 105951, 522133, 2159077, 10964223, 61134955, 293587801, 1641566913, 10124731987, 55014334903, 335177088285, 2251814156701, 13587321392743, 89436553249347, 647267633012833, 4276528756374265, 30198747030078651

Offset: 0

Seiichi Manyama, Nov 21 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..673

Cf. A122852, A330044, A357532.

PARI
```
a(n) = sum(k=0, n\3, (n-k)!/(n-3*k)!);
```

a(n) = (3 * (2*n-1) * a(n-1) - n * a(n-2) + 2 * (n-1) * n * (2*n-3) * a(n-3) + 2 * (2*n-3))/(9 * (n-1)) for n > 2.

a(n) ~ sqrt(Pi) * 2^(2*n/3 + 1) * n^(2*n/3 + 1/2) / (3^(2*n/3 + 3/2) * exp(2*n/3 - (2/3)^(1/3) * n^(1/3))) * (1 + 1/(2^(4/3) * 3^(5/3) * n^(1/3)) + 145/(2^(11/3) * 3^(10/3) * n^(2/3)) + 3349/(23328*n)). - Vaclav Kotesovec, Nov 25 2022

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

cp's OEIS Frontend

A330045 Expansion of e.g.f. exp(x) / (1 - x^4).

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A337725 a(n) = (3*n+1)! * Sum_{k=0..n} 1 / (3*k+1)!.

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A337726 a(n) = (3*n+2)! * Sum_{k=0..n} 1 / (3*k+2)!.

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A158757 Expansion of e.g.f. exp(t*x)/(1 - x^2/t^2 - t^3* x^3).

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A337750 a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k / (n-3*k)!.

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A334157 Row sums of array A158777.

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A358547 a(n) = Sum_{k=0..floor(n/3)} (n-k)!/(n-3*k)!.

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

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Mathematica

Sage

Formula

A337725 a(n) = (3n+1)! Sum_{k=0..n} 1 / (3*k+1)!.

A337726 a(n) = (3n+2)! Sum_{k=0..n} 1 / (3*k+2)!.

A158757 Expansion of e.g.f. exp(tx)/(1 - x^2/t^2 - t^3 x^3).

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