A330045 -id:A330045 - 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

A337727 a(n) = (4n)! Sum_{k=0..n} 1 / (4*k)!.

Original entry on oeis.org

1, 25, 42001, 498971881, 21795091762081, 2534333270094778681, 646315807872650838343345, 317599587988620621961919733001, 274101148417699141578015206369183041, 387502275541069630431671657548241448722521, 849931991080760484603611346800010863970028660561

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100733, A330045, A332890, A337725, A337726, A337728, A337729, A337730.

Table[(4 n)! Sum[1/(4 k)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n)! SeriesCoefficient[(1/2) (Cos[x] + Cosh[x])/(1 - x^4), {x, 0, 4 n}], {n, 0, 10}]
Table[Floor[(1/2) (Cos[1] + Cosh[1]) (4 n)!], {n, 0, 10}]

a(n) = (4*n)!*sum(k=0, n, 1/(4*k)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (cos(x) + cosh(x)) / (1 - x^4) = 1 + 25*x^4/4! + 42001*x^8/8! + 498971881*x^12/12! + ...

a(n) = floor(c * (4*n)!), where c = (cos(1) + cosh(1)) / 2 = A332890.

A337728 a(n) = (4n+1)! Sum_{k=0..n} 1 / (4*k+1)!.

Original entry on oeis.org

1, 121, 365905, 6278929801, 358652470233121, 51516840824285500441, 15640512874253077933887601, 8915467710633236496186345872425, 8755702529258688898174686554391144001, 13878488965077362598718732163634314533105081, 33731389859841228248933904149069928786421237268881

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A330045, A334363, A337725, A337726, A337727, A337729, A337730.

Table[(4 n + 1)! Sum[1/(4 k + 1)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n + 1)! SeriesCoefficient[(1/2) (Sin[x] + Sinh[x])/(1 - x^4), {x, 0, 4 n + 1}], {n, 0, 10}]
Table[Floor[(1/2) (Sin[1] + Sinh[1]) (4 n + 1)!], {n, 0, 10}]

a(n) = (4*n+1)!*sum(k=0, n, 1/(4*k+1)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (sin(x) + sinh(x)) / (1 - x^4) = x + 121*x^5/5! + 365905*x^9/9! + 6278929801*x^13/13! + ...

a(n) = floor(c * (4*n+1)!), where c = (sin(1) + sinh(1)) / 2 = A334363.

A337729 a(n) = (4n+2)! Sum_{k=0..n} 1 / (4*k+2)!.

Original entry on oeis.org

1, 361, 1819441, 43710250585, 3210080802962401, 563561785768079119561, 202205968733586788098486801, 132994909755454702268136738753721, 148026526435655214537290625514621562305, 262237873172349351865682580536682974917045801, 704454843460345510903820429747302209179158476142321

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A330045, A334364, A337725, A337726, A337727, A337728, A337730.

Table[(4 n + 2)! Sum[1/(4 k + 2)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n + 2)! SeriesCoefficient[(1/2) (Cosh[x] - Cos[x])/(1 - x^4), {x, 0, 4 n + 2}], {n, 0, 10}]
Table[Floor[(1/2) (Cosh[1] - Cos[1]) (4 n + 2)!], {n, 0, 10}]

a(n) = (4*n+2)!*sum(k=0, n, 1/(4*k+2)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (cosh(x) - cos(x)) / (1 - x^4) = x^2/2! + 361*x^6/6! + 1819441*x^10/10! + 43710250585*x^14/14! + ...

a(n) = floor(c * (4*n+2)!), where c = (cosh(1) - cos(1)) / 2 = A334364.

A337730 a(n) = (4n+3)! Sum_{k=0..n} 1 / (4*k+3)!.

Original entry on oeis.org

1, 841, 6660721, 218205219961, 20298322381652065, 4313799472548696853801, 1816972337837511114820981201, 1372104830641374893468212163747161, 1724241814377177346127894133451232399041, 3403694723384093133512770088891935585284510985

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A330045, A334365, A337725, A337726, A337727, A337728, A337729.

Table[(4 n + 3)! Sum[1/(4 k + 3)!, {k, 0, n}], {n, 0, 9}]
Table[(4 n + 3)! SeriesCoefficient[(1/2) (Sinh[x] - Sin[x])/(1 - x^4), {x, 0, 4 n + 3}], {n, 0, 9}]
Table[Floor[(1/2) (Sinh[1] - Sin[1]) (4 n + 3)!], {n, 0, 9}]

a(n) = (4*n+3)!*sum(k=0, n, 1/(4*k+3)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (sinh(x) - sin(x)) / (1 - x^4) = x^3/3! + 841*x^7/7! + 6660721*x^11/11! + 218205219961*x^15/15! + ...

a(n) = floor(c * (4*n+3)!), where c = (sinh(1) - sin(1)) / 2 = A334365.

A337751 a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k / (n-4*k)!.

Original entry on oeis.org

1, 1, 1, 1, -23, -119, -359, -839, 38641, 359857, 1809361, 6644881, -459055079, -6175146119, -43468088663, -217686301559, 20051525850721, 352724346317281, 3192296431410721, 20250050516224417, -2331591425921837879, -50665325105014242839, -560439561498466178759

Offset: 0

Ilya Gutkovskiy, Sep 18 2020

sign

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

Cf. A182386, A330045, A337749, A337750.

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

a(n) = n!*sum(k=0, n\4, (-1)^k / (n-4*k)!); \\ Michel Marcus, Sep 18 2020

G.f.: Sum_{k>=0} (-1)^k * (4*k)! * x^(4*k) / (1 - x)^(4*k+1).
E.g.f.: exp(x) / (1 + x^4).
a(0) = a(1) = a(2) = a(3) = 1; a(n) = 1 - n * (n-1) * (n-2) * (n-3) * a(n-4).

A158777 Irregular array T(n,k), read by rows: row n is the polynomial expansion in t of p(x,t) = exp(tx)/(1 - x/t - t^4 x^4) with weighting factors t^n*n!.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 2, 0, 1, 6, 0, 6, 0, 3, 0, 1, 24, 0, 24, 0, 12, 0, 4, 0, 25, 120, 0, 120, 0, 60, 0, 20, 0, 245, 0, 121, 720, 0, 720, 0, 360, 0, 120, 0, 2190, 0, 1446, 0, 361, 5040, 0, 5040, 0, 2520, 0, 840, 0, 20370, 0, 15162, 0, 5047, 0, 841, 40320, 0, 40320, 0, 20160, 0, 6720, 0

Offset: 0

Roger L. Bagula, Mar 26 2009

Row sums are A334157: {1, 2, 5, 16, 89, 686, 5917, 54860, 588401, 7370074, ...}.
Outer diagonal is A330045: {1, 1, 1, 1, 25, 121, 361, 841, 42001, 365905, ...}.
From Petros Hadjicostas, Apr 15 2020: (Start)
To prove the general formula below for T(n,2*m), let v = x/t in the equation Sum_{n,k >= 0} (T(n,k)/n!) * (x/t)^n * t^k = p(x,t). We get Sum_{n,k >= 0} (T(n,k)/n!) * v^n * t^k = exp(t^2*v)/(1 - v - t^8*v^4).
Using the Taylor expansions of exp(t^2*v) and 1/(1 - v - t^8*v^4) around v = 0 (from array A180184), we get that Sum_{n,k >= 0} (T(n,k)/n!) * v^n * t^k = (Sum_{n >= 0} (t^2*v)^n/n!) * (Sum_{n >= 0} Sum_{h=0..floor(n/4)} binomial(n - 3*h, h)*t^(8*h)*v^n).
Using the Cauchy product of the above two series, for each n >= 0, we get Sum_{k >= 0} (T(n,k)/n!)*t^k = Sum_{l=0..n} Sum_{h=0..floor(l/4)} (binomial(l - 3*h, h)/(n-l)!)*t^(8*h+2*n-2*l). This implies that T(n,k) = 0 for odd k >= 1.
Letting k = 2*m = 8*h + 2*n - 2*l and s = n - l, we get Sum_{m >= 0} (T(n, 2*m)/n!)*t^(2*m) = Sum_{m >= 0} Sum_{s=0..m with 4|(m-s)} (binomial(n - s - 3*(m - s)/4, (m - s)/4)/s!)*t^(2*m) (and also that T(n,2*j) = 0 for j > m). Equating coefficients, we get the formula for T(n,2*m) shown below. (End)

			Array T(n,k) (with n >= 0 and 0 <= k <= 2*n) begins as follows:
     1;
     1, 0,    1;
     2, 0,    2, 0,    1;
     6, 0,    6, 0,    3, 0,   1;
    24, 0,   24, 0,   12, 0,   4, 0,    25;
   120, 0,  120, 0,   60, 0,  20, 0,   245, 0,   121;
   720, 0,  720, 0,  360, 0, 120, 0,  2190, 0,  1446, 0,  361;
  5040, 0, 5040, 0, 2520, 0, 840, 0, 20370, 0, 15162, 0, 5047, 0, 841;
  ...

Cf. A000142, A001710, A001715, A180184, A330045, A334157.

# Triangle T(n, k) without the zeros (even k):
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;
for n1 from 0 to 20 do
seq(W(n1,m1), m1=0..n1); end do; # Petros Hadjicostas, Apr 15 2020

(* Generates the sequence in the data section *)
Table[Expand[t^n*n!*SeriesCoefficient[Series[Exp[t*x]/(1 - x/t - t^4*x^4), {x, 0, 20}], n]], {n, 0, 10}];
a = Table[CoefficientList[Expand[t^n*n!*SeriesCoefficient[Series[Exp[t*x]/(1 - x/t - t^4*x^4), {x, 0, 20}], n]], t], {n, 0, 10}];
Flatten[%]
(* Generates row sums *)
Table[Apply[Plus, CoefficientList[Expand[t^n*n!*SeriesCoefficient[Series[Exp[t*x]/( 1 - x/t - t^4*x^4), {x, 0, 20}], n]], t]], {n, 0, 10}];

T(n,k) = [t^k] (t^n * n! * ([x^n] p(x,t))), where p(x,t) = exp(t*x)/(1 - x/t - t^4*x^4).
From Petros Hadjicostas, Apr 15 2020: (Start)
Sum_{n,k >= 0} (T(n,k)/n!) * (x/t)^n * t^k = p(x,t).
T(n,0) = n! = A000142(n) for n >= 0; T(n,2) = n! for n >= 1; T(n,4) = n!/2 = A001710(n) for n >= 2; T(n,6) = n!/6 = A001715(n) for n >= 3.
T(n,2*m) = n! * Sum_{s = 0..m with 4|(m-s)} binomial(n - s - 3*(m-s)/4, (m-s)/4)/s! for n >= 0 and 0 <= m <= n.
T(n,2*n) = A330045(n) for n >= 0. (End)

Various sections edited by Petros Hadjicostas, Apr 13 2020

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).

cp's OEIS Frontend

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

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A337727 a(n) = (4*n)! * Sum_{k=0..n} 1 / (4*k)!.

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A337728 a(n) = (4*n+1)! * Sum_{k=0..n} 1 / (4*k+1)!.

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A337729 a(n) = (4*n+2)! * Sum_{k=0..n} 1 / (4*k+2)!.

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A337730 a(n) = (4*n+3)! * Sum_{k=0..n} 1 / (4*k+3)!.

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A337751 a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k / (n-4*k)!.

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A158777 Irregular array T(n,k), read by rows: row n is the polynomial expansion in t of p(x,t) = exp(t*x)/(1 - x/t - t^4 * x^4) with weighting factors t^n*n!.

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A334157 Row sums of array A158777.

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A337727 a(n) = (4n)! Sum_{k=0..n} 1 / (4*k)!.

A337728 a(n) = (4n+1)! Sum_{k=0..n} 1 / (4*k+1)!.

A337729 a(n) = (4n+2)! Sum_{k=0..n} 1 / (4*k+2)!.

A337730 a(n) = (4n+3)! Sum_{k=0..n} 1 / (4*k+3)!.

A158777 Irregular array T(n,k), read by rows: row n is the polynomial expansion in t of p(x,t) = exp(tx)/(1 - x/t - t^4 x^4) with weighting factors t^n*n!.