A293050 -id:A293050 - cp's OEIS frontend

A293049 Expansion of e.g.f. exp(x^3/(1 - x)).

1, 0, 0, 6, 24, 120, 1080, 10080, 100800, 1149120, 14515200, 199584000, 2973801600, 47740492800, 820928908800, 15049152518400, 292919058432000, 6031865968128000, 130990787582054400, 2991455760887193600, 71659101232502784000, 1796424431562528768000

Offset: 0

Seiichi Manyama, Sep 29 2017

nonn

For n > 4, a(n) is a multiple of 10. - Muniru A Asiru, Oct 09 2017

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=2 of A293053.
E.g.f.: Product_{i>k} exp(x^i): A000262 (k=0), A052845 (k=1), this sequence (k=2), A293050 (k=3).
Cf. A361533, A361572.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=3..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 30 2017
seq(factorial(k)*coeftayl(exp(x^3/(1-x)), x = 0, k),k=0..50); # Muniru A Asiru, Oct 09 2017

CoefficientList[Series[E^(x^3/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

x='x+O('x^66); Vec(serlaplace(exp(x^3/(1-x))))

E.g.f.: Product_{i>2} exp(x^i).
a(n) ~ n^(n-1/4) * exp(-5/2 + 2*sqrt(n) - n) / sqrt(2). - Vaclav Kotesovec, Sep 30 2017
a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + 6*binomial(n-1,2) * a(n-3) - 12*binomial(n-1,3) * a(n-4) for n > 3. - Seiichi Manyama, Mar 15 2023
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,n-3*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k * a(n-k)/(n-k)!. (End)

A293053 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>k} exp(x^i).

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 0, 2, 13, 1, 0, 0, 6, 73, 1, 0, 0, 6, 36, 501, 1, 0, 0, 0, 24, 240, 4051, 1, 0, 0, 0, 24, 120, 1920, 37633, 1, 0, 0, 0, 0, 120, 1080, 17640, 394353, 1, 0, 0, 0, 0, 120, 720, 10080, 183120, 4596553, 1, 0, 0, 0, 0, 0, 720, 5040, 100800, 2116800, 58941091

Offset: 0

Seiichi Manyama, Sep 29 2017

			Square array begins:
    1,   1,   1,   1, ...
    1,   0,   0,   0, ...
    3,   2,   0,   0, ...
   13,   6,   6,   0, ...
   73,  36,  24,  24, ...
  501, 240, 120, 120, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000262, A052845, A293049, A293050.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
A(n,n-1) gives A000142(n).
Cf. A293024, A293119.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(n-j, k)*binomial(n-1, j-1)*j!, j=1+k..n))
    end:
seq(seq(A(n,d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 29 2017

A[0, ] = 1; A[n, k_] /; n <= k = 0; A[n_, k_] := A[n, k] = Sum[(i+1)! Binomial[n-1, i] A[n-1-i, k], {i, k, n-1}];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Nov 07 2020 *)

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (k..i - 1).inject(0){|s, j| s + f(j + 1) * ncr(i - 1, j) * ary[i - 1 - j]}}
  ary
end
def A293053(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A293053(20)

E.g.f. of column k: exp(x^(k+1)/(1-x)).
A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = Sum_{i=k..n-1} (i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.
A(n,k) = 2*(n-1) * A(n-1,k) - (n-1)*(n-2) * A(n-2,k) + (k+1)!*binomial(n-1,k) * A(n-1-k,k) - k*(k+1)!*binomial(n-1,k+1) * A(n-2-k,k) for n > k+1. - Seiichi Manyama, Mar 15 2023

A361545 Expansion of e.g.f. exp(x^4/(24 * (1-x))).

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 30, 210, 1715, 15750, 160650, 1801800, 22043175, 292116825, 4168464300, 63725161500, 1039028615625, 17998106626500, 330068683444500, 6388785205803000, 130156170633113625, 2783924007745505625, 62375052003905891250, 1460924768552182683750

Offset: 0

Seiichi Manyama, Mar 15 2023

Cf. A185369, A361533, A361547.

Cf. A293050.

RecurrenceTable[{3 (-4 + n) (-3 + n) (-2 + n) (-1 + n) a[-5 + n] - 4 (-3 + n) (-2 + n) (-1 + n) a[-4 + n] + 24 (-2 + n) (-1 + n) a[-2 + n] - 48 (-1 + n) a[-1 + n] + 24 a[n] == 0, a[1] == 0, a[2] == 0, a[3] == 0, a[4] == 1, a[5] == 5}, a, {n, 0, 25}] (* Vaclav Kotesovec, Aug 28 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/(24*(1-x)))))

a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + binomial(n-1,3) * a(n-4) - 3*binomial(n-1,4) * a(n-5) for n > 4.
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k)/(24^k * k!).
a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} k * a(n-k)/(n-k)!. (End)
a(n) ~ 2^(-5/4) * 3^(-1/4) * exp(-7/48 + sqrt(n/6) - n) * n^(n - 1/4). - Vaclav Kotesovec, Aug 28 2025

cp's OEIS Frontend

A293049 Expansion of e.g.f. exp(x^3/(1 - x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A293053 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>k} exp(x^i).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Ruby

Formula

A361545 Expansion of e.g.f. exp(x^4/(24 * (1-x))).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula