A053482 -id:A053482 - cp's OEIS frontend

A181783 Array described in comments to A053482, here read by increasing antidiagonals. See comments below.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 4, 1, 1, 1, 16, 21, 7, 1, 1, 1, 65, 142, 63, 11, 1, 1, 1, 326, 1201, 709, 151, 16, 1, 1, 1, 1957, 12336, 9709, 2521, 311, 22, 1, 1, 1, 13700, 149989, 157971, 50045, 7186, 575, 29, 1, 1, 1, 109601, 2113546, 2993467, 1158871, 193765, 17536, 981, 37, 1

Offset: 0

Richard Choulet, Dec 23 2012

We denote by a(n,k) the number in row number n >= 0 and column number k >= 0. The recurrence which defines the array is a(n,k) = n*(k-1)*a(n-1,k) + a(n,k-1). The initial values are given by a(n,0) = 1 = a(0,k) for all n >= 0 and k >= 0.

			Array read row after row:
  1, 1,    1,      1,       1,        1,         1, ...
  1, 1,    2,      4,       7,       11,        16, ...
  1, 1,    5,     21,      63,      151,       311, ...
  1, 1,   16,    142,     709,     2521,      7186, ...
  1, 1,   65,   1201,    9709,    50045,    193765, ...
  1, 1,  326,  12336,  157971,  1158871,   6002996, ...
  1, 1, 1957, 149989, 2993467, 30806371, 210896251, ...
  ...
A(4,3) = 1201.

Cf. A000522, A053482, A185106, A371898.

A181783 := proc(n,k)
    option remember;
    if n =0 or k = 0 then
        1;
    else
        n*(k-1)*procname(n-1,k)+procname(n,k-1) ;
    end if;
end proc:
seq(seq(A181783(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Mar 02 2016

T[n_, k_] := T[n, k] = If[n == 0 || k == 0, 1, n (k - 1) T[n - 1, k] + T[n, k - 1]];
Table[T[n - k, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 10 2023 *)

If we consider the e.g.f. Psi(k) of column number k we have: Psi(k)(z) = Psi(k-1)(z)/(1-(k-1)*z) with Psi(1)(z) = exp(z). Then Psi(k)(z) = exp(z)/Product_{j=0..k-1} (1 - j*z). We conclude that a(n,k) = n!*Sum_{m=0..n} Sum_{j=1..k-1} (-1)^(k-1-j)*j^(m+k-2)/((n-m)!*(j-1)!*(k-1-j)!). It seems after the recurrence (and its proof) in A053482 that:
A(n,k) = -Sum_{j=1..k-1} s1(k,k-j)*n*(n-1)*...*(n-k+1)*a(n-j,k) + 1 where s1(m,n) are the classical Stirling numbers of the first kind.
A(n,1) = 1 for every n.
A(1,k) = 1 + k*(k-1)/2 for every k.
A(n, k+1) = A371898(n+k, k) * n! / ((n+k)! * k!). - Werner Schulte, Apr 14 2024

Edited by N. J. A. Sloane, Dec 24 2012

A185106 Column 4 of A181783.

Original entry on oeis.org

1, 7, 63, 709, 9709, 157971, 2993467, 64976353, 1593358809, 43632348319, 1321213523191, 43869502390077, 1585770335098693, 62013234471100459, 2609265444024424179, 117558236422872707161, 5647316731308685308337, 288166881285968665526583, 15566545814457889774570159, 887503412305357492886020789

Offset: 0

Richard Choulet, Dec 26 2012

A181783 is written as follows:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 4, 7, 11, 16, ...
1, 1, 5, 21, 63, 151, 311, ...
1, 1, 16, 142, 709, 2521, ...
1, 1, 65, 1201, 9709, ...
A000522 and A053482 are respectively the columns number 2 and 3 of this array. Our sequence gives the column number 4 (the fifth).

G. C. Greubel, Table of n, a(n) for n = 0..375

Cf. A181783, A000522, A053482.

a(0,1):=1:for p from 2 to 15 do for n from 0 to 20 do a(n,0):=1 :a(n,p):=n!*sum('1/(n-m)!*sum('k^(p-2)*(-1)^(p-1-k)*k^m/((k-1)!*(p-1-k)!)','k'=1..(p-1))','m'=0..n):od:seq(a(n,p),n=0..20):od;

CoefficientList[Series[E^x/((1-x)*(1-2x)*(1-3x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)

x='x+O('x^50); Vec(serlaplace(exp(x)/((1-x)*(1-2*x)*(1-3*x)))) \\ G. C. Greubel, Jun 22 2017

Recurrence relation: a(n)= 6*n*a(n-1) -11*n*(n-1)*a(n-2) +6*n*(n-1)*(n-2)*a(n-3) +1 (or following A053482 for a linear homogeneous recurrence) a(n)= (6n+1)*a(n-1) -(11n+6)*(n-1)*a(n-2) +(6n+11)*(n-1)*(n-2)*a(n-3) -6*(n-1)*(n-2)*(n-3)*a(n-4).
E.g.f: exp(z)/((1-z)*(1-2*z)*(1-3*z)), as explained in A181783.
With p=4, a(n)=a(n,p)=n!*sum('1/(n-m)!*sum('k^(p-2)*(-1)^(p-1-k)*k^m/((k-1)!*(p-1-k)!)','k'=1..(p-1))','m'=0..n)
a(n) ~ n! * exp(1/3)*3^(n+2)/2. - Vaclav Kotesovec, Oct 02 2013

A367962 Triangle read by rows. T(n, k) = Sum_{j=0..k} (n!/j!).

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 6, 12, 15, 16, 24, 48, 60, 64, 65, 120, 240, 300, 320, 325, 326, 720, 1440, 1800, 1920, 1950, 1956, 1957, 5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700, 40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601

Offset: 0

Peter Luschny, Dec 06 2023

			  [0]   1;
  [1]   1,    2;
  [2]   2,    4,    5;
  [3]   6,   12,   15,   16;
  [4]  24,   48,   60,   64,   65;
  [5] 120,  240,  300,  320,  325,  326;
  [6] 720, 1440, 1800, 1920, 1950, 1956, 1957;

Cf. A094587, A000142 (T(n, 0)), A052849 (T(n, 1)), A000522 (T(n, n)), A007526 (T(n,n-1)), A038154 (T(n, n-2)), A355268 (T(n, n/2)), A367963(n) (T(2*n, n)/n!).

Cf. A001339 (row sums), A087208 (alternating row sums), A082030 (accumulated sums), A053482, A331689.

T := (n, k) -> add(n!/j!, j = 0..k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Module[{n=1},NestList[Append[n#,1+Last[#]n++]&,{1},10]] (* or *)
Table[Sum[n!/j!,{j,0,k}],{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 07 2023 *)

from functools import cache
@cache
def a_row(n: int) -> list[int]:
    if n == 0: return [1]
    row = a_row(n - 1) + [0]
    for k in range(n): row[k] *= n
    row[n] = row[n - 1] + 1
    return row

def T(n, k): return sum(falling_factorial(n, n - j) for j in range(k + 1))
for n in range(9): print([T(n, k) for k in range(n + 1)])

T(n, k) = A094587(n, k) * A000522(k).
T(n, k) = e * (n! / k!) * Gamma(k + 1, 1).
Sum_{k=0..n} T(n, k) * 2^(n - k) = A053482(n).
Sum_{k=0..n} T(n, k) * binomial(n, k) = A331689(n).
Recurrence: T(n, n) = T(n, n-1) + 1 starting with T(0, 0) = 1.
For k <> n: T(n, k) = n * T(n-1, k).

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A181783 Array described in comments to A053482, here read by increasing antidiagonals. See comments below.

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A185106 Column 4 of A181783.

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A367962 Triangle read by rows. T(n, k) = Sum_{j=0..k} (n!/j!).

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