A084757 For n, k > 0, let T(n, k) be given by T(n, 1) = n and T(n, k+1) = k*T(n, k) + 1. a(n) is the sum of the n-th antidiagonal.

1, 4, 11, 31, 106, 466, 2577, 17151, 132666, 1165310, 11438525, 123981551, 1469997610, 18919751410, 262644893329, 3911200633719, 62186842823250, 1051369907752254, 18832837831656989, 356278889320409303

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..450

Formula

E.g.f.: exp(x)*(exp(-1)*(Ei(1) - Ei(1-x))*x + 1 - log(1-x) + 1/(1-x)) - 1. - Vladeta Jovovic, Jan 06 2005

From G. C. Greubel, May 14 2023: (Start)

a(n) = Sum_{k=1..n} (A000522(n-k) + (n-k)!*(k-1)). [Corrected by Sean A. Irvine, May 08 2026]

a(n) = Sum_{k=0..n-1} floor(e*k!) + n*A003422(n) - A003422(n+1). (End)

Example

The array, A084756(n,k), begins
  1, 2,  5, 16,  65, 326, 1957, ...
  2, 3,  7, 22,  89, 446, 2677, ...
  3, 4,  9, 28, 113, 566, 3397, ...
  4, 5, 11, 34, 137, 686, 4117, ...
  ...
The antidiagonal rows and sums are:
    1                       :   1;
    2,  2                   :   4;
    5,  3,  3               :  11;
   16,  7,  4,  4           :  31;
   65, 22,  9,  5, 5        : 106;
  326, 89, 28, 11, 6, 6     : 466;
  ...

Mathematica

A084756[n_, k_]:= Floor[E*(n-1)!] + (k-1)*(n-1)!;
A084757[n_]:= -1 + Sum[A084756[n-j+1, j], {j, n}];
Table[A084757[n], {n, 40}] (* G. C. Greubel, May 14 2023 *)

Prog

(Magma)
A084756:= func< n, k | Floor(Exp(1)*Factorial(n-1)) + (k-1)*Factorial(n-1) >;
A084757:= func< n | -1 + (&+[A084756(n-k+1, k): k in [1..n]]) >;
[A084757(n): n in [1..40]]; // G. C. Greubel, May 14 2023
(SageMath)
def A084756(n, k): return floor(e*factorial(n-1)) + (k-1)*factorial(n-1) - int(n==1)
def A084757(n): return sum( A084756(n-k+1, k) for k in range(1, n+1) )
[A084757(n) for n in range(1, 41)] # G. C. Greubel, May 14 2023

Crossrefs

Cf. A000522, A003422, A084756.

Sequence in context: A389172 A134597 A076730 * A353425 A155962 A027153

Adjacent sequences: A084754 A084755 A084756 * A084758 A084759 A084760

Keyword

nonn,easy

Author

Amarnath Murthy, Jun 17 2003

Extensions

Edited and extended by David Wasserman, Jan 05 2005

Status

approved