A143411 Square array, read by antidiagonals: form the Euler-Seidel matrix for the sequence {2^k*k!} and then divide column k by 2^k*k!.

1, 3, 1, 13, 5, 1, 79, 33, 7, 1, 633, 277, 61, 9, 1, 6331, 2849, 643, 97, 11, 1, 75973, 34821, 7993, 1225, 141, 13, 1, 1063623, 493825, 114751, 17793, 2071, 193, 15, 1, 17017969, 7977173, 1870837, 292681, 34361, 3229, 253, 17, 1

Offset: 0

Author

Peter Bala, Aug 19 2008

Keywords

• easy
• nonn
• tabl

Comments

This table is closely connected to the constant 1/sqrt(e). The row, column and diagonal entries of this table occur in series acceleration formulas for 1/sqrt(e). For a similar table based on the differences of the sequence {2^k*k!} and related to the constant sqrt(e), see A143410. For other arrays similarly related to constants see A086764 (for e), A143409 (for 1/e), A008288 (for log(2)), A108625 (for zeta(2)) and A143007 (for zeta(3)).

Examples

			The Euler-Seidel matrix for the sequence {2^k*k!} begins
  ========================================
  n\k|     0     1     2     3     4     5
  ========================================
  0  |     1     2     8    48   384  3840
  1  |     3    10    56   432  4224
  2  |    13    66   488  4656
  3  |    79   554  5144
  4  |   633  5698
  5  |  6331
  ...
.
  Dividing the k-th column by 2^k*k! gives
  ========================================
  n\k|     0     1     2     3     4     5
  ========================================
  0  |     1     1     1     1     1     1
  1  |     3     5     7     9    11
  2  |    13    33    61    97
  3  |    79   277   643
  4  |   633  2849
  5  |  6331
  ...
		

Links

• G. C. Greubel, Antidiagonals n = 0..50, flattened
• D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
• Eric Weisstein's World of Mathematics, Poisson-Charlier polynomial.

Crossrefs

Cf. A008288, A065919 (main diagonal), A076571, A086764, A108625, A143007, A143409, A143410.

Programs

Magma

A:= func< n,k | (&+[Binomial(n,j)*Factorial(k+j)*2^j/Factorial(k): j in [0..n]]) >; // Array
A143411:= func< n,k | A(n-k,k) >; // antidiagonal triangle
[A143411(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023

Maple

with combinat: T := (n, k) -> 1/k!*add(2^j*binomial(n,j)*(k+j)!, j = 0..n): for n from 0 to 9 do seq(T(n, k), k = 0..9) end do;

Mathematica

A[n_, k_]:= (1/k!)*Sum[Binomial[n,j]*(k+j)!*2^j, {j,0,n}]; (* array *)
A143411[n_, k_]:= A[n-k,k]; (* antidiagonals *)
Table[A143411[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 05 2023 *)

SageMath

def A(n,k): return sum(binomial(n,j)*factorial(j+k)*2^j/factorial(k) for j in range(n+1)) # array
def A143411(n,k): return A(n-k,k) # antidiagonal triangle
flatten([[A143411(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023

Formula

T(n,k) = (1/k!)*Sum_{j = 0..n} 2^j*binomial(n,j)*(k+j)!.

Relation with Poisson-Charlier polynomials c_n(x,a):

T(n,k) = (-1)^n*c_n(-(k+1),1/2).

Recurrence relations:

T(n,k) = 2*n*T(n-1,k) + T(n,k-1);

T(n,k) = 2*(n+k)*T(n-1,k) + T(n-1,k-1);

T(n,k) = 2*(k+1)*T(n-1,k+1) + T(n-1,k).

Recurrence for row n entries: 2*k*T(n,k) = (2*n+2*k-1)*T(n,k-1) + T(n,k-2).

E.g.f. for column k: exp(y)/(1 - 2*y)^(k+1).

E.g.f. for array: exp(y)/(1 - x - 2*y) = (1 + x + x^2 + ...) + (3 + 5*x + 7*x^2 + ...)*y + (13 + 37*x + 61*x^2 + ...)*y^2/2! + ... .

Series acceleration formulas for 1/sqrt(e):

Row n: 1/sqrt(e) = 2^n*n!*(1/T(n,0) - 1/(2*1!*T(n,0)*T(n,1)) + 1/(2^2*2!*T(n,1)*T(n,2)) - 1/(2^3*3!*T(n,2)*T(n,3)) + ...). For example, row 3 gives 1/sqrt(e) = 48*(1/79 - 1/(2*79*277) + 1/(8*277*643) - 1/(48*643*1225) + ...).

Column k: 1/sqrt(e) = (1 - (1/2)/1! + (1/2)^2/2! - ... + (-1/2)^k/k!) + (-1)^(k+1)/(2^k*k!)*( Sum_{n = 0..inf} 2^n*n!/(T(n,k)*T(n+1,k)) ). For example, column 3 gives 1/sqrt(e) = 29/48 + 1/48*( 1/(1*9) + 2/(9*97) + 8/(97*1225) + 48/(1225*17793) + ... ).

Main diagonal: 1/sqrt(e) = 1 - 2*( 1/(1*5) - 1/(5*61) + 1/(61*1225) - ... ). See A065919.