0,2

The Euler-Seidel matrix for the sequence {k!} is array A076571 read as a square, whose k_th column entries have a common factor of k!. Removing these common factors gives the current table. This table is closely connected to the constant 1/e. The row, column and diagonal entries of this table occur in series acceleration formulas for 1/e. For a similar table based on the differences of the sequence {k!} and related to the constant e, see A086764. For other arrays similarly related to constants see A143410 (for sqrt(e)), A143411 (for 1/sqrt(e)), A008288 (for log(2)), A108625 (for zeta(2)) and A143007 (for zeta(3)).

D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

Eric Weisstein's World of Mathematics Poisson-Charlier polynomial

T(n,k) = 1/k!*sum {j = 0..n} C(n,j)*(k+j)!. T(n,k) = (n+k)!/k!*Num_Pade(n,k), where Num_Pade(n,k) denotes the numerator of the Pade approximation for the function exp(x) of degree (n,k) evaluated at x = 1. Recurrence relations: T(n,k) = T(n-1,k) + (k+1)*T(n-1,k+1); T(n,k) = (n+k)*T(n-1,k) + T(n-1,k-1). E.g.f for column k: exp(y)/(1-y)^(k+1). E.g.f. for array: exp(y)/(1-x-y) = (1 + x + x^2 + ...) + (2 + 3*x + 4*x^2 + ...)*y + (5 + 11*x + 19*x^2 + ...)*y^2/2! + ... . Row n lists the values of the Poisson-Charlier polynomial x^(n) + C(n,1)*x^(n-1) + C(n,2)*x^(n-2) + ... + C(n,n) for x = 1,2,3,..., where x^(m) denotes the rising factorial x*(x+1)*...*(x+m-1). Main diagonal is A001517. Series formulas for 1/e: Row n: 1/e = n!*[1/T(n,0) - 1/(1!*T(n,0)*T(n,1)) + 1/(2!*T(n,1)*T(n,2)) - 1/(3!*T(n,2)*T(n,3)) + ...]. Column k: k!/e = A000166(k) + (-1)^(k+1)*[0!/(T(0,k)*T(1,k)) + 1!/(T(1,k)*T(2,k)) + 2!/(T(2,k)*T(3,k)) + ...]. Main diagonal: 1/e = 1 - 2* sum {n = 0..inf} (-1)^n/(T(n,n)*T(n+1,n+1)) = 1 - 2*[1/(1*3) - 1/(3*19) + 1/(19*193) - ...]. Second subdiagonal: 1/e = 2*[1^2/(1*5) - 2^2/(5*49) + 3^2/(49*685) - ...]. Compare with A143413.

The Euler-Seidel matrix for the sequence {k!} begins

==============================================

n\k|.....0.....1.....2.....3.....4.....5.....6

==============================================

0..|.....1.....1.....2.....6....24...120...720

1..|.....2.....3.....8....30...144...840

2..|.....5....11....38...174...984

3..|....16....49...212..1158

4..|....65...261..1370

5..|...326..1631

6..|..1957

...

Dividing the k_th column by k! gives

==============================================

n\k|.....0.....1.....2.....3.....4.....5.....6

==============================================

0..|.....1.....1.....1.....1.....1.....1.....1

1..|.....2.....3.....4.....5.....6.....7

2..|.....5....11....19....29....41

3..|....16....49...106...193

4..|....65...261...685

5..|...326..1631

6..|..1957

...

Examples of series formula for 1/e:

Row 2: 1/e = 2*[1/5 - 1/(1!*5*11) + 1/(2!*11*19) - 1/(3!*19*29) + ...].

Column 4: 24/e = 9 - [0!/(1*6) + 1!/(6*41) + 2!/(41*316) + ...].

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

Cf. A008288, A076571, A086764, A108625, A143007, A143410, A143411, A143413, A001517 (main diagonal), A028387 (row 2), A000522 (column 0), A001339 (column 1), A082030 (column 2), A095000 (column 3), A095177 (column 4).

Sequence in context: A048471 A067345 A160185 this_sequence A067418 A067323 A106534

Adjacent sequences: A143406 A143407 A143408 this_sequence A143410 A143411 A143412

easy,nonn,tabl

Peter Bala (pbala(AT)toucansurf.com), Aug 14 2008