0,2

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)).

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} 2^j*C(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.

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

6..|..

...

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

6..|..

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;

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

Sequence in context: A016479 A134768 A113139 this_sequence A096773 A118384 A133176

Adjacent sequences: A143408 A143409 A143410 this_sequence A143412 A143413 A143414

easy,nonn,tabl

Peter Bala (pbala(AT)toucansurf.com), Aug 19 2008