OFFSET
0,4
COMMENTS
This table is closely connected to the constant sqrt(e). The row, column and diagonal entries of this table occur in series acceleration formulas for sqrt(e). For a similar table based on the Euler-Seidel matrix of the sequence {2^k*k!} and related to the constant 1/sqrt(e), see A143411. 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)).
LINKS
Eric Weisstein's World of Mathematics, Poisson-Charlier polynomial.
FORMULA
T(n,k) = ((-1)^n/k!)*Sum {j = 0..n} (-2)^j*C(n,j)*(k+j)!.
Relation with Poisson-Charlier polynomials c_n(x,a): T(n,k) = 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 + ...) + (1 + 3*x + 5*x^2 + ...)*y + (5 + 17*x + 37*x^2 + ...)*y^2/2! + ... .
Series acceleration formulas for sqrt(e):
Row n: sqrt(e) = 2^n*n!*(1/T(n,0) + (-1)^n*(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 sqrt(e) = 48*(1/29 - 1/(2*29*131) - 1/(8*131*353) - 1/(48*353*743) - ...).
Column k: sqrt(e) = (1 + (1/2)/1! + (1/2)^2 / 2! + ... + (1/2)^k/k!) + 1/(2^k*k!) * Sum_{n>= 0} ((-2)^n *n!/(T(n,k)*T(n+1,k))). For example, column 3 gives sqrt(e) = 79/48 + (1/48)*(1/(1*7) - 2/(7*65) + 8/(65*743) - 48/(743*10049) + ...).
Main diagonal: sqrt(e) = 1 + 2*(1/(1*3) - 1/(3*37) + 1/(37*743) - ...). See A143412.
T(n, k) = (-1)^n*(-1/2)^(k + 1)*KummerU(k + 1, k + n + 2, -1/2). - Peter Luschny, Jan 02 2020
EXAMPLE
Table of differences of {2^k*k!}
=====================================================
Column 0 1 2 3 4 5
=====================================================
Sequence 2^k*k! 1 2 8 48 384 3840
First differences 1 6 40 336 3456
Second differences 5 34 296 3120
Third differences 29 262 2824
Fourth differences 233 2562
...
Remove the common factor 2^k*k! from k-th column entries:
====================================
n\k| 0 1 2 3 4
====================================
0 | 1 1 1 1 1
1 | 1 3 5 7 9
2 | 5 17 37 65 101
3 | 29 131 353 743 1349
4 | 233 1281 4105 10049 20841
MAPLE
T := (n, k) -> (-1)^n/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;
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Aug 19 2008
STATUS
approved