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A143410 Form the difference table of the sequence {2^k*k!}, then divide k-th column entries by 2^k*k!. Array read by ascending antidiagonals, T(n, k) for n, k >= 0. 4
1, 1, 1, 5, 3, 1, 29, 17, 5, 1, 233, 131, 37, 7, 1, 2329, 1281, 353, 65, 9, 1, 27949, 15139, 4105, 743, 101, 11, 1, 391285, 209617, 56189, 10049, 1349, 145, 13, 1, 6260561, 3325923, 883885, 156679, 20841, 2219, 197, 15, 1, 112690097, 59475329, 15700313 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..47.

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..inf}((-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

Cf. A008288, A076571, A086764, A108625, A143007, A143409, A143411, A143412.

Sequence in context: A157891 A173644 A115991 * A114344 A317674 A201333

Adjacent sequences:  A143407 A143408 A143409 * A143411 A143412 A143413

KEYWORD

easy,nonn,tabl

AUTHOR

Peter Bala, Aug 19 2008

STATUS

approved

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Last modified January 21 11:01 EST 2020. Contains 331105 sequences. (Running on oeis4.)