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A143410
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Form the difference table of the sequence {2^k*k!}, then divide k-th column entries by 2^k*k!.
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4
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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; internal format)
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OFFSET
| 0,4
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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)).
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LINKS
| Eric Weisstein's World of Mathematics Poisson-Charlier polynomial
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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.
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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:
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n\k|...0......1......2......3......4
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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
...
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MAPLE
| with combinat: 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;
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CROSSREFS
| Cf. A008288, A076571, A086764, A108625, A143007, A143409, A143411.
Sequence in context: A157891 A173644 A115991 * A114344 A201333 A088324
Adjacent sequences: A143407 A143408 A143409 * A143411 A143412 A143413
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Peter Bala (pbala(AT)toucansurf.com), Aug 19 2008
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