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A028421 Array of numbers f(n,k) from n-th differences of sequence {1/x^2}; n-th difference is n!*P(x)/(D^2) where P(x) is a degree-n polynomial: P(n) = Sum_k { f(n,k)*x^k } and D = x(x+1) ...(x+n-1)(x+n). 30
1, 1, 2, 2, 6, 3, 6, 22, 18, 4, 24, 100, 105, 40, 5, 120, 548, 675, 340, 75, 6, 720, 3528, 4872, 2940, 875, 126, 7, 5040, 26136, 39396, 27076, 9800, 1932, 196, 8, 40320, 219168, 354372, 269136, 112245, 27216, 3822, 288, 9, 362880, 2053152 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

From Johannes W. Meijer, Oct 07 2009: (Start)

The higher order exponential integrals E(x,m,n) are defined in A163931 and the general formula of the asymptotic expansion of E(x,m,n) can be found in A163932.

We used the general formula and the asymptotic expansion of E(x,m=1,n), see A130534, to determine that E(x,m=2,n) ~ (exp(-x)/x^2)*(1 - (1+2*n)/x + (2+6*n+3*n^2)/x^2 - (6 + 22*n + 18*n^2+ 4*n^3)/x^3 + ... ) which can be verified with the EA(x,2,n) formula, see A163932. The coefficients in the denominators of this expansion lead to the sequence given above.

The asymptotic expansion of E(x,m=2,n) leads for n from one to ten to known sequences, see the cross-references. With these sequences one can form the triangles A165674 (left hand columns) and A093905 (right hand columns).

(End)

For connections to an operator relation between log(x) and x^n(d/dx)^n, see A238363. - Tom Copeland, Feb 28 2014

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

f(n, k) = (k+1)*s[ n+1, k+1 ] (for n >= k) where s[ n, k ] is an unsigned stirling number of the first kind.

E.g.f.: -log(1-x)/(1-x)^y. - Vladeta Jovovic, Oct 12 2003

The e.g.f. y = x + (1 + 2*t)*x^2/2! + (2 + 6*t + 3*t^2)*x^3/3! + ... has series reversion with respect to x equal to y - (1 + 2*t)*y^2/2! + (1 + 3*t)^2*y^3/3! - (1 + 4*t)^3*y^4/4! + .... This is an e.g.f. for a signed version of A139526. - Peter Bala, Jul 18 2013

MAPLE

with(combinat): A028421 := proc(n, k): (-1)^(n+k) * binomial(k+1, 1)* stirling1(n+1, k+1) end: seq(seq(A028421(n, k), k=0..n), n=0..8); # (* Johannes W. Meijer, Oct 07 2009, Revised Sep 09 2012 *)

MATHEMATICA

f[n_, k_] = (k + 1) StirlingS1[n + 1, k + 1] // Abs; Flatten[Table[f[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 47]] (* Jean-Fran├žois Alcover, Jun 01 2011, after formula *)

CROSSREFS

Row sums give A000254.

From Johannes W. Meijer, Oct 07 2009: (Start)

A000142, A052517, 3*A000399, 5*A000482 are the first four left hand columns; A000027, A002411 are the first two right hand columns.

The asymptotic expansion of E(x,m=2,n) leads to A000254 (n=1), A001705 (n=2), A001711 (n=3), A001716 (n=4), A001721 (n=5), A051524 (n=6), A051545 (n=7), A051560 (n=8), A051562 (n=9), A051564 (n=10), A093905 (triangle) and A165674 (triangle).

Cf. A163931 (E(x,m,n)), A130534 (m=1), A163932 (m=3), A163934 (m=4), A074246 (E(x,m=2,n+1)).

(End)

Cf. A061356, A139526.

Sequence in context: A227608 A276484 A100641 * A263003 A081745 A240578

Adjacent sequences:  A028418 A028419 A028420 * A028422 A028423 A028424

KEYWORD

tabl,nonn

AUTHOR

Peter Wiggen (wiggen(AT)math.psu.edu)

STATUS

approved

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Last modified February 19 10:20 EST 2018. Contains 299330 sequences. (Running on oeis4.)