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A111537 Column 1 of triangle A111536. 6
1, 2, 8, 44, 296, 2312, 20384, 199376, 2138336, 24936416, 314142848, 4252773824, 61594847360, 950757812864, 15586971531776, 270569513970944, 4959071121374720, 95721139472072192, 1941212789888952320, 41271304403571227648, 918030912312297752576, 21325054720042613565440 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums of triangle in A200659. - Philippe Deléham, Nov 21 2011.

REFERENCES

A. N. Khovanskii. The Application of Continued Fractions and Their Generalizations to Problem in Approximation Theory. Groningen: Noordhoff, Netherlands, 1963. See p.141 (10.19)

H. P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..250

Herman P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.

FORMULA

a(n) = A111536(n+1, 1) = 2*A111536(n, 0) = 2*A111529(n) for n>=1.

G.f.: log(sum_{n>=0} (n+1)!*x^n) = sum_{n>=1} a(n)*x^n/n.

a(n+1) = (n+3)!-2*(n+2)!-sum((n-k+1)!*a(k+1),k=0..n-1)

a(n+1) is the moment of order n for the measure of density x*exp(-x)/((x*exp(-x)*Ei(x)-1)^2+(Pi*x*exp(-x))^2) on the interval 0..infinity

G.f.: 1/(1-2x/(1-2x/(1-3x/(1-3x/(1-4x/(1-4x/(1-5x/(1-...(continued fraction). - Philippe Deléham, Nov 21 2011.

G.f. (1-U(0))/x; where U(k) = 1-x*(k+1)/(1-x*(k+2)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Jun 29 2012

G.f. -1 + 1/x + U(0)/x where U(k) = 2*x - 1 + 2*x*k - x^2*(k+1)*(k+2)/U(k+1), U(0)=x - W(1,1;-x)/W(1,2;-x), W(a,b,x)= 1 - a*b*x/1! + a*(a+1)*b*(b+1)*x^2/2! -...+ a*(a+1)*...*(a+n-1)*b*(b+1)*...*(b+n-1)*x^n/n! +...; see [A.N.Khovanskii, p.141 (10.19)]; (continued fraction, 1-step). - Sergei N. Gladkovskii, Aug 15 2012

G.f.: 1/Q(0), where Q(k)= 1 + k*x - x*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013

G.f.: 1/x - 1/( x*G(0)), where G(k)= 1 - x*(k+1)/(x -  1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 03 2013

MAPLE

a:= proc(n) option remember; `if`(n=0, 1,

      n*(n+1)! -add((n-k+1)!*a(k), k=1..n-1))

    end:

seq(a(n), n=0..30);  # Alois P. Heinz, May 06 2013

MATHEMATICA

a[n_] := a[n] = If[n==0, 1, n*(n+1)! - Sum[(n-k+1)!*a[k], {k, 1, n-1}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 13 2017, after Alois P. Heinz *)

PROG

(PARI) {a(n)=if(n<0, 0, (matrix(n+2, n+2, m, j, if(m==j, 1, if(m==j+1, -m+1, -(m-j-1)*polcoeff(log(sum(i=0, m, (i+1)!/1!*x^i)), m-j-1))))^-1)[n+2, 2])}

CROSSREFS

Cf. A111536, A111529, A200659.

Sequence in context: A179489 A240165 A318977 * A051045 A112912 A303613

Adjacent sequences:  A111534 A111535 A111536 * A111538 A111539 A111540

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 06 2005

STATUS

approved

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Last modified November 12 04:21 EST 2019. Contains 329051 sequences. (Running on oeis4.)