A111537 Column 1 of triangle A111536.

1, 2, 8, 44, 296, 2312, 20384, 199376, 2138336, 24936416, 314142848, 4252773824, 61594847360, 950757812864, 15586971531776, 270569513970944, 4959071121374720, 95721139472072192, 1941212789888952320, 41271304403571227648, 918030912312297752576, 21325054720042613565440

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_{k=0..n-1} (n-k+1)!*a(k+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/(1-2*x/(1-2*x/(1-3*x/(1-3*x/(1-4*x/(1-4*x/(1-5*x/(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

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} a(k) * a(n-k-1). - Ilya Gutkovskiy, Jul 05 2020

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: A240165 A357832 A318977 * A051045 A112912 A303613

Adjacent sequences: A111534 A111535 A111536 * A111538 A111539 A111540

Keyword

nonn

Author

Paul D. Hanna, Aug 06 2005

Status

approved