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A155585 a(n) = Sum_{k=0..n-1} (-1)^(k)*C(n-1,k)*a(n-1-k)*a(k) for n>0 with a(0)=1. 19
1, 1, 0, -2, 0, 16, 0, -272, 0, 7936, 0, -353792, 0, 22368256, 0, -1903757312, 0, 209865342976, 0, -29088885112832, 0, 4951498053124096, 0, -1015423886506852352, 0, 246921480190207983616, 0, -70251601603943959887872, 0, 23119184187809597841473536, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Factorials have a similar recurrence: f(n) = Sum_{k=0..n-1} C(n-1,k)*f(n-1-k)*f(k), n>0.

Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times sum(k=0..n  binomial(n,k)*k^q ), then A155585(x)= sum(k=0..x-1 T(x,k)*(-1)^k). See Mathematica code below. - John M. Campbell, Nov 16 2011

For the difference table and the relation to the Seidel triangle see A239005. - Paul Curtz, Mar 06 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

H. M. Edwards, A normal form for elliptic curves, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 3, 393-422. see section 12, pp. 410-411.

FORMULA

E.g.f.: exp(x)*sech(x) = exp(x)/cosh(x). (See A009006.) - Paul Barry, Mar 15 2006

Sequence of absolute values is A009006 (e.g.f. 1+tan(x)).

O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 + 2*k*x). - Paul D. Hanna, Jul 20 2011

a(n) = 2^n*E_{n}(1) where E_{n}(x) are the Euler polynomials. - Peter Luschny, Jan 26 2009

a(n) = EL_{n}(-1) where EL_{n}(x) are the Eulerian polynomials. - Peter Luschny, Aug 03 2010

a(n) = (4^n-2^n)*B_n(1)/n, where B_{n}(x) are the Bernoulli polynomials (B_n(1) = B_n for n <> 1). - Peter Luschny, Apr 22 2009

G.f.: 1/(1-x+x^2/(1-x+4*x^2/(1-x+9*x^2/(1-x+16*x^2/(1-...))))) (continued fraction). - Paul Barry, Mar 30 2010

G.f.: -log(x/(exp(x)-1))/x = sum(n>=0, a(n)*x^n/(2^(n+1)*(2^(n+1)-1)*n!). - Vladimir Kruchinin, Nov 05 2011

E.g.f.: exp(x)/cosh(x) = 2/(1+exp(-2*x)) = 2/(G(0) + 1) ; G(k) = 1 - 2*x/(2*k + 1 - x*(2*k+1)/(x - (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 10 2011

E.g.f. is x(t,1) + y(t,1) where x(t,a) and y(t,a) satisfy y(t,a)^2 = (a^2 - x(t,a)^2) / (1 - a^2 * x(t,a)^2) and dx(t,a) / dt = y(t,a) * (1 - a * x(t,a)^2) and are the elliptic functions of Edwards. - Michael Somos, Jan 16 2012

E.g.f.: 1/(1 - x/(1+x/(1 - x/(3+x/(1 - x/(5+x/(1 - x/(7+x/(1 - x/(9+x/(1 +...))))))))))), a continued fraction. - Paul D. Hanna, Feb 11 2012

E.g.f. satisfies: A(x) = Sum_{n>=0} Integral( A(-x) dx )^n / n!. - Paul D. Hanna, Nov 25 2013

a(n) = -2^(n+1)*Li_{-n}(-1). - Peter Luschny, Jun 28 2012

a(n) = sum_{k=1..n} sum_{j=0..k} (-1)^(j+1)*binomial(n+1,k-j)*j^n for n>0. - Peter Luschny, Jul 23 2012

G.f.: 1 + x/T(0) where T(k)=  1 + (k+1)*(k+2)*x^2/T(k+1)); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 25 2012

E.g.f.: exp(x)/cosh(x)= 1 + x/S(0) where S(k)= (2*k+1) + x^2/S(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 25 2012

E.g.f.: 1+x/(U(0)+x) where U(k)=  4*k+1 - x/(1 + x/(4*k+3 - x/(1 + x/U(k+1))));(continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 08 2012

E.g.f.: 1+tanh(x)=4*x/(G(0)+2*x) where G(k)=1 - (k+1)/(1 - 2*x/(2*x + (k+1)^2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 07 2012

G.f.: 1 / (1 - b(1)*x / (1 - b(2)*x / (1 - b(3)*x / ... ))) where b = A001057. - Michael Somos, Jan 03 2013

G.f.: 1 + x/G(0) where G(k) = 1 + 2*x^2*(2*k+1)^2 - x^4*(2*k+1)*(2*k+2)^2*(2*k+3)/G(k+1); (continued fraction due to Stieltjes). - Sergei N. Gladkovskii, Jan 13 2013

E.g.f.: 1 + x/( G(0) + x ) where G(k) = 1 - 2*x/(1 + (k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 02 2013

G.f.: 2 - 1/Q(0) where Q(k) = 1 + x*(k+1)/( 1 - x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Apr 02 2013

G.f.: 2 - 1/Q(0), where Q(k)= 1 + x*k^2 + x/(1 - x*(k+1)^2/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 17 2013

G.f.: 1/Q(0), where Q(k)= 1 - 2*x + x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 20 2013

G.f.: 1/Q(0), where Q(k)= 1 - x*(k+1)/(1 + x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013

E.g.f.: 1 + x*Q(0), where Q(k) = 1 - x^2/( x^2 + (2*k+1)*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013

G.f.: 2 - T(0)/(1+x), where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 + (1+x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 11 2013

E.g.f.: 1/(x - Q(0)), where Q(k) = 4*k^2 - 1 + 2*x + x^2*(2*k-1)*(2*k+3)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 16 2013

From Paul Curtz, Mar 06 2014 : (Start)

a(2n) = A000007(n).

a(2n+1) = (-1)^n*A000182(n+1).

a(n) is the binomial transform of A122045(n).

a(n) is the row sum of A081658. For fractional Euler numbers see A238800.

a(n) + A122045(n) = 2, 1, -1, -2, 5, 16,... = -A163982(n).

a(n) - A122045(n) = -A163747(n).

a(n) is the Akiyama-Tanigawa transform applied to 1, 0, -1/2, -1/2, -1/4, 0,... = A046978(n+3)/A016116(n). (End)

EXAMPLE

E.g.f.: 1 + x - 2*x^3/3! + 16*x^5/5! - 272*x^7/7! + 7936*x^9/9! -+... = exp(x)/cosh(x).

O.g.f.: 1 + x - 2*x^3 + 16*x^5 - 272*x^7 + 7936*x^9 - 353792*x^11 +-...

O.g.f.: 1 + x/(1+2*x) + 2!*x^2/((1+2*x)*(1+4*x)) + 3!*x^3/((1+2*x)*(1+4*x)*(1+6*x)) +...

MAPLE

A155585 := n -> 2^n*euler(n, 1): # Peter Luschny, Jan 26 2009

MATHEMATICA

a[m_] := Sum[(-2)^(m - k) k! StirlingS2[m, k], {k, 0, m}] (* Peter Luschny, Apr 29 2009 *)

poly[q_] :=  2^(q-n)/n*FunctionExpand[Sum[Binomial[n, k]*k^q, {k, 0, n}]]; T[q_, r_] :=  First[Take[CoefficientList[poly[q], n], {r+1, r+1}]]; Table[Sum[T[x, k]*(-1)^k, {k, 0, x-1}], {x, 1, 16}] (* John M. Campbell, Nov 16 2011 *)

f[n_] := (-1)^n 2^(n+1) PolyLog[-n, -1]; f[0] = -f[0]; Array[f, 27, 0] (* Robert G. Wilson v, Jun 28 2012 *)

PROG

(PARI) {a(n)=if(n==0, 1, sum(k=0, n-1, (-1)^(k)*binomial(n-1, k)*a(n-1-k)*a(k)))}

(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(exp(X)/cosh(X), n)}

(PARI) {a(n)=polcoeff(sum(m=0, n, m!*x^m/prod(k=1, m, 1+2*k*x+x*O(x^n))), n)} \\ Paul D. Hanna, Jul 20 2011

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( 1 + sinh(x + A) / cosh(x + A), n))} /* Michael Somos, Jan 16 2012 */

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, intformal(subst(A, x, -x)+x*O(x^n))^k/k!)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Nov 25 2013

for(n=0, 30, print1(a(n), ", "))

(Sage)

def A155585(n) :

    if n == 0 : return 1

    return add(add((-1)^(j+1)*binomial(n+1, k-j)*j^n for j in (0..k)) for k in (1..n))

[A155585(n) for n in (0..26)] # Peter Luschny, Jul 23 2012

def A155585_list(n) :  Akiyama-Tanigawa algorithm

    A = [0]*(n+1); R = []

    for m in range(n+1) :

        d = divmod(m+3, 4)

        A[m] = 0 if d[1] == 0 else (-1)^d[0]/2^(m//2)

        for j in range(m, 0, -1) :

            A[j - 1] = j * (A[j - 1] - A[j])

        R.append(A[0])

    return R

A155585_list(30)  # Peter Luschny, Mar 09 2014

CROSSREFS

Cf. A001057, A009006, A102573.

Equals row sums of A119879. - Johannes W. Meijer, Apr 20 2011

Sequence in context: A146558 A025600 A009006 * A236755 A057375 A009045

Adjacent sequences:  A155582 A155583 A155584 * A155586 A155587 A155588

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jan 24 2009

STATUS

approved

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Last modified November 28 02:24 EST 2014. Contains 250286 sequences.