A110505 Numerators of unsigned columns of triangle A110504: a(n) = n!*A110504(n,0) = (-1)^k*n!*A110504(n+k,k) for all k >= 0.

0, 1, 3, 7, 30, 144, 876, 6084, 48816, 438624, 4389120, 48263040, 579242880, 7529552640, 105417365760, 1581231456000, 25299906508800, 430096581734400, 7741753102540800, 147093162635059200, 2941864569520128000

Offset

0,3

Comments

Triangle A110504 equals the matrix logarithm of triangle A110503.

Triangle A110503 shifts one column left under matrix inverse.

Lim_{n->infinity} a(n)/n! = Pi*2*sqrt(3)/9 = 1.209199576...

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Formula

E.g.f.: (2+x-x^2)/(2*(1-x)) * arccos(1-x^2/2) / sqrt(1-x^2/4).

E.g.f. A(x) satisfies:

(1) A(x)*A(-x) = -arccos(1-1/2*x^2)^2 = Sum_{n>=0} -x^(2*n+2)/( C(2*n+1, n)*(n+1)^2 ).

(2) 1/(1-x) = Sum_{n>=1} A(x)^floor((n+1)/2) * A(-x)^floor(n/2)/n!.

a(2*n+1) = (2*n+1)!*(1 + Sum_{k=1..n} (1/binomial(2*k+1, k))/(k+1)).

a(2*n+2) = (2*n+2)!*(1 + 1/2 - Sum_{k=1..n} 1/binomial(2*k+2, k)/k) = n!*(1 + 1/2 - 1/3 + 1/12 - 1/20 + 1/60 - 1/105 + 1/280 -+ ...).

Recurrence: 4*a(n) = 2*(2*n-1)*a(n-1) + (n-2)*(n+1)*a(n-2) - (n-3)*(n-2)*n*a(n-3). - Vaclav Kotesovec, May 09 2014

Example

E.g.f.: A(x) = x + 3*x^2/2! + 7*x^3/3! + 30*x^4/4! + 144*x^5/5! + 876*x^6/6! + ...
where A(x) satisfies: A(x)*A(-x) = -arccos(1-1/2*x^2)^2, and
arccos(1-1/2*x^2)^2 = Sum_{n>=0} x^(2*n+2)/( C(2*n+1, n)*(n+1)^2 ) = x^2 + 1/12*x^4 + 1/90*x^6 + 1/560*x^8 + 1/3150*x^10 + ...

Prog

(PARI) /* From relation to unsigned columns of triangle A110504: */
{a(n)=local(M=matrix(n+1, n+1, r, c, if(r>=c, if(r==c || c%2==1, 1, if(r%2==0 && r==c+2, -2, -1))))); n!*sum(i=1, #M, -(M^0-M)^i/i)[n+1, 1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* As Partial Sums of Series: */
a(n)=if(n<1, 0, n!*(1+sum(n=2, n, (-1)^n/(binomial(n-2, n\2-1)*n*(n-1)/((n+1)\2)))))
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A110503 (triangular matrix), A110504 (matrix logarithm), A002544.

Sequence in context: A182322 A132105 A019021 * A354518 A009128 A053648

Adjacent sequences: A110502 A110503 A110504 * A110506 A110507 A110508

Keyword

frac,nonn

Author

Paul D. Hanna, Jul 23 2005

Status

approved