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A110505
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Numerators of unsigned columns of triangle A110504: a(n) = n!*A110504(n,0) = (-1)^k*n!*A110504(n+k,k) for all k>=0.
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10
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0, 1, 3, 7, 30, 144, 876, 6084, 48816, 438624, 4389120, 48263040, 579242880, 7529552640, 105417365760, 1581231456000, 25299906508800, 430096581734400, 7741753102540800, 147093162635059200, 2941864569520128000
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OFFSET
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0,3
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COMMENTS
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Triangle A110504 equals the matrix logarithm of triangle A110503. Triangle A110503 shifts one column left under matrix inverse. Limit a(n)/n! = Pi*2*sqrt(3)/9 = 1.209199576...
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LINKS
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Table of n, a(n) for n=0..20.
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FORMULA
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E.g.f.: (2+x-x^2)/2/(1-x)*acos(1-1/2*x^2)/sqrt(1-1/4*x^2). E.g.f. A(x) satisfies: A(x)*A(-x) = -acos(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 -... Also, 1/(1-x) = Sum_{n>=1} A(x)^[(n+1)/2] * A(-x)^[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 -+...).
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PROG
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(PARI) {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]} (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)))))}
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CROSSREFS
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Cf. A110503 (triangular matrix), A110504 (matrix logarithm), A002544.
Sequence in context: A182322 A132105 A019021 * A009128 A053648 A138862
Adjacent sequences: A110502 A110503 A110504 * A110506 A110507 A110508
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KEYWORD
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frac,nonn
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AUTHOR
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Paul D. Hanna, Jul 23 2005
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STATUS
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approved
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