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A112934
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a(0) = 1; a(n+1) = Sum_{k, 0<=k<=n} a(k)*A001147(n-k), where A001147 = double factorial numbers.
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14
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1, 1, 2, 6, 26, 158, 1282, 13158, 163354, 2374078, 39456386, 737125446, 15279024026, 347786765150, 8621313613954, 231139787526822, 6663177374810266, 205503866668090750, 6751565903597571842
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Contribution from Paul Barry, Dec 04 2009: (Start)
The g.f. of a(n+1) is 1/(1-2x/(1-x/(1-4x/(1-3x/(1-6x/(1-5x/(1-.... (continued fraction).
The Hankel transform of a(n+1) is A137592. (End)
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FORMULA
| INVERT transform of double factorials (A001147), shifted right one place, where g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^2]/A(x)^2.
G.f. satisfies: A(x) = 1+x + 2*x^2*[d/dx A(x)]/A(x) (log derivative). G.f.: A(x) = 1+x +2*x^2/(1-3*x -2*2*1*x^2/(1-7*x -2*3*3*x^2/(1-11*x -2*4*5*x^2/(1-15*x -... -2*n*(2*n-3)*x^2/(1-(4*n-1)*x -...)))) (continued fraction). G.f.: A(x) = 1/(1-x/(1 -1*x/(1-2*x/(1 -3*x/(1-4*x(1 -...))))))) (continued fraction).
a(n) = Sum_{k,0<=k<=n}A111106(n,k) . - Philippe DELEHAM, Jun 20 2006
a(n) = upper left term in M^n, M = the production matrix:
1, 1
1, 1, 2
1, 1, 2, 3
1, 1, 2, 3, 4
1, 1, 2, 3, 4, 5
...
- Gary W. Adamson, Jul 08 2011
a(n) = (2*n - 1) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
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EXAMPLE
| A(x) = 1 + x + 2*x^2 + 6*x^3 + 26*x^4 + 158*x^5 + 1282*x^6 +...
1/A(x) = 1 - x - x^2 - 3*x^3 - 15*x^4 - 105*x^5 -... -A001147(n)*x^(n+1)-...
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MATHEMATICA
| a[0] = 1; a[n_] := a[n] = Sum[a[k]*(2n - 2k - 3)!!, {k, 0, n - 1}]; Table[ a[n], {n, 0, 19}] (from Robert G. Wilson v, Oct 12 2005)
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PROG
| (PARI) {a(n)=local(F=1+x+x*O(x^n)); for(i=1, n, F=1+x+2*x^2*deriv(F)/F); return(polcoeff(F, n, x))}
(PARI) {a(n) = local(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2*k - 1) * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */
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CROSSREFS
| Cf. A001147, A112935 (log derivative); A112936, A112937, A112938, A112939, A112940, A112941, A112942, A112943.
Sequence in context: A123306 A099758 A099760 * A135922 A103367 A047863
Adjacent sequences: A112931 A112932 A112933 * A112935 A112936 A112937
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KEYWORD
| nonn
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr) and Paul D. Hanna (pauldhanna(AT)juno.com), Oct 09 2005
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