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Search: id:A000995
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| A000995 |
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Shifts left two terms under the binomial transform.
(Formerly M1228 N0471)
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+0
8
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| 0, 1, 0, 1, 2, 4, 10, 29, 90, 295, 1030, 3838, 15168, 63117, 275252, 1254801, 5968046, 29551768, 152005634, 810518729, 4472244574, 25497104007, 149993156234, 909326652914, 5674422994544, 36408092349897, 239942657880360
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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The binomial transform of A000995 has g.f. x*c(x)^2/(1+x^2*c(x)^2). - Paul Barry (pbarry(AT)wit.ie), Oct 06 2007
Equals row sums of triangle A137854 such that A000995(3) = 1 = first row of triangle A137854. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 15 2008
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REFERENCES
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S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
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FORMULA
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Since this satisfies a recurrence similar to that of the Bell numbers (A000110), the asymptotic behavior is presumably just as complicated - see A000110 for details.
However, A000994(n)/A000995(n) [ e.g. 77464/63117 ] -> 1.228..., the constant in A051148 and A051149.
O.g.f.: A(x) = Sum_{n>=0} x^(2*n+1)/Product_{k=0..n} (1-k*x)^2 . - Paul D. Hanna (pauldhanna(AT)juno.com), Oct 28 2006
G.f.: (1+2x^2*c(x)^2)/(1+x^2*c(x^2)), c(x) the g.f. of A000108; - Paul Barry (pbarry(AT)wit.ie), Oct 06 2007
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EXAMPLE
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A(x) = x + x^3/(1-x)^2 + x^5/((1-x)*(1-2x))^2 + x^7/((1-x)*(1-2x)*(1-3x))^2 +...
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MAPLE
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A000995 := proc(n) local k; option remember; if n <= 1 then n else n + add(binomial(n, k)*A000995(k - 2), k = 2 .. n); fi; end;
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PROGRAM
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(PARI) a(n)=polcoeff(sum(k=0, n, x^(2*k+1)/prod(j=0, k, 1-j*x+x*O(x^n))^2), n) - Paul D. Hanna (pauldhanna(AT)juno.com), Oct 28 2006
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CROSSREFS
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Cf. A000994, A051139, A051140.
Cf. A137854.
Sequence in context: A060315 A005505 A135334 this_sequence A010359 A086631 A047051
Adjacent sequences: A000992 A000993 A000994 this_sequence A000996 A000997 A000998
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KEYWORD
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nonn,eigen,easy,nice
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AUTHOR
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njas
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EXTENSIONS
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More terms from Paul D. Hanna (pauldhanna(AT)juno.com), Oct 28 2006
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