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A052649
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Expansion of e.g.f. (2+x-x^2)/(1-x)^2.
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6
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2, 5, 14, 54, 264, 1560, 10800, 85680, 766080, 7620480, 83462400, 997920000, 12933043200, 180583603200, 2702527027200, 43153254144000, 732297646080000, 13160434839552000, 249692574523392000, 4987449116762112000, 104614786351595520000, 2299092397726924800000
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OFFSET
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0,1
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COMMENTS
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a(1) is 5 and gives the row number in the table of 0-origin permutations of order 3 in which the first 3 items are reversed. Row 5 of this table is 2 1 0. a(2) is 14 and gives the row number in the table of 0-origin permutations of order 4 in which the first three items are reversed. Row 14 of this table is 2 1 0 3.... a(6) is 10800 and gives the row number in the table of 0-origin permutations of order 8 in which the first 3 items are reversed. Row 10800 of this table is 2 1 0 3 4 5 6 7. Et cetera. - Eugene McDonnell (eemcd(AT)mac.com), Dec 03 2004
In factorial base representation (A007623) the terms of this sequence are written as: 10, 21, 210, 2100, 21000, 210000, ... From a(1) = 5 = "21" onward each term begins always with "21", which is then followed by n-1 zeros. - Antti Karttunen, Sep 24 2016
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LINKS
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FORMULA
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a(n) = (3+2*n)*n!.
E.g.f.: -(-x+x^2-2)/(-1+x)^2.
Recurrence: a(0)=2, a(1)=5, (-7*n-5-2*n^2)*a(n)+(3+2*n)*a(n+1)=0 for n>=1.
Sum_{n>=0} 1/a(n) = 1/6 + e/2 - erfi(1)*sqrt(Pi)/4, where erfi is the imaginary error function.
Sum_{n>=0} (-1)^n/a(n) = 1/6 - 1/(2*e) + erf(1)*sqrt(Pi)/4, where erf is the error function. (End)
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MAPLE
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spec := [S, {S=Prod(Sequence(Z), Union(Z, Sequence(Z), Sequence(Z)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
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f[n_] := (3 + 2 n) n!; f[0] = 2; Array[f, 19, 0]
a[n_] := a[n] = a[n - 1]*n (2 n + 3)/(2 n + 1); a[0] = 2; a[1] = 5; Array[ a, 19, 0] (* Robert G. Wilson v *)
With[{nn=20}, CoefficientList[Series[(2+x-x^2)/(1-x)^2, {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Nov 09 2017 *)
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PROG
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(PARI) a(n)=if(n<=1, [2, 5][n+1], a(n-1)*(n*(2*n+3))/(2*n+1) );
for(n=0, 11, print1(a(n), ", "))
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CROSSREFS
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Row 4 of A276955 (from a(1)=5 onward).
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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STATUS
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approved
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