Search: id:A001515
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%I A001515 M1803 N0713
%S A001515 1,2,7,37,266,2431,27007,353522,5329837,90960751,1733584106,
%T A001515 36496226977,841146804577,21065166341402,569600638022431,
%U A001515 16539483668991901,513293594376771362,16955228098102446847
%N A001515 Bessel polynomial y_n(x) evaluated at x=1.
%C A001515 For some applications it is better to start this sequence with an extra 
               1 at the beginning: 1, 1, 2, 2, 37, 266, 2431, 27007, 353522, 5329837, 
               ... (again with offset 0). This sequence now has its own entry - 
               see A144301.
%C A001515 Number of partitions of {1,..,k}, n<=k<=2n, into n blocks with no more 
               than 2 elements per block. Restated, number of ways to use the elements 
               of {1,..,k}, n<=k<=2n, once each to form a collection of n sets, 
               each having 1 or 2 elements. - Bob Proctor, Apr 18 2005, Jun 26 2006. 
               E.g. for n=2 we get: (k=2): {1,2}; (k=3): {1,23}, {2,13}, {3,12}; 
               (k=4): {12,34}, {13,24}, {14,23}, for a total of a(2) = 7 partitions.
%C A001515 Equivalently, number of sequences of n unlabeled items such that each 
               item occurs just once or twice (cf. A105749). - David Applegate, 
               Dec 08 2008
%C A001515 Numerator of (n+1)-th convergent to 1+tanh(1). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Dec 20 2002
%C A001515 The following Maple lines show how this sequence and A144505, A144498, 
               A001514, A144513, A144506, A144514, A144507, A144301 are related.
%C A001515 f0:=proc(n) local k; add((n+k)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f0(n),
               n=0..10)];
%C A001515 # that is A001515
%C A001515 f1:=proc(n) local k; add((n+k+1)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f1(n),
               n=0..10)];
%C A001515 # that is A144498
%C A001515 f2:=proc(n) local k; add((n+k+2)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f2(n),
               n=0..10)];
%C A001515 # that is A144513; divided by 2 gives A001514
%C A001515 f3:=proc(n) local k; add((n+k+3)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f3(n),
               n=0..10)];
%C A001515 # that is A144514; divided by 6 gives A144506
%C A001515 f4:=proc(n) local k; add((n+k+4)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f4(n),
               n=0..10)];
%C A001515 # that divided by 24 gives A144507
%C A001515 a(n) is also the numerator of the continued fraction sequence beginning 
               with 2 followed by 3 and the remining odd numbers: [2,3,5,7,9,11,
               13,...]. [From Gil Broussard (gilbroussard(AT)bellsouth.net), Oct 
               07 2009]
%D A001515 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001515 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001515 Frink, O. and H. L. Krall, A new class of orthogonal polynomials, Trans. 
               Amer. Math. Soc. 65,100-115, 1945 [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), 
               Feb 15 2009]
%D A001515 E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.
%D A001515 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A001515 T. D. Noe, <a href="b001515.txt">Table of n, a(n) for n=0..100</a>
%H A001515 David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">
               The Gift Exchange Problem</a> (arXiv:0907.0513, 2009)
%H A001515 P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, 
               <a href="http://arXiv.org/abs/quant-ph/0501155">Boson normal ordering 
               via substitutions and Sheffer-type polynomials</a>
%H A001515 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               On generalizations of Stirling number triangles</a>, J. Integer Seqs., 
               Vol. 3 (2000), #00.2.4.
%H A001515 <a href="Sindx_Be.html#Bessel">Index entries for sequences related to 
               Bessel functions or polynomials</a>
%H A001515 <a href="Sindx_Par.html#partN">Index entries for related partition-counting 
               sequences</a>
%F A001515 The following formulae can all be found in (or are easily derived from 
               formulae in ) Grosswald's book.
%F A001515 a(0) = 1, a(1) = 2; thereafter a(n) = (2*n-1)*a(n-1) + a(n-2).
%F A001515 E.g.f. exp(1-sqrt(1-2*x))/sqrt(1-2*x).
%F A001515 a(n) = Sum_{ k = 0..n } binomial(n+k,2*k)*(2*k)!/(k!*2^k).
%F A001515 Equivalently, a(n) = Sum_{ k = 0..n } (n+k)!/((n-k)!*k!*2^k) = Sum_{ 
               k = n..2n } k!/( (2n-k)!*((k-n)!*2^(k-n)).
%F A001515 Equivalently, as the value of a hypergeometric function: a(n) = 2F0[ 
               n+1, -n ; - ; -1/2].
%F A001515 a(n) ~ exp(1)*(2n)!/(n!*2^n) as n -> oo. [See Grosswald, p. 124]
%F A001515 a(n) = A144301(n+1).
%F A001515 G.f.: 1/(1-x-x/(1-x-2x/(1-x-3x/(1-x-4x/(1-x-5x/(1-.... (continued fraction). 
               [From Paul Barry (pbarry(AT)wit.ie), Feb 08 2009]
%e A001515 The first few Bessel polynomials are (cf. A001497, A001498):
%e A001515 y_0 = 1
%e A001515 y_1 = 1+x
%e A001515 y_2 = 1+3x+3x^2
%e A001515 y_3 = 1+6x+15x^2+15x^3, etc.
%p A001515 A001515 := proc(n) option remember; if n=0 then 1 elif n=1 then 2 else 
               (2*n-1)*A001515(n-1)+A001515(n-2); fi; end;
%p A001515 A001515:=proc(n) local k; add( (n+k)!/((n-k)!*k!*2^k),k=0..n); end;
%p A001515 A001515:= n-> hypergeom( [n+1,-n],[],-1/2);
%p A001515 bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); 
               end;
%Y A001515 See A144301 for other formulae and comments.
%Y A001515 Row sums of Bessel triangle A001497 as well as of A001498.
%Y A001515 Cf. A000806, A001514, A001517, A144505.
%Y A001515 a(n) = A105749(n)/n!. See also A143990.
%Y A001515 Partial sums: A105748. First differences: A144498.
%Y A001515 Replace "sets" by "lists" in comment: A001517.
%Y A001515 Sequence in context: A072597 A125515 A135920 this_sequence A144301 A083659 
               A036247
%Y A001515 Adjacent sequences: A001512 A001513 A001514 this_sequence A001516 A001517 
               A001518
%K A001515 nonn,easy,nice
%O A001515 0,2
%A A001515 N. J. A. Sloane (njas(AT)research.att.com).
%E A001515 Extensively edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 
               07 2008

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