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A001515 Bessel polynomial y_n(x) evaluated at x=1.

%I A001515 M1803 N0713 #155 Feb 25 2025 10:35:53
%S A001515 1,2,7,37,266,2431,27007,353522,5329837,90960751,1733584106,
%T A001515 36496226977,841146804577,21065166341402,569600638022431,
%U A001515 16539483668991901,513293594376771362,16955228098102446847,593946277027962411007,21992967478132711654106,858319677924203716921141
%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, 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_, 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 this sequence
%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 remaining odd numbers: [2,3,5,7,9,11,13,...]. - _Gil Broussard_, Oct 07 2009
%C A001515 Also, number of scenarios in the Gift Exchange Game when a gift can be stolen at most once. - _N. J. A. Sloane_, Jan 25 2017
%D A001515 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%D A001515 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001515 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001515 Seiichi Manyama, <a href="/A001515/b001515.txt">Table of n, a(n) for n = 0..404</a> (first 101 terms from T. D. Noe)
%H A001515 Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1701.08394">Analysis of the Gift Exchange Problem</a>, arXiv:1701.08394 [math.CO], 2017.
%H A001515 Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="/A144416/a144416.txt">On-Line Appendix I to "Analysis of the gift exchange problem"</a>, giving Type D recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
%H A001515 Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="/A144416/a144416_1.txt">On-Line Appendix II to "Analysis of the gift exchange problem"</a>, giving Type C recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
%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 [math.CO], 2009.
%H A001515 Veronica Bitonti, Bishal Deb, and Alan D. Sokal, <a href="https://arxiv.org/abs/2412.10214">Thron-type continued fractions (T-fractions) for some classes of increasing trees</a>, arXiv:2412.10214 [math.CO], 2024. See p. 58.
%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>, arXiv:quant-ph/0501155, 2005.
%H A001515 Dmitry Efimov, <a href="https://arxiv.org/abs/1904.08651">The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials</a>, arXiv:1904.08651 [math.CO], 2019.
%H A001515 Andrew Francis and Michael Hendriksen, <a href="https://arxiv.org/abs/2502.14223">Counting spinal phylogenetic networks</a>, arXiv:2502.14223 [q-bio.PE], 2025. See p. 11.
%H A001515 O. Frink and H. L. Krall, <a href="http://dx.doi.org/10.1090/S0002-9947-1949-0028473-1">A new class of orthogonal polynomials</a>, Trans. Amer. Math. Soc. 65,100-115, 1945. [From _Roger L. Bagula_, Feb 15 2009]
%H A001515 E. Grosswald, <a href="http://dx.doi.org/10.1002/zamm.19800601018">Bessel Polynomials</a>, Lecture Notes Math., Vol. 698, 1978.
%H A001515 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A001515 Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="http://dx.doi.org/10.1016/j.aml.2012.02.009">On the Stirling numbers associated with the meromorphic Weyl algebra</a>, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From _N. J. A. Sloane_, Sep 15 2012
%H A001515 Wojciech Mlotkowski and Anna Romanowicz, <a href="http://www.math.uni.wroc.pl/~pms/files/33.2/Article/33.2.19.pdf">A family of sequences of binomial type</a>, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
%H A001515 Robert A. Proctor, <a href="http://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way for Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.
%H A001515 J. Riordan, <a href="/A001519/a001519_1.pdf">Letter to N. J. A. Sloane, Jul. 1968</a>
%H A001515 J. Riordan, <a href="/A001820/a001820.pdf">Notes to N. J. A. Sloane, Jul. 1968</a>
%H A001515 N. J. A. Sloane, <a href="/A001514/a001514.pdf">Letter to J. Riordan, Nov. 1970</a>
%H A001515 <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%H A001515 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A001515 The following formulas can all be found in (or are easily derived from formulas in) Grosswald's book.
%F A001515 D-finite with recurrence: 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 a(n) = Hypergeometric2F0( [n+1, -n] ; - ; -1/2).
%F A001515 a(n) = A105749(n)/n!.
%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-2*x/(1-x-3*x/(1-x-4*x/(1-x-5*x/(1-.... (continued fraction). - _Paul Barry_, Feb 08 2009
%F A001515 From _Michael Somos_, Apr 08 2012: (Start)
%F A001515 a(-1 - n) = a(n).
%F A001515 (a(n+1) + a(n+2))^2 = a(n)*a(n+2) + a(n+1)*a(n+3) for all integer n. (End)
%F A001515 G.f.: 1/G(0) where G(k) = 1 - x - x*(2*k+1)/(1 - x - 2*x*(k+1)/G(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Oct 05 2012
%F A001515 E.g.f.: E(0)/(2*sqrt(1-2*x)), where E(k) = 1 + 1/(1 - 2*x/(2*x + (k+1)*(1+sqrt(1-2*x))/E(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 23 2013
%F A001515 G.f.: T(0)/(1-x), where T(k) = 1 - (k+1)*x/((k+1)*x - (1-x)^2/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 15 2013
%F A001515 a(n) = (2*BesselI(1/2, 1)+BesselI(3/2, 1))*BesselK(n+1/2, 1). - _Jean-François Alcover_, Feb 03 2014
%F A001515 a(n) = exp(1)*sqrt(2/Pi)*BesselK(1/2+n,1). - _Gerry Martens_, Jul 22 2015
%F A001515 From _Peter Bala_, Apr 14 2017: (Start)
%F A001515 a(n) = (1/n!)*Integral_{x = 0..inf} exp(-x)*x^n*(1 + x/2)^n dx.
%F A001515 E.g.f.: d/dx( exp(x*c(x/2)) ) = 1 + 2*x + 7*x^2/2! + 37*x^3/3! + ..., where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
%F A001515 From _G. C. Greubel_, Aug 16 2017: (Start)
%F A001515 a(n) = (1/2)_{n} * 2^n * hypergeometric1f1(-n; -2*n; 2).
%F A001515 G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 2*t/(1-t)^2). (End)
%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 + 3*x +  3*x^2
%e A001515   y_3 = 1 + 6*x + 15*x^2 + 15*x^3, etc.
%e A001515 G.f. = 1 + 2*x + 7*x^2 + 37*x^3 + 266*x^4 + 2431*x^5 + 27007*x^6 + 353522*x^7 + ...
%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;
%t A001515 RecurrenceTable[{a[0]==1,a[1]==2,a[n]==(2n-1)a[n-1]+a[n-2]},a[n], {n,25}] (* _Harvey P. Dale_, Jun 18 2011 *)
%t A001515 Table[Sum[BellY[n+1, k, (2 Range[n+1] - 3)!!], {k, n+1}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%o A001515 (PARI) {a(n) = if( n<0, n = -1 - n); sum( k=0, n, (2*n - k)! / (k! * (n-k)!) * 2^(k-n))} /* _Michael Somos_, Apr 08 2012 */
%o A001515 (Haskell)
%o A001515 a001515 = sum . a001497_row -- _Reinhard Zumkeller_, Nov 24 2014
%o A001515 (Magma) [(&+[Binomial(n+j, 2*j)*Catalan(j)*Factorial(j+1)/2^j: j in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Sep 26 2023
%o A001515 (SageMath) [sum(binomial(n+j,2*j)*binomial(2*j,j)*factorial(j)//2^j for j in range(n+1)) for n in range(31)] # _G. C. Greubel_, Sep 26 2023
%Y A001515 Cf. A000108, A000806, A001514, A105749, A143990, A144505, A144506, A144507, A144513, A144514.
%Y A001515 See A144301 for other formulas and comments.
%Y A001515 Row sums of Bessel triangle A001497 as well as of A001498.
%Y A001515 Partial sums: A105748.
%Y A001515 First differences: A144498.
%Y A001515 Replace "sets" with "lists" in comment: A001517.
%Y A001515 The gift scenarios sequences when a gift can be stolen at most s times, for s = 1..9, are this sequence, A144416, A144508, A144509, A149187, A281358, A281359, A281360, A281361.
%K A001515 nonn,easy,nice
%O A001515 0,2
%A A001515 _N. J. A. Sloane_
%E A001515 Extensively edited by _N. J. A. Sloane_, Dec 07 2008