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

1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, 90960751, 1733584106, 36496226977, 841146804577, 21065166341402, 569600638022431, 16539483668991901, 513293594376771362, 16955228098102446847, 593946277027962411007, 21992967478132711654106, 858319677924203716921141

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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.

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.

Equivalently, number of sequences of n unlabeled items such that each item occurs just once or twice (cf. A105749). - David Applegate, Dec 08 2008

Numerator of (n+1)-th convergent to 1+tanh(1). - Benoit Cloitre, Dec 20 2002

The following Maple lines show how this sequence and A144505, A144498, A001514, A144513, A144506, A144514, A144507, A144301 are related.

f0:=proc(n) local k; add((n+k)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f0(n),n=0..10)];

# that is this sequence

f1:=proc(n) local k; add((n+k+1)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f1(n),n=0..10)];

# that is A144498

f2:=proc(n) local k; add((n+k+2)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f2(n),n=0..10)];

# that is A144513; divided by 2 gives A001514

f3:=proc(n) local k; add((n+k+3)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f3(n),n=0..10)];

# that is A144514; divided by 6 gives A144506

f4:=proc(n) local k; add((n+k+4)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f4(n),n=0..10)];

# that divided by 24 gives A144507

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

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

Examples

			The first few Bessel polynomials are (cf. A001497, A001498):
  y_0 = 1
  y_1 = 1 +   x
  y_2 = 1 + 3*x +  3*x^2
  y_3 = 1 + 6*x + 15*x^2 + 15*x^3, etc.
G.f. = 1 + 2*x + 7*x^2 + 37*x^3 + 266*x^4 + 2431*x^5 + 27007*x^6 + 353522*x^7 + ...
		

References

• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..404 (first 101 terms from T. D. Noe)
• Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
• Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix I to "Analysis of the gift exchange problem", giving Type D recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
• Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix II to "Analysis of the gift exchange problem", giving Type C recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
• David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
• Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
• P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffer-type polynomials, arXiv:quant-ph/0501155, 2005.
• Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2019.
• Andrew Francis and Michael Hendriksen, Counting spinal phylogenetic networks, arXiv:2502.14223 [q-bio.PE], 2025. See p. 11.
• O. Frink and H. L. Krall, A new class of orthogonal polynomials, Trans. Amer. Math. Soc. 65,100-115, 1945. [From _Roger L. Bagula_, Feb 15 2009]
• E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.
• Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
• Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From _N. J. A. Sloane_, Sep 15 2012
• Wojciech Mlotkowski and Anna Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
• Robert A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
• J. Riordan, Letter to N. J. A. Sloane, Jul. 1968
• J. Riordan, Notes to N. J. A. Sloane, Jul. 1968
• N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
• Index entries for sequences related to Bessel functions or polynomials
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000108, A000806, A001514, A105749, A143990, A144505, A144506, A144507, A144513, A144514.

See A144301 for other formulas and comments.

Row sums of Bessel triangle A001497 as well as of A001498.

Partial sums: A105748.

First differences: A144498.

Replace "sets" with "lists" in comment: A001517.

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.

Programs

Haskell

a001515 = sum . a001497_row -- Reinhard Zumkeller, Nov 24 2014

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

Maple

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;
A001515:=proc(n) local k; add( (n+k)!/((n-k)!*k!*2^k),k=0..n); end;
A001515:= n-> hypergeom( [n+1,-n],[],-1/2);
bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end;

Mathematica

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 *)
Table[Sum[BellY[n+1, k, (2 Range[n+1] - 3)!!], {k, n+1}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

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 */

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

Formula

The following formulas can all be found in (or are easily derived from formulas in) Grosswald's book.

D-finite with recurrence: a(0) = 1, a(1) = 2; thereafter a(n) = (2*n-1)*a(n-1) + a(n-2).

E.g.f.: exp(1-sqrt(1-2*x))/sqrt(1-2*x).

a(n) = Sum_{ k = 0..n } binomial(n+k,2*k)*(2*k)!/(k!*2^k).

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)).

a(n) = Hypergeometric2F0( [n+1, -n] ; - ; -1/2).

a(n) = A105749(n)/n!.

a(n) ~ exp(1)*(2n)!/(n!*2^n) as n -> oo. [See Grosswald, p. 124]

a(n) = A144301(n+1).

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

From Michael Somos, Apr 08 2012: (Start)

a(-1 - n) = a(n).

(a(n+1) + a(n+2))^2 = a(n)*a(n+2) + a(n+1)*a(n+3) for all integer n. (End)

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

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

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

a(n) = (2*BesselI(1/2, 1)+BesselI(3/2, 1))*BesselK(n+1/2, 1). - Jean-François Alcover, Feb 03 2014

a(n) = exp(1)*sqrt(2/Pi)*BesselK(1/2+n,1). - Gerry Martens, Jul 22 2015

From Peter Bala, Apr 14 2017: (Start)

a(n) = (1/n!)*Integral_{x = 0..inf} exp(-x)*x^n*(1 + x/2)^n dx.

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)

From G. C. Greubel, Aug 16 2017: (Start)

a(n) = (1/2)_{n} * 2^n * hypergeometric1f1(-n; -2*n; 2).

G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 2*t/(1-t)^2). (End)

Extensions

Extensively edited by N. J. A. Sloane, Dec 07 2008