0,2

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.

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 (benoit7848c(AT)orange.fr), 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 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)];

# 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 remining odd numbers: [2,3,5,7,9,11,13,...]. [From Gil Broussard (gilbroussard(AT)bellsouth.net), Oct 07 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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]

E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

T. D. Noe, Table of n, a(n) for n=0..100

David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffer-type polynomials

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Index entries for sequences related to Bessel functions or polynomials

Index entries for related partition-counting sequences

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

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)).

Equivalently, as the value of a hypergeometric function: a(n) = 2F0[ n+1, -n ; - ; -1/2].

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-2x/(1-x-3x/(1-x-4x/(1-x-5x/(1-.... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 08 2009]

The first few Bessel polynomials are (cf. A001497, A001498):

y_0 = 1

y_1 = 1+x

y_2 = 1+3x+3x^2

y_3 = 1+6x+15x^2+15x^3, etc.

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;

See A144301 for other formulae and comments.

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

Cf. A000806, A001514, A001517, A144505.

a(n) = A105749(n)/n!. See also A143990.

Partial sums: A105748. First differences: A144498.

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

Sequence in context: A072597 A125515 A135920 this_sequence A144301 A083659 A036247

Adjacent sequences: A001512 A001513 A001514 this_sequence A001516 A001517 A001518

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Extensively edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 07 2008