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A001515
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Bessel polynomial y_n(x) evaluated at x=1.
(Formerly M1803 N0713)
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41
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1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, 90960751, 1733584106, 36496226977, 841146804577, 21065166341402, 569600638022431, 16539483668991901, 513293594376771362, 16955228098102446847
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OFFSET
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0,2
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COMMENTS
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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, 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 remaining odd numbers: [2,3,5,7,9,11,13,...]. [Gil Broussard, Oct 07 2009]
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REFERENCES
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Frink, O. 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.
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
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).
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LINKS
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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
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FORMULA
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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). [Paul Barry, Feb 08 2009]
a(-1 - n) = a(n) and (a(n+1) + a(n+2))^2 = a(n)*a(n+2) + a(n+1)*a(n+3) for all integer n. - Michael Somos, Apr 08 2012
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
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EXAMPLE
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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.
1 + 2*x + 7*x^2 + 37*x^3 + 266*x^4 + 2431*x^5 + 27007*x^6 + 353522*x^7 + ...
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MAPLE
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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;
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MATHEMATICA
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RecurrenceTable[{a[0]==1, a[1]==2, a[n]==(2n-1)a[n-1]+a[n-2]}, a[n], {n, 25}] (* Harvey P. Dale, June 18 2011 *)
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PROG
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(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 */
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CROSSREFS
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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: A125515 A135920 * A144301 A083659 A036247 A107877
Adjacent sequences: A001512 A001513 A001514 * A001516 A001517 A001518
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Extensively edited by N. J. A. Sloane, Dec 07 2008
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STATUS
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approved
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