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A122848
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Exponential Riordan array (1, x(1+x/2)).
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19
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1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 0, 15, 10, 1, 0, 0, 0, 15, 45, 15, 1, 0, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0
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OFFSET
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0,9
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COMMENTS
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Entries are Bessel polynomial coefficients. Row sums are A000085. Diagonal sums are A122849. Inverse is A122850. Product of A007318 and A122848 gives A100862.
T(n,k) is the number of self inverse permutations of {1,2,...,n} having exactly k cycles. - Geoffrey Critzer, May 08 2012
Bessel numbers of the second kind. For relations to the Hermite polynomials and the Catalan (A033184 and A009766) and Fibonacci (A011973, A098925, and A092865) matrices, see Yang and Qiao. - Tom Copeland, Dec 18 2013.
Also the inverse Bell transform of the double factorial of odd numbers Product_{k= 0..n-1} (2*k+1) (A001147). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015
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LINKS
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G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
P. Bala, Generalized Dobinski formulas
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From R. J. Mathar, Mar 20 2009]
S. Yang and Z. Qiao, The Bessel Numbers and Bessel Matrices, Journal of Mathematical Research & Exposition, July, 2011, Vol. 31, No. 4, pp. 627-636. [From Tom Copeland, Dec 18 2013]
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FORMULA
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Number triangle T(n,k) = k!*C(n,k)/((2k-n)!*2^(n-k)).
T(n,k) = A001498(k,n-k). - Michael Somos, Oct 03 2006
E.g.f.: exp(y(x+x^2/2)). - Geoffrey Critzer, May 08 2012
Triangle equals the matrix product A008275*A039755. Equivalently, the n-th row polynomial R(n,x) is given by the Type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} P(n,2*k+1)*(x/2)^k/k!, where P(n,x) = x*(x-1)*...*(x-n+1) denotes the falling factorial polynomial. Cf. A113278. - Peter Bala, Jun 23 2014
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EXAMPLE
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Triangle begins:
1
0 1
0 1 1
0 0 3 1
0 0 3 6 1
0 0 0 15 10 1
0 0 0 15 45 15 1
0 0 0 0 105 105 21 1
0 0 0 0 105 420 210 28 1
0 0 0 0 0 945 1260 378 36 1
From Gus Wiseman, Jan 12 2021: (Start)
As noted above, a(n) is the number of set partitions of {1..n} into k singletons or pairs. This is also the number of set partitions of subsets of {1..n} into n - k pairs. In the first case, row n = 5 counts the following set partitions:
{{1},{2,3},{4,5}} {{1},{2},{3},{4,5}} {{1},{2},{3},{4},{5}}
{{1,2},{3},{4,5}} {{1},{2},{3,4},{5}}
{{1,2},{3,4},{5}} {{1},{2,3},{4},{5}}
{{1,2},{3,5},{4}} {{1,2},{3},{4},{5}}
{{1},{2,4},{3,5}} {{1},{2},{3,5},{4}}
{{1},{2,5},{3,4}} {{1},{2,4},{3},{5}}
{{1,3},{2},{4,5}} {{1},{2,5},{3},{4}}
{{1,3},{2,4},{5}} {{1,3},{2},{4},{5}}
{{1,3},{2,5},{4}} {{1,4},{2},{3},{5}}
{{1,4},{2},{3,5}} {{1,5},{2},{3},{4}}
{{1,4},{2,3},{5}}
{{1,4},{2,5},{3}}
{{1,5},{2},{3,4}}
{{1,5},{2,3},{4}}
{{1,5},{2,4},{3}}
In the second case, we have:
{{1,2},{3,4}} {{1,2}} {}
{{1,2},{3,5}} {{1,3}}
{{1,2},{4,5}} {{1,4}}
{{1,3},{2,4}} {{1,5}}
{{1,3},{2,5}} {{2,3}}
{{1,3},{4,5}} {{2,4}}
{{1,4},{2,3}} {{2,5}}
{{1,4},{2,5}} {{3,4}}
{{1,4},{3,5}} {{3,5}}
{{1,5},{2,3}} {{4,5}}
{{1,5},{2,4}}
{{1,5},{3,4}}
{{2,3},{4,5}}
{{2,4},{3,5}}
{{2,5},{3,4}}
(End)
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MAPLE
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# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n<2, 1, 0), 9); # Peter Luschny, Jan 27 2016
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MATHEMATICA
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t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten
(* Second program: *)
rows = 12;
t = Join[{1, 1}, Table[0, rows]];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
sbs[{}]:={{}}; sbs[set:{i_, ___}]:=Join@@Function[s, (Prepend[#1, s]&)/@sbs[Complement[set, s]]]/@Cases[Subsets[set], {i}|{i, _}];
Table[Length[Select[sbs[Range[n]], Length[#]==k&]], {n, 0, 6}, {k, 0, n}] (* Gus Wiseman, Jan 12 2021 *)
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PROG
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(PARI) {T(n, k)=if(2*k<n||k>n, 0, n!/(2*k-n)!/(n-k)!*2^(k-n))} /* Michael Somos, Oct 03 2006 */
(Sage) # uses[inverse_bell_transform from A265605]
multifact_2_1 = lambda n: prod(2*k + 1 for k in (0..n-1))
inverse_bell_matrix(multifact_2_1, 9) # Peter Luschny, Dec 31 2015
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CROSSREFS
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Cf. A001497, A008275, A039755, A096713, A104556, A113278, A130757.
Row sums are A000085.
Column sums are A001515.
Same as A049403 but with a first column k = 0.
The same set partitions counted by number of pairs are A100861.
Reversing rows gives A111924 (without column k = 0).
A047884 counts standard Young tableaux by size and greatest row length.
A238123 counts standard Young tableaux by size and least row length.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
Cf. A000110, A000258, A320732, A321729, A339741, A339887.
Sequence in context: A085604 A306268 A144357 * A272481 A054548 A059202
Adjacent sequences: A122845 A122846 A122847 * A122849 A122850 A122851
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Paul Barry, Sep 14 2006
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STATUS
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approved
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