A144299 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,0) for n >= 0.
1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 3, 0, 0, 1, 10, 15, 0, 0, 0, 1, 15, 45, 15, 0, 0, 0, 1, 21, 105, 105, 0, 0, 0, 0, 1, 28, 210, 420, 105, 0, 0, 0, 0, 1, 36, 378, 1260, 945, 0, 0, 0, 0, 0, 1, 45, 630, 3150, 4725, 945, 0, 0, 0, 0, 0, 1, 55, 990, 6930, 17325, 10395, 0, 0, 0, 0, 0, 0
Offset: 0
Examples
Triangle begins:
n:
0: 1
1: 1 0
2: 1 1 0
3: 1 3 0 0
4: 1 6 3 0 0
5: 1 10 15 0 0 0
6: 1 15 45 15 0 0 0
7: 1 21 105 105 0 0 0 0
8: 1 28 210 420 105 0 0 0 0
9: 1 36 378 1260 945 0 0 0 0 0
...
The row sums give A000085.
For some purposes it is convenient to rotate the triangle by 45 degrees:
1 0 0 0 0 0 0 0 0 0 0 0 ...
1 1 0 0 0 0 0 0 0 0 0 ...
1 3 3 0 0 0 0 0 0 0 ...
1 6 15 15 0 0 0 0 0 ...
1 10 45 105 105 0 0 0 ...
1 15 105 420 945 945 0 ...
1 21 210 1260 4725 10395 ...
1 28 378 3150 17325 ...
1 36 630 6930 ...
1 45 990 ...
...
The latter triangle is important enough that it has its own entry, A144331. Here the column sums give A000085 and the rows sums give A001515.
If the entries in the rotated triangle are denoted by b1(n,k), n >= 0, k <= 2n, then we have the recurrence b1(n, k) = b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2).
Then b1(n,k) is the number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1 or 2.
References
- E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.
Links
- Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
- David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
- J. Y. Choi and J. D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.
- Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
- Toufik Mansour, Matthias Schork, and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
- T. Mansour and M. Shattuck, Partial matchings and pattern avoidance, Appl. Anal. Discrete Math. 7 (2013) 25-50.
Crossrefs
Programs
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Haskell
a144299 n k = a144299_tabl !! n !! k a144299_row n = a144299_tabl !! n a144299_tabl = [1] : [1, 0] : f 1 [1] [1, 0] where f i us vs = ws : f (i + 1) vs ws where ws = (zipWith (+) (0 : map (i *) us) vs) ++ [0] -- Reinhard Zumkeller, Jan 01 2014 -
Magma
A144299:= func< n,k | k le Floor(n/2) select Factorial(n)/(Factorial(n-2*k)*Factorial(k)*2^k) else 0 >; [A144299(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 29 2023
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Maple
Maple code producing the rotated version: b1 := proc(n, k) option remember; if n = k then 1; elif k < n then 0; elif n < 1 then 0; else b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2); end if; end proc; for n from 0 to 12 do lprint([seq(b1(n,k),k=0..2*n)]); od:
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Mathematica
T[n_,0]=0; T[1,1]=1; T[2,1]=1; T[n_, k_]:= T[n-1,k-1] + (n-1)T[n-2,k-1]; Table[T[n,k], {n,12}, {k,n,1,-1}]//Flatten (* Robert G. Wilson v *) Table[If[k<=Floor[n/2],n!/((n-2 k)! k! 2^k),0], {n, 0, 12},{k,0,n}]//Flatten (* Stefano Spezia, Jun 15 2023 *) -
SageMath
def A144299(n,k): return factorial(n)/(factorial(n-2*k)*factorial(k)*2^k) if k <= (n//2) else 0 flatten([[A144299(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 29 2023
Formula
T(n, k) = T(n-1, k-1) + (n-1)*T(n-2, k-1).
E.g.f.: Sum_{k >= 0} Sum_{n = 0..2k} T(n,k) y^k x^n/n! = exp(y(x+x^2/2)). (The coefficient of y^k is the e.g.f. for the k-th row of the rotated triangle shown below.)
T(n, k) = n!/((n - 2*k)!*k!*2^k) for 0 <= k <= floor(n/2) and 0 otherwise. - Stefano Spezia, Jun 15 2023
From G. C. Greubel, Sep 29 2023: (Start)
T(n, 1) = A000217(n-1).
Sum_{k=0..n} T(n,k) = A000085(n).
Sum_{k=0..n} (-1)^k*T(n,k) = A001464(n). (End)
Extensions
Offset fixed by Reinhard Zumkeller, Jan 01 2014
Comments