A132062 Sheffer triangle (1,1-sqrt(1-2*x)). Extended Bessel triangle A001497.

1, 0, 1, 0, 1, 1, 0, 3, 3, 1, 0, 15, 15, 6, 1, 0, 105, 105, 45, 10, 1, 0, 945, 945, 420, 105, 15, 1, 0, 10395, 10395, 4725, 1260, 210, 21, 1, 0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1, 0, 2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1, 0

Offset: 0

Author

Wolfdieter Lang Sep 14 2007

Keywords

• nonn
• easy
• tabl

Comments

This is a Jabotinsky type exponential convolution triangle related to A001147 (double factorials). For Jabotinsky type triangles See the D. E. Knuth reference given under A039692.

The subtriangle (n>=m>=1) is A001497(n,m) (Bessel).

For the combinatorial interpretation in terms of unordered forests of increasing plane trees see the W. Lang comment and example under A001497.

This is a special type of Sheffer triangle. See the S. Roman reference given under A048854 (the notation here differs).

This triangle (or the A001497 subtriangle) appears as generalized Stirling numbers of the second kind, S2p(-1,n,m):=S2(-k;m,m)*(-1)^(n-m) for k=1, eqs. (27)-(29) of the W. Lang reference.

Also the Bell transform of the double factorial of odd numbers A001147. For the Bell transform of the double factorial of even numbers A000165 see A039683. For the definition of the Bell transform see A264428. - Peter Luschny, Dec 20 2015

Examples

			[1]
[0,      1]
[0,      1,      1]
[0,      3,      3,     1]
[0,     15,     15,     6,     1]
[0,    105,    105,    45,    10,    1]
[0,    945,    945,   420,   105,   15,   1]
[0,  10395,  10395,  4725,  1260,  210,  21,  1]
[0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1]
		

References

• 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
• Steven Roman, The Umbral Calculus, Pure and Applied Mathematics, 111, Academic Press, 1984. (p. 78) [Emanuele Munarini, Oct 10 2017]

Links

• Leonard Carlitz, A Note on the Bessel Polynomials, Duke Math. J. 24 (2) (1957), 151-162. [_Emanuele Munarini_, Oct 10 2017]
• 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]
• Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
• Wolfdieter Lang, First 10 rows.
• Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.

Crossrefs

Columns m=1: A001147.

Row sums give [1, A001515]. Alternating row sums give [1, -A000806].

Cf. A122850. - R. J. Mathar, Mar 20 2009

Cf. A039683, A264428.

Programs

Maple

# The function BellMatrix is defined in A264428.
BellMatrix(n -> doublefactorial(2*n-1), 9); # Peter Luschny, Jan 27 2016
# Alternative:
egf := exp(y*(1 - sqrt(1 - 2*x))): serx := series(egf, x, 12):
coefx := n -> n!*coeff(serx, x, n): row := n -> seq(coeff(coefx(n), y, k), k = 0..n): for n from 0 to 8 do row(n) od;  # Peter Luschny, Apr 25 2024

Mathematica

Table[If[k <= n, Binomial[2n-2k,n-k] Binomial[2n-k-1,k-1] (n-k)!/2^(n-k), 0], {n, 0, 6}, {k, 0, n}] // Flatten (* Emanuele Munarini, Oct 10 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[(2#-1)!!&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

Sage

# uses[bell_transform from A264428]
def A132062_row(n):
    a = sloane.A001147
    dblfact = a.list(n)
    return bell_transform(n, dblfact)
[A132062_row(n) for n in (0..9)] # Peter Luschny, Dec 20 2015

Formula

a(n,m)=0 if n

E.g.f. m-th column ((x*f2p(1;x))^m)/m!, m>=0. with f2p(1;x):=1-sqrt(1-2*x)= x*c(x/2) with the o.g.f.of A000108 (Catalan).

From Emanuele Munarini, Oct 10 2017: (Start)

a(n,k) = binomial(2*n-2*k,n-k)*binomial(2*n-k-1,k-1)*(n-k)!/2^(n-k).

The row polynomials p_n(x) (studied by Carlitz) satisfy the recurrence: p_{n+2}(x) - (2*n+1)*p_{n+1}(x) - x^2*p_n(x) = 0. (End)

T(n, k) = n! [y^k] [x^n] exp(y*(1 - sqrt(1 - 2*x))). - Peter Luschny, Apr 25 2024

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