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

%I A132062 #42 Apr 25 2024 13:33:04
%S A132062 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,
%T A132062 15,1,0,10395,10395,4725,1260,210,21,1,0,135135,135135,62370,17325,
%U A132062 3150,378,28,1,0,2027025,2027025,945945,270270,51975,6930,630,36,1,0
%N A132062 Sheffer triangle (1,1-sqrt(1-2*x)). Extended Bessel triangle A001497.
%C A132062 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.
%C A132062 The subtriangle (n>=m>=1) is A001497(n,m) (Bessel).
%C A132062 For the combinatorial interpretation in terms of unordered forests of increasing plane trees see the W. Lang comment and example under A001497.
%C A132062 This is a special type of Sheffer triangle. See the S. Roman reference given under A048854 (the notation here differs).
%C A132062 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.
%C A132062 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
%D A132062 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
%D A132062 Steven Roman, The Umbral Calculus, Pure and Applied Mathematics, 111, Academic Press, 1984. (p. 78)  [_Emanuele Munarini_, Oct 10 2017]
%H A132062 Leonard Carlitz, <a href="https://doi.org/10.1215/S0012-7094-57-02421-3">A Note on the Bessel Polynomials</a>, Duke Math. J. 24 (2) (1957), 151-162. [_Emanuele Munarini_, Oct 10 2017]
%H A132062 H. Han and S. Seo, <a href="http://dx.doi.org/10.1016/j.ejc.2007.12.002">Combinatorial proofs of inverse relations and log-concavity for Bessel numbers</a>, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]
%H A132062 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A132062 Wolfdieter Lang, <a href="/A132062/a132062.txt">First 10 rows</a>.
%H A132062 Robert S. Maier, <a href="https://arxiv.org/abs/2308.10332">Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers</a>, arXiv:2308.10332 [math.CO], 2023. See p. 18.
%F A132062 a(n,m)=0 if n<m; a(n,0)=1 if n=1, else 0; a(n,m)=(2*(n-1)-m)*a(n-1,m) + a(n-1,m-1).
%F A132062 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).
%F A132062 From _Emanuele Munarini_, Oct 10 2017: (Start)
%F A132062 a(n,k) = binomial(2*n-2*k,n-k)*binomial(2*n-k-1,k-1)*(n-k)!/2^(n-k).
%F A132062 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)
%F A132062 T(n, k) = n! [y^k] [x^n] exp(y*(1 - sqrt(1 - 2*x))). - _Peter Luschny_, Apr 25 2024
%e A132062 [1]
%e A132062 [0,      1]
%e A132062 [0,      1,      1]
%e A132062 [0,      3,      3,     1]
%e A132062 [0,     15,     15,     6,     1]
%e A132062 [0,    105,    105,    45,    10,    1]
%e A132062 [0,    945,    945,   420,   105,   15,   1]
%e A132062 [0,  10395,  10395,  4725,  1260,  210,  21,  1]
%e A132062 [0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1]
%p A132062 # The function BellMatrix is defined in A264428.
%p A132062 BellMatrix(n -> doublefactorial(2*n-1), 9); # _Peter Luschny_, Jan 27 2016
%p A132062 # Alternative:
%p A132062 egf := exp(y*(1 - sqrt(1 - 2*x))): serx := series(egf, x, 12):
%p A132062 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
%t A132062 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 *)
%t A132062 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A132062 rows = 10;
%t A132062 M = BellMatrix[(2#-1)!!&, rows];
%t A132062 Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)
%o A132062 (Sage) # uses[bell_transform from A264428]
%o A132062 def A132062_row(n):
%o A132062     a = sloane.A001147
%o A132062     dblfact = a.list(n)
%o A132062     return bell_transform(n, dblfact)
%o A132062 [A132062_row(n) for n in (0..9)] # _Peter Luschny_, Dec 20 2015
%Y A132062 Columns m=1: A001147.
%Y A132062 Row sums give [1, A001515]. Alternating row sums give [1, -A000806].
%Y A132062 Cf. A122850. - _R. J. Mathar_, Mar 20 2009
%Y A132062 Cf. A039683, A264428.
%K A132062 nonn,easy,tabl
%O A132062 0,8
%A A132062 _Wolfdieter Lang_ Sep 14 2007