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A013988 Triangle read by rows, the inverse Bell transform of n!*binomial(5,n) (without column 0).

%I A013988 #43 Oct 03 2023 13:14:33
%S A013988 1,5,1,55,15,1,935,295,30,1,21505,7425,925,50,1,623645,229405,32400,
%T A013988 2225,75,1,21827575,8423415,1298605,103600,4550,105,1,894930575,
%U A013988 358764175,59069010,5235405,271950,8330,140,1,42061737025,17398082625,3016869625,289426830,16929255,621810,14070,180,1
%N A013988 Triangle read by rows, the inverse Bell transform of n!*binomial(5,n) (without column 0).
%C A013988 Previous name was: Triangle of numbers related to triangle A049224; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
%C A013988 T(n, m) = S2p(-5; n,m), a member of a sequence of triangles including S2p(-1; n,m) = A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1; n,m) = A008277(n,m) (Stirling 2nd kind). T(n, 1) = A008543(n-1).
%C A013988 For the definition of the Bell transform see A264428 and the link. - _Peter Luschny_, Jan 16 2016
%H A013988 G. C. Greubel, <a href="/A013988/b013988.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A013988 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.
%H A013988 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) 09.8.3
%H A013988 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 A013988 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%H A013988 <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%F A013988 T(n, m) = n!*A049224(n, m)/(m!*6^(n-m));
%F A013988 T(n+1, m) = (6*n-m)*T(n, m) + T(n, m-1), for n >= m >= 1, with T(n, m) = 0, n<m, and T(n, 0) = 0, T(1, 1) = 1.
%F A013988 E.g.f. of m-th column: ((1 - (1-6*x)^(1/6))^m)/m!.
%F A013988 Sum_{k=1..n} T(n, k) = A028844(n).
%e A013988 Triangle begins as:
%e A013988           1;
%e A013988           5,         1;
%e A013988          55,        15,        1;
%e A013988         935,       295,       30,       1;
%e A013988       21505,      7425,      925,      50,      1;
%e A013988      623645,    229405,    32400,    2225,     75,     1;
%e A013988    21827575,   8423415,  1298605,  103600,   4550,   105,    1;
%e A013988   894930575, 358764175, 59069010, 5235405, 271950,  8330,  140,   1;
%t A013988 (* First program *)
%t A013988 rows = 10;
%t A013988 b[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, rows}]];
%t A013988 A = Table[b[n, m], {n, 1, rows}, {m, 1, rows}] // Inverse // Abs;
%t A013988 A013988 = Table[A[[n, m]], {n, 1, rows}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *)
%t A013988 (* Second program *)
%t A013988 T[n_, k_]:= T[n, k]= If[k==0, 0, If[k==n, 1, (6*(n-1) -k)*T[n-1,k] +T[n-1, k-1]]];
%t A013988 Table[T[n,k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Oct 03 2023 *)
%o A013988 (Sage) # uses[inverse_bell_matrix from A264428]
%o A013988 # Adds 1,0,0,0, ... as column 0 at the left side of the triangle.
%o A013988 inverse_bell_matrix(lambda n: factorial(n)*binomial(5, n), 8) # _Peter Luschny_, Jan 16 2016
%o A013988 (Magma)
%o A013988 function T(n,k) // T = A013988
%o A013988   if k eq 0 then return 0;
%o A013988   elif k eq n then return 1;
%o A013988   else return (6*(n-1)-k)*T(n-1,k) + T(n-1,k-1);
%o A013988   end if;
%o A013988 end function;
%o A013988 [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Oct 03 2023
%Y A013988 Cf. A008277, A008543, A049224, A264428.
%Y A013988 Cf. A028844 (row sums).
%Y A013988 Triangles with the recurrence T(n,k) = (m*(n-1)-k)*T(n-1,k) + T(n-1,k-1): A010054 (m=1), A001497 (m=2), A004747 (m=3), A000369 (m=4), A011801 (m=5), this sequence (m=6).
%K A013988 easy,nonn,tabl
%O A013988 1,2
%A A013988 _Wolfdieter Lang_
%E A013988 New name from _Peter Luschny_, Jan 16 2016