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A011801 Triangle read by rows, the inverse Bell transform of n!*binomial(4,n) (without column 0).

%I A011801 #47 Oct 03 2023 13:14:46
%S A011801 1,4,1,36,12,1,504,192,24,1,9576,3960,600,40,1,229824,100656,17160,
%T A011801 1440,60,1,6664896,3048192,563976,54600,2940,84,1,226606464,107255232,
%U A011801 21095424,2256576,142800,5376,112,1,8837652096,4302305280,887785920,102332160,7254576,325584,9072,144,1
%N A011801 Triangle read by rows, the inverse Bell transform of n!*binomial(4,n) (without column 0).
%C A011801 Previous name was: Triangle of numbers related to triangle A049223; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
%C A011801 T(n, m) = S2p(-4; 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) = A008546(n-1).
%C A011801 For the definition of the Bell transform see A264428 and the link. - _Peter Luschny_, Jan 16 2016
%H A011801 G. C. Greubel, <a href="/A011801/b011801.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A011801 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 A011801 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 A011801 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%H A011801 <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%F A011801 T(n, m) = n!*A049223(n, m)/(m!*5^(n-m)).
%F A011801 T(n+1, m) = (5*n-m)*T(n, m) + T(n, m-1), for n >= m >= 1, with T(n, m) = 0, for n < m, and T(n, 0) = 0, T(1, 1) = 1.
%F A011801 E.g.f. of n-th column: (1/n!)*( 1 - (1-5*x)^(1/5) )^n.
%F A011801 Sum_{k=1..n} T(n, k) = A028575(n).
%e A011801 Triangle starts:
%e A011801           1;
%e A011801           4,         1;
%e A011801          36,        12,        1;
%e A011801         504,       192,       24,       1;
%e A011801        9576,      3960,      600,      40,      1;
%e A011801      229824,    100656,    17160,    1440,     60,     1;
%e A011801     6664896,   3048192,   563976,   54600,   2940,    84,    1;
%e A011801   226606464, 107255232, 21095424, 2256576, 142800,  5376,  112,   1;
%t A011801 (* First program *)
%t A011801 T[n_, m_] /; n>=m>=1:= T[n, m]= (5*(n-1)-m)*T[n-1, m] + T[n-1, m-1]; T[n_, m_] /; n<m=0; T[_, 0]=0; T[1,1]=1;
%t A011801 Table[T[n,m], {n,10}, {m,n}]//Flatten (* _Jean-François Alcover_, Jun 20 2018 *)
%t A011801 (* Second program *)
%t A011801 rows = 10;
%t A011801 b[n_, m_]:= BellY[n, m, Table[k! Binomial[4, k], {k, 0, rows}]];
%t A011801 T= Table[b[n, m], {n,rows}, {m,rows}]//Inverse//Abs;
%t A011801 A011801= Table[T[[n, m]], {n,rows}, {m,n}]//Flatten (* _Jean-François Alcover_, Jun 22 2018 *)
%o A011801 (Sage) # uses[inverse_bell_matrix from A264428]
%o A011801 # Adds 1,0,0,0, ... as column 0 at the left side of the triangle.
%o A011801 inverse_bell_matrix(lambda n: factorial(n)*binomial(4, n), 8) # _Peter Luschny_, Jan 16 2016
%o A011801 (Magma)
%o A011801 function T(n,k) // T = A011801
%o A011801   if k eq 0 then return 0;
%o A011801   elif k eq n then return 1;
%o A011801   else return (5*(n-1)-k)*T(n-1,k) + T(n-1,k-1);
%o A011801   end if;
%o A011801 end function;
%o A011801 [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Oct 03 2023
%Y A011801 Cf. A008277, A008546, A049223, A264428.
%Y A011801 Cf. A028575 (row sums).
%Y A011801 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), this sequence (m=5), A013988 (m=6).
%K A011801 easy,nonn,tabl
%O A011801 1,2
%A A011801 _Wolfdieter Lang_
%E A011801 New name from _Peter Luschny_, Jan 16 2016