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A039692 Jabotinsky-triangle related to A039647.

%I A039692 #51 Jan 11 2025 10:29:26
%S A039692 1,3,1,8,9,1,42,59,18,1,264,450,215,30,1,2160,4114,2475,565,45,1,
%T A039692 20880,43512,30814,9345,1225,63,1,236880,528492,420756,154609,27720,
%U A039692 2338,84,1,3064320,7235568,6316316,2673972,594489,69552,4074,108,1
%N A039692 Jabotinsky-triangle related to A039647.
%C A039692 Triangle gives the nonvanishing entries of the Jabotinsky matrix for F(z)= A(z)/z = 1/(1-z-z^2) where A(z) is the g.f. of the Fibonacci numbers A000045. (Notation of F(z) as in Knuth's paper.)
%C A039692 E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x)=1, are exponential convolution polynomials: E(n,x+y) = Sum_{k=0..n} binomial(n,k)*E(k,x)*E(n-k,y) (cf. Knuth's paper with E(n,x)= n!*F(n,x)).
%C A039692 E.g.f. for E(n,x): (1 - z - z^2)^(-x).
%C A039692 Explicit a(n,m) formula: see Knuth's paper for f(n,m) formula with f(k)= A039647(n).
%C A039692 E.g.f. for the m-th column sequence: ((-log(1 - z - z^2))^m)/m!.
%C A039692 Also the Bell transform of n!*(F(n)+F(n+2)), F(n) the Fibonacci numbers. For the definition of the Bell transform see A264428 and the link. - _Peter Luschny_, Jan 16 2016
%H A039692 Vincenzo Librandi, <a href="/A039692/b039692.txt">Rows n = 1..50, flattened</a>
%H A039692 D. E. Knuth, <a href="http://arxiv.org/abs/math/9207221">Convolution polynomials</a>, Mathematica J. 2.1 (1992), no. 4, 67-78.
%H A039692 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%F A039692 a(n, 1)= A039647(n)=(n-1)!*L(n), L(n) := A000032(n) (Lucas); a(n, m) = Sum_{j=1..n-m+1} binomial(n-1, j-1)*A039647(j)*a(n-j, m-1), n >= m >= 2.
%F A039692 Conjectured row sums: sum_{m=1..n} a(n,m) = A005442(n). - _R. J. Mathar_, Jun 01 2009
%F A039692 T(n,m) = n! * Sum_{k=m..n} stirling1(k,m)*binomial(k,n-k)*(-1)^(k+m)/k!. - _Vladimir Kruchinin_, Mar 26 2013
%e A039692 1;
%e A039692 3, 1;
%e A039692 8, 9, 1;
%e A039692 42, 59, 18, 1;
%e A039692 264, 450, 215, 30, 1;
%p A039692 A000032 := proc(n) option remember; coeftayl( (2-x)/(1-x-x^2),x=0,n) ; end: A039647 := proc(n) (n-1)!*A000032(n) ; end: A039692 := proc(n,m) option remember ; if m = 1 then A039647(n) ; else add( binomial(n-1,j-1)*A039647(j)*procname(n-j,m-1),j=1..n-m+1) ; fi; end: # _R. J. Mathar_, Jun 01 2009
%t A039692 t[n_, m_] := n!*Sum[StirlingS1[k, m]*Binomial[k, n-k]*(-1)^(k+m)/k!, {k, m, n}]; Table[t[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 21 2013, after _Vladimir Kruchinin_ *)
%o A039692 (Maxima) T(n,m) := n!*sum((stirling1(k,m)*binomial(k,n-k))*(-1)^(k+m)/k!,k,m,n); /* _Vladimir Kruchinin_, Mar 26 2013 */
%o A039692 (PARI)
%o A039692 T(n,m) = n!*sum(k=m,n, (stirling(k,m,1)*binomial(k,n-k))*(-1)^(k+m)/k!);
%o A039692 for(n=1,10,for(k=1,n,print1(T(n,k),", "));print());
%o A039692 /* _Joerg Arndt_, Mar 27 2013 */
%o A039692 (Sage) # uses[bell_matrix from A264428]
%o A039692 # Adds 1,0,0,0, ... as column 0 to the left side of the triangle.
%o A039692 bell_matrix(lambda n: factorial(n)*(fibonacci(n)+fibonacci(n+2)), 8) # _Peter Luschny_, Jan 16 2016
%Y A039692 Cf. A039647, A000032, A000045. Another version of this triangle is in A194938.
%K A039692 nonn,tabl
%O A039692 1,2
%A A039692 _Wolfdieter Lang_