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A193971 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=x*p(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=(x+1)^n.

%I A193971 #16 Mar 28 2020 06:38:09
%S A193971 2,3,5,4,11,9,5,19,26,14,6,29,55,50,20,7,41,99,125,85,27,8,55,161,259,
%T A193971 245,133,35,9,71,244,476,574,434,196,44,10,89,351,804,1176,1134,714,
%U A193971 276,54,11,109,485,1275,2190,2562,2058,1110,375,65,12,131,649,1925
%N A193971 Triangular array:  the fission of (p(n,x)) by (q(n,x)), where p(n,x)=x*p(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=(x+1)^n.
%C A193971 See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.
%e A193971 First six rows:
%e A193971 2
%e A193971 3...5
%e A193971 4...11....9
%e A193971 5...19...26...14
%e A193971 6...29...55...50...20
%e A193971 7...41...99...125..85...27
%p A193971 # The function 'fission' is defined in A193842.
%p A193971 p := (n,x) -> `if`(n=0,1,x*p(n-1,x)+n+1);
%p A193971 q := (n,x) -> (x+1)^n;
%p A193971 A193971_row := n -> fission(p, q, n);
%p A193971 for n from 0 to 5 do A193971_row(n) od; # _Peter Luschny_, Jul 23 2014
%t A193971 z = 11;
%t A193971 p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 1;
%t A193971 q[n_, x_] := (x + 1)^n
%t A193971 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A193971 p1[n_, 0] := p[n, x] /. x -> 0;
%t A193971 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A193971 h[n_] := CoefficientList[d[n, x], {x}]
%t A193971 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A193971 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193971 *)
%t A193971 TableForm[Table[h[n], {n, 0, z}]]
%t A193971 Flatten[Table[h[n], {n, -1, z}]]  (* A193972 *)
%o A193971 (Sage) # uses[fission from A193842]
%o A193971 p = lambda n,x: x*p(n-1,x)+n+1 if n > 0 else 1
%o A193971 q = lambda n,x: (x+1)^n
%o A193971 A193971_row = lambda n: fission(p, q, n);
%o A193971 for n in range(7): A193971_row(n) # _Peter Luschny_, Jul 23 2014
%Y A193971 Cf. A193842, A193972.
%K A193971 nonn,tabl
%O A193971 0,1
%A A193971 _Clark Kimberling_, Aug 10 2011