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A194938 Triangle read by rows: coefficients of polynomials p(x,n) defined by 1/(1-t-t^2)^x = Sum_{n=1..oo} p(x,n)*t^n/n!.

%I A194938 #23 Mar 25 2020 03:09:54
%S A194938 1,0,1,0,3,1,0,8,9,1,0,42,59,18,1,0,264,450,215,30,1,0,2160,4114,2475,
%T A194938 565,45,1,0,20880,43512,30814,9345,1225,63,1,0,236880,528492,420756,
%U A194938 154609,27720,2338,84,1,0,3064320,7235568,6316316,2673972,594489,69552
%N A194938 Triangle read by rows: coefficients of polynomials p(x,n) defined by 1/(1-t-t^2)^x = Sum_{n=1..oo} p(x,n)*t^n/n!.
%C A194938 A039692 is a similar triangle but without the leading column.
%C A194938 1/(1-t-t^2) is the g.f. for the Fibonacci numbers (A000045).
%C A194938 Row sums: A005442(n-1).
%C A194938 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. - _Peter Luschny_, Jan 21 2016
%D A194938 Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 149-150
%e A194938 Triangle begins
%e A194938 1;
%e A194938 0, 1;
%e A194938 0, 3, 1;
%e A194938 0, 8, 9, 1;
%e A194938 0, 42, 59, 18, 1;
%e A194938 0, 264, 450, 215, 30, 1;
%e A194938 0, 2160, 4114, 2475, 565, 45, 1;
%e A194938 0, 20880, 43512, 30814, 9345, 1225, 63, 1;
%e A194938 0, 236880, 528492, 420756, 154609, 27720, 2338, 84, 1;
%e A194938 0, 3064320, 7235568, 6316316, 2673972, 594489, 69552, 4074, 108, 1;
%e A194938 0, 44634240, 110499696, 103889700, 49087520, 12803175, 1887753, 154350,6630,135,1;
%p A194938 # The function BellMatrix is defined in A264428.
%p A194938 with(combinat): g := n -> factorial(n)*(fibonacci(n)+fibonacci(n+2)):
%p A194938 BellMatrix(g, 10); # _Peter Luschny_, Jan 21 2016
%t A194938 p[t_] = 1/(1 - t - t^2)^x; Table[ ExpandAll[n!SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]
%t A194938 (* Second program *)
%t A194938 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A194938 B = BellMatrix[Function[n, n!*(Fibonacci[n] + Fibonacci[n+2])], rows = 12];
%t A194938 Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%o A194938 (Sage) # uses[bell_matrix from A264428]
%o A194938 bell_matrix(lambda n: factorial(n)*(fibonacci(n)+fibonacci(n+2)), 8) # _Peter Luschny_, Jan 21 2016
%Y A194938 Cf. A000045, A005442, A039692.
%K A194938 nonn,tabl
%O A194938 1,5
%A A194938 _Roger L. Bagula_, Apr 17 2008
%E A194938 Edited by _N. J. A. Sloane_, Aug 28 2011