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%I #20 Feb 13 2022 23:11:57
%S 1,1,1,2,3,2,3,9,12,6,5,24,56,60,24,8,62,228,414,360,120,13,156,864,
%T 2400,3480,2520,720,21,387,3132,12606,27360,32640,20160,5040,34,951,
%U 11034,62220,190704,335160,337680,181440,40320,55,2323,38136,294588,1229760,2997120,4394880,3820320,1814400,362880
%N Triangle, rows = inverse binomial transforms of A073133 columns.
%C Left border of the triangle = Fibonacci numbers, right border = factorials. Companion triangle A117937 is generated from Lucas polynomials, using analogous operations.
%C Note that binomial transforms are defined from offset 1 here. - _R. J. Mathar_, Aug 16 2019
%H G. C. Greubel, <a href="/A117936/b117936.txt">Rows n = 1..50 of the triangle, flatten</a>
%F Inverse binomial transforms of A073133 columns. Such columns are f(x), Fibonacci polynomials.
%e First few columns of A073133 are: (1, 1, 1, ...); (1, 2, 3, ...); (2, 5, 10, 17, ...); (3, 12, 33, 72, ...). As sequences, these are f(x), Fibonacci polynomials: (1); (x); (x^2 + 1); (x^3 + 2*x); (x^4 + 3*x^2 + 1); (x^5 + 4*x^3 + 3*x); ... For example, f(x), x = 1,2,3,... using (x^4 + 3*x^2 + 1) generates Column 5 of A073133: (5, 29, 109, 305, ...).
%e Inverse binomial transforms of the foregoing columns generates the triangle rows:
%e 1;
%e 1, 1;
%e 2, 3, 2;
%e 3, 9, 12, 6;
%e 5, 24, 56, 60, 24;
%e 8, 62, 228, 414, 360, 120;
%e ...
%p A117936 := proc(n,k)
%p add( A073133(i+1,n)*binomial(k-1,i)*(-1)^(i-k-1),i=0..k-1) ;
%p end proc:
%p seq(seq(A117936(n,k),k=1..n),n=1..13) ; # _R. J. Mathar_, Aug 16 2019
%t (* A = A073133 *) A[_, 1] = 1; A[n_, k_] := A[n, k] = If[k < 0, 0, n A[n, k - 1] + A[n, k - 2]];
%t T[n_, k_] := Sum[A[i+1, n] Binomial[k-1, i] (-1)^(i - k - 1), {i, 0, k-1}];
%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Apr 01 2020, from Maple *)
%o (Sage)
%o @CachedFunction
%o def A073133(n,k): return 0 if (k<0) else 1 if (k==1) else n*A073133(n,k-1) + A073133(n,k-2)
%o def A117936(n,k): return sum( (-1)^(j-k+1)*binomial(k-1, j)*A073133(j+1,n) for j in (0..k-1) )
%o flatten([[A117936(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Oct 23 2021
%Y Cf. A073133, A117937, A117938.
%Y Cf. A006684 (column 2), A309717 (column 3 halved).
%K nonn,tabl,easy
%O 1,4
%A _Gary W. Adamson_, Apr 03 2006
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