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A117937 Triangle, rows = inverse binomial transforms of A117938 columns.

%I A117937 #14 Jun 28 2025 18:58:36
%S A117937 1,1,1,3,3,2,4,10,12,6,7,27,58,60,24,11,71,240,420,360,120,18,180,920,
%T A117937 2460,3504,2520,720,29,449,3360,13020,27720,32760,20160,5040,47,1107,
%U A117937 11898,64620,194184,337680,338400,181440,40320,76,2710,41268,307194,1257120,3029760,4415040
%N A117937 Triangle, rows = inverse binomial transforms of A117938 columns.
%C A117937 A117936 is the companion triangle using analogous Fibonacci polynomials. Left border of A117936 = the Lucas numbers; right border = factorials.
%C A117937 [Note that most of the comments here and in many related sequences by the same author refer to some unusual definition of binomial transforms for sequences starting at index 1. - _R. J. Mathar_, Jul 05 2012]
%F A117937 Rows of the triangle are inverse binomial transforms of A117938 columns. A117938 columns are generated from f(x), Lucas polynomials: (1); (x); (x^2 + 2); (x^3 + 3x); (x^4 + 4x + 2);...
%e A117937 First few rows of the triangle are:
%e A117937 1;
%e A117937 1, 1;
%e A117937 3, 3, 2;
%e A117937 4, 10, 12, 6;
%e A117937 7, 27, 58, 60, 24;
%e A117937 11, 71, 240, 420, 360, 120;
%e A117937 ...
%e A117937 For example, row 4: (4, 10, 12, 6) = the inverse binomial transform of column 4 of A117938: (4, 14, 36, 76, 140...), being f(x), x =1,2,3...using the Lucas polynomial x^3 + 3x.
%p A117937 A117937 := proc(n,k)
%p A117937     add( A117938(n+i,n)*binomial(k-1,i)*(-1)^(1+i-k),i=0..k-1) ;
%p A117937 end proc:
%p A117937 seq(seq(A117937(n,k),k=1..n),n=1..13) ; # _R. J. Mathar_, Aug 16 2019
%Y A117937 Cf. A117936, A117938.
%K A117937 nonn,tabl,easy
%O A117937 1,4
%A A117937 _Gary W. Adamson_, Apr 04 2006