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A213948 Triangle, by rows, generated from the INVERT transforms of (1, 1, 2, 4, 8, 16, ...).

%I A213948 #15 Dec 11 2019 07:29:54
%S A213948 1,1,1,1,2,2,1,4,4,4,1,7,10,8,8,1,12,24,20,16,16,1,20,52,56,40,32,32,
%T A213948 1,33,112,144,112,80,64,64,1,54,238,344,320,224,160,128,128,1,88,496,
%U A213948 828,848,640,448,320,256,256
%N A213948 Triangle, by rows, generated from the INVERT transforms of (1, 1, 2, 4, 8, 16, ...).
%C A213948 Row sums = A001519, the odd-indexed Fibonacci terms. The triangle is a companion to A213947, having row sums of the even-indexed Fibonacci terms.
%F A213948 Create an array in which the n-th row is the INVERT transform of the first n terms in the sequence (1, 1, 2, 4, 8, 16, 32, ...):
%F A213948   1,    1,    1,    1,    1,    1,
%F A213948   1,    2,    3,    5,    8,   13,    (essentially A000045)
%F A213948   1,    2,    5,    9,   18,   37,    (essentially A077947)
%F A213948   1,    2,    5,   13,   26,   57,
%F A213948 Terms of the n-th row of the triangle are the finite differences downwards the n-th column of this array.
%e A213948 First few rows of the triangle are:
%e A213948   1;
%e A213948   1,   1;
%e A213948   1,   2,   2;
%e A213948   1,   4,   4,   4;
%e A213948   1,   7,  20,   8,   8;
%e A213948   1,  12,  24,  20,  16,  16;
%e A213948   1,  20,  52,  56,  40,  32,  32;
%e A213948   1,  33, 112, 144, 112,  80,  64,  64;
%e A213948   1,  54, 238, 344, 320, 224, 160, 128, 128;
%e A213948   1,  88, 496, 828, 848, 640, 448, 320, 256, 256;
%e A213948   ...
%p A213948 read("transforms") ;
%p A213948 A213948i := proc(n,k)
%p A213948     if n = 1 then
%p A213948         L := [1,seq(0,i=0..k)] ;
%p A213948     else
%p A213948         L := [1,seq(2^i,i=0..n-2),seq(0,i=0..k)] ;
%p A213948     end if;
%p A213948     INVERT(L) ;
%p A213948     op(k,%) ;
%p A213948 end proc:
%p A213948 A213948 := proc(n,k)
%p A213948     if k = 1 then
%p A213948         1;
%p A213948     else
%p A213948         A213948i(k,n)-A213948i(k-1,n) ;
%p A213948     end if;
%p A213948 end proc: # _R. J. Mathar_, Jun 30 2012
%Y A213948 Cf. A001519, A213947, A000071 (2nd column), A020714 (subdiagonal), A005009 (subdiagonal).
%K A213948 nonn,tabl
%O A213948 1,5
%A A213948 _Gary W. Adamson_, Jun 25 2012