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

%I

%S 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 1,33,112,144,112,80,64,64,1,54,238,344,320,224,160,128,128,1,88,496,

%U 828,848,640,448,320,256,256

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

%C 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 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 1, 1, 1, 1, 1, 1,

%F 1, 2, 3, 5, 8, 13, (essentially A000045)

%F 1, 2, 5, 9, 18, 37, (essentially A077947)

%F 1, 2, 5, 13, 26, 57,

%F Terms of the n-th row of the triangle are the finite differences downwards the n-th column of this array.

%e First few rows of the triangle are:

%e 1;

%e 1, 1;

%e 1, 2, 2;

%e 1, 4, 4, 4;

%e 1, 7, 20, 8, 8;

%e 1, 12, 24, 20, 16, 16;

%e 1, 20, 52, 56, 40, 32, 32;

%e 1, 33, 112, 144, 112, 80, 64, 64;

%e 1, 54, 238, 344, 320, 224, 160, 128, 128;

%e 1, 88, 496, 828, 848, 640, 448, 320, 256, 256;

%e ...

%p read("transforms") ;

%p A213948i := proc(n,k)

%p if n = 1 then

%p L := [1,seq(0,i=0..k)] ;

%p else

%p L := [1,seq(2^i,i=0..n-2),seq(0,i=0..k)] ;

%p end if;

%p INVERT(L) ;

%p op(k,%) ;

%p end proc:

%p A213948 := proc(n,k)

%p if k = 1 then

%p 1;

%p else

%p A213948i(k,n)-A213948i(k-1,n) ;

%p end if;

%p end proc: # _R. J. Mathar_, Jun 30 2012

%Y Cf. A001519, A213947, A000071 (2nd column), A020714 (subdiagonal), A005009 (subdiagonal).

%K nonn,tabl,changed

%O 1,5

%A _Gary W. Adamson_, Jun 25 2012

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Last modified December 16 01:32 EST 2019. Contains 330013 sequences. (Running on oeis4.)