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

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, 1, 33, 112, 144, 112, 80, 64, 64, 1, 54, 238, 344, 320, 224, 160, 128, 128, 1, 88, 496, 828, 848, 640, 448, 320, 256, 256

Offset: 1

Author

Gary W. Adamson, Jun 25 2012

Keywords

• nonn
• tabl

Comments

Row sums = A001519, the odd-indexed Fibonacci terms. The triangle is a companion to A213947, having row sums of the even-indexed Fibonacci terms.

Examples

			First few rows of the triangle are:
  1;
  1,   1;
  1,   2,   2;
  1,   4,   4,   4;
  1,   7,  20,   8,   8;
  1,  12,  24,  20,  16,  16;
  1,  20,  52,  56,  40,  32,  32;
  1,  33, 112, 144, 112,  80,  64,  64;
  1,  54, 238, 344, 320, 224, 160, 128, 128;
  1,  88, 496, 828, 848, 640, 448, 320, 256, 256;
  ...
		

Crossrefs

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

Programs

Maple

read("transforms") ;
A213948i := proc(n,k)
    if n = 1 then
        L := [1,seq(0,i=0..k)] ;
    else
        L := [1,seq(2^i,i=0..n-2),seq(0,i=0..k)] ;
    end if;
    INVERT(L) ;
    op(k,%) ;
end proc:
A213948 := proc(n,k)
    if k = 1 then
        1;
    else
        A213948i(k,n)-A213948i(k-1,n) ;
    end if;
end proc: # R. J. Mathar, Jun 30 2012

Formula

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, ...):

1, 1, 1, 1, 1, 1,

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

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

1, 2, 5, 13, 26, 57,

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