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%I #13 Feb 08 2022 22:18:19
%S 1,1,2,1,4,3,1,10,6,4,1,20,21,8,5,1,42,57,28,10,6,1,84,150,88,35,12,7,
%T 1,170,390,252,110,42,14,8,1,340,990,712,335,132,49,16,9,1,682,2475,
%U 1992,975,402,154,56,18,10
%N Triangle read by rows: columns are finite differences of the INVERT transform of (1, 2, 3, ...) terms.
%C Create an array in which the n-th row is the output of the INVERT transform on the first n natural numbers followed by zeros:
%C 1, 1, 1, 1, 1, 1, 1, ...
%C 1, 3, 5, 11, 21, 43, 85, ... (A001045)
%C 1, 3, 8, 17, 42, 100, 235, ... (A101822)
%C 1, 3, 8, 21, 50, 128, 323, ...
%C ...
%C For example, row 3 is the INVERT transform of (1, 2, 3, 0, 0, 0, ...). Then, take finite differences of column terms starting from the top; which become the rows of the triangle.
%e First few rows of the triangle:
%e 1;
%e 1, 2;
%e 1, 4, 3;
%e 1, 10, 6, 4;
%e 1, 20, 21, 8, 5;
%e 1, 42, 57, 28, 10, 6;
%e 1, 84, 150, 88, 35, 12, 7;
%e 1, 170, 390, 252, 110, 42, 14, 8;
%e 1, 340, 990, 712, 335, 132, 49, 16, 9;
%e 1, 682, 2475, 1992, 975, 402, 154, 56, 18, 10;
%e 1, 1364, 6138, 5464, 2805, 1200, 469, 176, 63, 20, 11;
%e ...
%p read("transforms") ;
%p A213947i := proc(n,k)
%p L := [seq(i,i=1..n),seq(0,i=0..k)] ;
%p INVERT(L) ;
%p op(k,%) ;
%p end proc:
%p A213947 := proc(n,k)
%p if k = 1 then
%p 1;
%p else
%p A213947i(k,n)-A213947i(k-1,n) ;
%p end if;
%p end proc: # _R. J. Mathar_, Jun 30 2012
%Y Cf. A001906 (row sums), A026644 (2nd column).
%K nonn,tabl
%O 1,3
%A _Gary W. Adamson_, Jun 25 2012
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