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A213947 Triangle read by rows: columns are finite differences of the INVERT transform of (1, 2, 3, ...) terms.

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