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

%I

%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 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 are:

%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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Last modified June 6 18:59 EDT 2020. Contains 334832 sequences. (Running on oeis4.)