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A213947 Triangle read by rows: columns are finite differences of the INVERT transform of (1, 2, 3,...) terms 2
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, 1, 170, 390, 252, 110, 42, 14, 8, 1, 340, 990, 712, 335, 132, 49, 16, 9, 1, 682, 2475, 1992, 975, 402, 154, 56, 18, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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:

1,...1,...1,...1,...1,...1,...1,...

1,...3,...5,..11,..21,..43,..85,... (A001045)

1,...3,...8,..17,..42,.100,.235,... (A101822)

1,...3,...8,..21,..50,.128,.323,...

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.

LINKS

Table of n, a(n) for n=1..55.

EXAMPLE

First few rows of the triangle are:

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;

1, 170, 390, 252, 110, 42, 14, 8;

1, 340, 990, 712, 335, 132, 49, 16, 9;

1, 682, 2475, 1992, 975, 402, 154, 56 18, 10;

1, 1364, 6138, 5464, 2805, 1200, 469, 176, 63, 20, 11;

...

MAPLE

read("transforms") ;

A213947i := proc(n, k)

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

        INVERT(L) ;

        op(k, %) ;

end proc:

A213947 := proc(n, k)

        if k = 1 then

                1;

        else

        A213947i(k, n)-A213947i(k-1, n) ;

        end if;

end proc: # R. J. Mathar, Jun 30 2012

CROSSREFS

Cf. A001906 (row sums), A026644 (2nd column)

Sequence in context: A319075 A264871 A067410 * A188403 A248929 A109977

Adjacent sequences:  A213944 A213945 A213946 * A213948 A213949 A213950

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Jun 25 2012

STATUS

approved

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Last modified March 28 07:59 EDT 2020. Contains 333079 sequences. (Running on oeis4.)