A118980 Triangle read by rows: rows = inverse binomial transforms of columns of A309220.

1, 2, 1, 6, 5, 2, 14, 22, 18, 6, 34, 85, 118, 84, 24, 82, 311, 660, 780, 480, 120, 198, 1100, 3380, 5964, 6024, 3240, 720, 478, 3809, 16380, 40740, 60480, 52920, 25200, 5040, 1154, 13005, 76518, 258804, 531864, 676080, 519840, 221760, 40320, 2786, 43978, 348462, 1564314, 4286880, 7444800, 8240400

Offset: 1

Author

Gary W. Adamson, May 07 2006

Keywords

• nonn
• tabl

Comments

First few columns of A309220:

1, 2, 6, 14, 34, ...

1, 3, 11, 36, 119, ...

1, 4, 18, 76, 322, ...

1, 5, 27, 140, 727, ...

1, 6, 38, 234, 1442, ...

1, 7, 51, 364, 2599, ...

1, 8, 66, 536, 4354, ...

...

Examples

			First few rows of the triangle:
   1;
   2,   1;
   6,   5,   2;
  14,  22,  18,   6;
  34,  85, 118,  84,  24;
  82, 311, 660, 780, 480, 120;
  ...
Column 3 of A309220 = (6, 11, 18, 27, 38, 51, ...), whose inverse binomial transform is (6, 5, 2).
		

Crossrefs

The leading column is A099425, and the rightmost two diagonals are A038720 and A000142.

Cf. A104509, A117938, A118981, A309220.

Programs

Maple

with(transforms);
M := 12;
T := [1];
S := series((1 + x^2)/(1-x-x^2 + x*y), x, 120):
for n from 1 to M do
R2 := expand(coeff(S, x, n));
R3 := [seq(abs(coeff(R2,y,n-i)),i=0..n)];
f := k-> add( R3[i]*k^(n-i+1), i=1..nops(R3) ):
s1 := [seq(f(i),i=1..3*n)];
s2 := BINOMIALi(s1);
s3 := [seq(s2[i],i=1..n+1)];
T := [op(T), op(s3)];
od:
T;  # N. J. A. Sloane, Aug 12 2019

Extensions

Edited and extended by N. J. A. Sloane, Aug 12 2019, guided by the comments of R. J. Mathar from Oct 30 2011