A118980 - OEIS

%I #31 Jul 21 2022 03:06:46

%S 1,2,1,6,5,2,14,22,18,6,34,85,118,84,24,82,311,660,780,480,120,198,

%T 1100,3380,5964,6024,3240,720,478,3809,16380,40740,60480,52920,25200,

%U 5040,1154,13005,76518,258804,531864,676080,519840,221760,40320,2786,43978,348462,1564314,4286880,7444800,8240400

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

%C First few columns of A309220:

%C   1, 2,  6,  14,   34, ...

%C   1, 3, 11,  36,  119, ...

%C   1, 4, 18,  76,  322, ...

%C   1, 5, 27, 140,  727, ...

%C   1, 6, 38, 234, 1442, ...

%C   1, 7, 51, 364, 2599, ...

%C   1, 8, 66, 536, 4354, ...

%C   ...

%e First few rows of the triangle:

%e    1;

%e    2,   1;

%e    6,   5,   2;

%e   14,  22,  18,   6;

%e   34,  85, 118,  84,  24;

%e   82, 311, 660, 780, 480, 120;

%e   ...

%e Column 3 of A309220 = (6, 11, 18, 27, 38, 51, ...), whose inverse binomial transform is (6, 5, 2).

%p with(transforms);

%p M := 12;

%p T := [1];

%p S := series((1 + x^2)/(1-x-x^2 + x*y), x, 120):

%p for n from 1 to M do

%p R2 := expand(coeff(S, x, n));

%p R3 := [seq(abs(coeff(R2,y,n-i)),i=0..n)];

%p f := k-> add( R3[i]*k^(n-i+1), i=1..nops(R3) ):

%p s1 := [seq(f(i),i=1..3*n)];

%p s2 := BINOMIALi(s1);

%p s3 := [seq(s2[i],i=1..n+1)];

%p T := [op(T), op(s3)];

%p od:

%p T;  # _N. J. A. Sloane_, Aug 12 2019

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

%Y Cf. A104509, A117938, A118981, A309220.

%K nonn,tabl

%O 1,2

%A _Gary W. Adamson_, May 07 2006

%E Edited and extended by _N. J. A. Sloane_, Aug 12 2019, guided by the comments of _R. J. Mathar_ from Oct 30 2011