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

%I A118980 #31 Jul 21 2022 03:06:46
%S A118980 1,2,1,6,5,2,14,22,18,6,34,85,118,84,24,82,311,660,780,480,120,198,
%T A118980 1100,3380,5964,6024,3240,720,478,3809,16380,40740,60480,52920,25200,
%U A118980 5040,1154,13005,76518,258804,531864,676080,519840,221760,40320,2786,43978,348462,1564314,4286880,7444800,8240400
%N A118980 Triangle read by rows: rows = inverse binomial transforms of columns of A309220.
%C A118980 First few columns of A309220:
%C A118980   1, 2,  6,  14,   34, ...
%C A118980   1, 3, 11,  36,  119, ...
%C A118980   1, 4, 18,  76,  322, ...
%C A118980   1, 5, 27, 140,  727, ...
%C A118980   1, 6, 38, 234, 1442, ...
%C A118980   1, 7, 51, 364, 2599, ...
%C A118980   1, 8, 66, 536, 4354, ...
%C A118980   ...
%e A118980 First few rows of the triangle:
%e A118980    1;
%e A118980    2,   1;
%e A118980    6,   5,   2;
%e A118980   14,  22,  18,   6;
%e A118980   34,  85, 118,  84,  24;
%e A118980   82, 311, 660, 780, 480, 120;
%e A118980   ...
%e A118980 Column 3 of A309220 = (6, 11, 18, 27, 38, 51, ...), whose inverse binomial transform is (6, 5, 2).
%p A118980 with(transforms);
%p A118980 M := 12;
%p A118980 T := [1];
%p A118980 S := series((1 + x^2)/(1-x-x^2 + x*y), x, 120):
%p A118980 for n from 1 to M do
%p A118980 R2 := expand(coeff(S, x, n));
%p A118980 R3 := [seq(abs(coeff(R2,y,n-i)),i=0..n)];
%p A118980 f := k-> add( R3[i]*k^(n-i+1), i=1..nops(R3) ):
%p A118980 s1 := [seq(f(i),i=1..3*n)];
%p A118980 s2 := BINOMIALi(s1);
%p A118980 s3 := [seq(s2[i],i=1..n+1)];
%p A118980 T := [op(T), op(s3)];
%p A118980 od:
%p A118980 T;  # _N. J. A. Sloane_, Aug 12 2019
%Y A118980 The leading column is A099425, and the rightmost two diagonals are A038720 and A000142.
%Y A118980 Cf. A104509, A117938, A118981, A309220.
%K A118980 nonn,tabl
%O A118980 1,2
%A A118980 _Gary W. Adamson_, May 07 2006
%E A118980 Edited and extended by _N. J. A. Sloane_, Aug 12 2019, guided by the comments of _R. J. Mathar_ from Oct 30 2011