OFFSET
1,3
COMMENTS
As pointed out by Peter Munn, A117938 gives the same triangle, except that it has an additional diagonal at the right. - N. J. A. Sloane, Aug 13 2019
EXAMPLE
The first few antidiagonals are:
1,
1,2,
1,3,6,
1,4,11,14,
1,5,18,36,34,
1,6,27,76,119,82,
1,7,38,140,322,393,198,
...
The first nine rows of A are
1, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...
1, 3, 11, 36, 119, 393, 1298, 4287, 14159, 46764, 154451, 510117, ...
1, 4, 18, 76, 322, 1364, 5778, 24476, 103682, 439204, 1860498, 7881196, ...
1, 5, 27, 140, 727, 3775, 19602, 101785, 528527, 2744420, 14250627, 73997555, ...
1, 6, 38, 234, 1442, 8886, 54758, 337434, 2079362, 12813606, 78960998, 486579594, ...
1, 7, 51, 364, 2599, 18557, 132498, 946043, 6754799, 48229636, 344362251, 2458765393, ...
1, 8, 66, 536, 4354, 35368, 287298, 2333752, 18957314, 153992264, 1250895426, 10161155672, ...
1, 9, 83, 756, 6887, 62739, 571538, 5206581, 47430767, 432083484, 3936182123, 35857722591, ...
1, 10, 102, 1030, 10402, 105050, 1060902, 10714070, 108201602, 1092730090, 11035502502, 111447755110, ...
MAPLE
M := 12;
A:=Array(1..2*M, 1..2*M, 0):
for i from 1 to M do A[i, 1]:=1; od:
S := series((1 + x^2)/(1-x-x^2 + x*y), x, 120): # this is g.f. for A104509
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) ): # this is the formula for the (n+1)-st column
s1 := [seq(f(i), i=1..M)];
for i from 1 to M do A[i, n+1]:=s1[i]; od:
od:
for i from 1 to M do lprint([seq(A[i, j], j=1..M)]); od:
# alternative by R. J. Mathar, Aug 13 2019 :
A104509 := proc(n, k)
(1+x^2)/(1-x-x^2+x*y) ;
coeftayl(%, x=0, n) ;
coeftayl(%, y=0, k) ;
end proc:
A309220 := proc(n::integer, k::integer)
local x;
add( abs(A104509(k-1, i))*x^i, i=0..k-1) ;
subs(x=n, %) ;
end proc:
seq( seq(A309220(d-k, k), k=1..d-1), d=2..13) ;
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved