A309220 Square array A read by antidiagonals: the columns are given by A(n,1)=1, A(n,2)=n+1, A(n,3) = n^2+2n+3, A(n,4) = n^3+3*n^2+6*n+4, A(n,5) = n^4+4*n^3+10*n^2+12*n+7, ..., whose coefficients are given by A104509 (see also A118981).

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, 1, 8, 51, 234, 727, 1364, 1298, 478, 1, 9, 66, 364, 1442, 3775, 5778, 4287, 1154, 1, 10, 83, 536, 2599, 8886, 19602, 24476, 14159, 2786, 1, 11, 102, 756, 4354, 18557

Offset: 1

Author

N. J. A. Sloane, Aug 12 2019, based on R. J. Mathar's 2011 analysis of A118980

Keywords

• nonn
• tabl

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

Examples

			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, ...
		

Crossrefs

Cf. A104509, A117938, A118980, A118981, A099425 (top row), A006497 (essentially the 2nd row), A014448 (essentially the 3rd row), A087130 (essentially the 4th row).

Programs

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) ;