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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).
5
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
1,3
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
Cf. A104509, A117938, A118980, A118981, A099425 (top row), A006497 (essentially the 2nd row), A014448 (essentially the 3rd row), A087130 (essentially the 4th row).
Sequence in context: A348113 A103280 A046899 * A225632 A035206 A210238
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Aug 12 2019, based on R. J. Mathar's 2011 analysis of A118980.
STATUS
approved