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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).

%I A309220 #19 Jul 09 2025 04:47:21
%S A309220 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,
%T A309220 393,198,1,8,51,234,727,1364,1298,478,1,9,66,364,1442,3775,5778,4287,
%U A309220 1154,1,10,83,536,2599,8886,19602,24476,14159,2786,1,11,102,756,4354,18557
%N 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).
%C A309220 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
%e A309220 The first few antidiagonals are:
%e A309220 1,
%e A309220 1,2,
%e A309220 1,3,6,
%e A309220 1,4,11,14,
%e A309220 1,5,18,36,34,
%e A309220 1,6,27,76,119,82,
%e A309220 1,7,38,140,322,393,198,
%e A309220 ...
%e A309220 The first nine rows of A are
%e A309220 1, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...
%e A309220 1, 3, 11, 36, 119, 393, 1298, 4287, 14159, 46764, 154451, 510117, ...
%e A309220 1, 4, 18, 76, 322, 1364, 5778, 24476, 103682, 439204, 1860498, 7881196, ...
%e A309220 1, 5, 27, 140, 727, 3775, 19602, 101785, 528527, 2744420, 14250627, 73997555, ...
%e A309220 1, 6, 38, 234, 1442, 8886, 54758, 337434, 2079362, 12813606, 78960998, 486579594, ...
%e A309220 1, 7, 51, 364, 2599, 18557, 132498, 946043, 6754799, 48229636, 344362251, 2458765393, ...
%e A309220 1, 8, 66, 536, 4354, 35368, 287298, 2333752, 18957314, 153992264, 1250895426, 10161155672, ...
%e A309220 1, 9, 83, 756, 6887, 62739, 571538, 5206581, 47430767, 432083484, 3936182123, 35857722591, ...
%e A309220 1, 10, 102, 1030, 10402, 105050, 1060902, 10714070, 108201602, 1092730090, 11035502502, 111447755110, ...
%p A309220 M := 12;
%p A309220 A:=Array(1..2*M,1..2*M,0):
%p A309220 for i from 1 to M do A[i,1]:=1; od:
%p A309220 S := series((1 + x^2)/(1-x-x^2 + x*y), x, 120): # this is g.f. for A104509
%p A309220 for n from 1 to M do
%p A309220 R2 := expand(coeff(S, x, n));
%p A309220 R3 := [seq(abs(coeff(R2,y,n-i)),i=0..n)];
%p A309220 f := k-> add( R3[i]*k^(n-i+1), i=1..nops(R3) ): # this is the formula for the (n+1)-st column
%p A309220 s1 := [seq(f(i),i=1..M)];
%p A309220 for i from 1 to M do A[i,n+1]:=s1[i]; od:
%p A309220 od:
%p A309220 for i from 1 to M do lprint([seq(A[i,j],j=1..M)]); od:
%p A309220 # alternative by _R. J. Mathar_, Aug 13 2019 :
%p A309220 A104509 := proc(n,k)
%p A309220     (1+x^2)/(1-x-x^2+x*y) ;
%p A309220     coeftayl(%,x=0,n) ;
%p A309220     coeftayl(%,y=0,k) ;
%p A309220 end proc:
%p A309220 A309220 := proc(n::integer,k::integer)
%p A309220     local x;
%p A309220     add( abs(A104509(k-1,i))*x^i,i=0..k-1) ;
%p A309220     subs(x=n,%) ;
%p A309220 end proc:
%p A309220 seq( seq(A309220(d-k,k),k=1..d-1),d=2..13) ;
%Y A309220 Cf. A104509, A117938, A118980, A118981, A099425 (top row), A006497 (essentially the 2nd row), A014448 (essentially the 3rd row), A087130 (essentially the 4th row).
%K A309220 nonn,tabl
%O A309220 1,3
%A A309220 _N. J. A. Sloane_, Aug 12 2019, based on _R. J. Mathar_'s 2011 analysis of A118980