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Array read by ascending antidiagonals: A(n,k) = k^n + k*n + 1.
1

%I #8 Aug 25 2024 18:49:16

%S 2,1,2,1,3,2,1,4,5,2,1,5,9,7,2,1,6,15,16,9,2,1,7,25,37,25,11,2,1,8,43,

%T 94,77,36,13,2,1,9,77,259,273,141,49,15,2,1,10,143,748,1045,646,235,

%U 64,17,2,1,11,273,2209,4121,3151,1321,365,81,19,2

%N Array read by ascending antidiagonals: A(n,k) = k^n + k*n + 1.

%F G.f. for the k-th column: (2*x^2 - 3*x - k^2 + k + 1)/((x - 1)^2*(x - k)).

%F E.g.f. for the k-th column: exp(x)*(1 + exp((k-1)*x) + k*x).

%F A(n,1) = n + 2.

%F A(2,n) = A000290(n+1).

%F A(n,n) = 2*A214647(n) + 1.

%e Array begins:

%e 2, 2, 2, 2, 2, 2, ...

%e 1, 3, 5, 7, 9, 11, ...

%e 1, 4, 9, 16, 25, 36, ...

%e 1, 5, 15, 37, 77, 141, ...

%e 1, 6, 25, 94, 273, 646, ...

%e 1, 7, 43, 259, 1045, 3151, ...

%e 1, 8, 77, 748, 4121, 15656, ...

%e ...

%t A[0,0]=2; A[n_,k_]:=k^n+k*n+1;Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

%Y Cf. A000290, A004247, A004248, A005408 (n=1), A005491 (n=3), A007395 (n=0), A054977 (k=0), A176691 (k=2), A176805 (k=3), A176916 (k=5), A176972 (k=7), A214647.

%Y Cf. A375578 (antidiagonal sums).

%K nonn,easy,tabl

%O 0,1

%A _Stefano Spezia_, Aug 19 2024