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Antidiagonal products of A319840.
0

%I #69 Jun 22 2024 22:39:09

%S 1,1,4,36,576,12800,360000,12192768,481890304,21743271936,

%T 1101996057600,61952000000000,3824628881965056,257164113195565056,

%U 18704075505689706496,1462975070062038220800,122444006400000000000000,10918111308394619734065152,1033255398127440061257744384

%N Antidiagonal products of A319840.

%C a(n) has trailing zeros iff n is congruent to 0 or 1 mod 5. Cf. A008851.

%C a(n) is a square iff n = 1 or congruent to {1, 3, 4} mod 5. Cf. A047206.

%C It appears that: (Start)

%C a(n) is a cube iff n = 0, 1, or is of the form (3*m - 4)^3 with m > 1 (A016791);

%C the only fourth powers in the sequence are 1 and a(9) = 21743271936 = 384^4;

%C the only fifth powers in the sequence are 1 and a(32) = 227200942336^5;

%C a(n) is a sixth power iff n = 0, 1, or is of the form (6*m - 10)^3 with m > 1;

%C the only seventh powers in the sequence are 1 and a(128) = 77458109039896212820250015287665035595218944^7. (End)

%F a(0) = a(1) = 1, and a(n) = n^2*2^(n-2)*(n - 1)^(n-2) for n > 1.

%t a[0]=a[1]=1; a[n_]:=n^2*2^(n-2)*(n-1)^(n-2); Array[a,19,0]

%Y Cf. A000079, A000169, A000290, A008851, A016791, A047206, A319840.

%K nonn

%O 0,3

%A _Stefano Spezia_, Jun 22 2024