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a(n) = Product_{k=0..9} floor((n+k)/10).
10

%I #23 Feb 02 2023 11:24:13

%S 0,0,0,0,0,0,0,0,0,0,1,2,4,8,16,32,64,128,256,512,1024,1536,2304,3456,

%T 5184,7776,11664,17496,26244,39366,59049,78732,104976,139968,186624,

%U 248832,331776,442368,589824,786432,1048576,1310720,1638400,2048000,2560000,3200000,4000000

%N a(n) = Product_{k=0..9} floor((n+k)/10).

%C For n >= 10, a(n) is the maximal product of ten positive integers with sum n.

%H <a href="/index/Rec#order_92">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 9, -18, 9, 0, 0, 0, 0, 0, 0, 0, -36, 72, -36, 0, 0, 0, 0, 0, 0, 0, 84, -168, 84, 0, 0, 0, 0, 0, 0, 0, -126, 252, -126, 0, 0, 0, 0, 0, 0, 0, 126, -252, 126, 0, 0, 0, 0, 0, 0, 0, -84, 168, -84, 0, 0, 0, 0, 0, 0, 0, 36, -72, 36, 0, 0, 0, 0, 0, 0, 0, -9, 18, -9, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).

%F a(n) = 2*a(n-1) - a(n-2) + 9*a(n-10) - 18*a(n-11) + 9*a(n-12) - 36*a(n-20) + 72*a(n-21) - 36*a(n-22) + 84*a(n-30) - 168*a(n-31) + 84*a(n-32) - 126*a(n-40) + 252*a(n-41) - 126*a(n-42) + 126*a(n-50) - 252*a(n-51) + 126*a(n-52) - 84*a(n-60) + 168*a(n-61) - 84*a(n-62) + 36*a(n-70) - 72*a(n-71) + 36*a(n-72) - 9*a(n-80) + 18*a(n-81) - 9*a(n-82) + a(n-90) - 2*a(n-91) + a(n-92).

%F Sum_{n>=10} 1/a(n) = 1 + zeta(10). - _Amiram Eldar_, Jan 10 2023

%F a(10*n) = n^10 (A008454). - _Bernard Schott_, Feb 02 2023

%t Table[Product[Floor[(n + k)/10], {k, 0, 9}], {n, 0, 50}]

%o (PARI) a(n) = prod(k=0, 9, (n+k)\10); \\ _Michel Marcus_, Jul 09 2022

%Y Maximal product of k positive integers with sum n, for k = 2..10: A002620 (k=2), A006501 (k=3), A008233 (k=4), A008382 (k=5), A008881 (k=6), A009641 (k=7), A009694 (k=8), A009714 (k=9), this sequence (k=10).

%Y Cf. A008454 (subsequence), A013668.

%K nonn

%O 0,12

%A _Wesley Ivan Hurt_, Jul 08 2022