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A193242
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a(n) = (C(n, floor(n/2)) + 2)^n for n >= 0.
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3
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1, 3, 16, 125, 4096, 248832, 113379904, 94931877133, 722204136308736, 9223372036854775808, 1117730665547154976408576, 214633637635011206805784100864, 397495155639882245867698528490622976, 1135797931555041090259334993227408493600768
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OFFSET
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0,2
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COMMENTS
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If the terms in each row of a Pascal's triangle (see the tabular presentation of A007318) 1, 1-1, 1-2-1, 1-3-3-1, 1-4-6-4-1, 1-5-10-10-5-1, 1-6-15-20-15-6-1, 1-7-21-35-35-21-7-1 are concatenated (if necessary) and considered as palindromes, represented in different bases, then A051920(n) for n>=0 could be considered as the smallest base radix for which those palindromes are composed of single digits/letters. Those palindromes will look like: 1, 11, 121, 1331, 14641, 15AA51, 16FKF61, 17LZZL71, ... . Conversion of such palindromes from their above mentioned bases to decimal yields this sequence of the consecutive ascending powers. Such powers are enumerations of the rows in a Pascal's triangle, counting from 0, namely: 1, 3, 16, 125, 4096, 248832, 113379904, 94931877133, ... (that is 1^0, 3^1, 4^2, 5^3, 8^4, 12^5, 22^6, 37^7, ...). In general those powers could be described as (A051920(n)+1)^n for n >= 0. Another property of the palindromes discussed above is that their digits/letters sum to 2^n.
Also (as noted by Robert Munafo) the terms of this sequence are a(n) = (A001405(n)+2)^n for n>=0.
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LINKS
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FORMULA
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MATHEMATICA
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Table[(Binomial[n, Floor[n/2]] + 2)^n, {n, 0, 20}] (* G. C. Greubel, Feb 20 2017 *)
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PROG
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(PARI) for(n=0, 20, print1((binomial(n, floor(n/2)) + 2)^n, ", ")) \\ G. C. Greubel, Feb 20 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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