A193242 a(n) = (C(n, floor(n/2)) + 2)^n for n >= 0.

1, 3, 16, 125, 4096, 248832, 113379904, 94931877133, 722204136308736, 9223372036854775808, 1117730665547154976408576, 214633637635011206805784100864, 397495155639882245867698528490622976, 1135797931555041090259334993227408493600768

Offset: 0

Author

Alexander R. Povolotsky, Feb 11 2013

Keywords

• nonn

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..59

Crossrefs

Cf. A007318, A051920, A001405.

Programs

Mathematica

Table[(Binomial[n, Floor[n/2]] + 2)^n, {n,0,20}] (* G. C. Greubel, Feb 20 2017 *)

PARI

for(n=0,20, print1((binomial(n, floor(n/2)) + 2)^n, ", ")) \\ G. C. Greubel, Feb 20 2017

Formula

a(n) = (A051920(n) + 1)^n.

a(n) = (A001405(n) + 2)^n.

Extensions

Corrected a(8) onward - G. C. Greubel, Feb 20 2017