A349736 Binomial coefficients C(m,k) such that C(m,k), C(m,k+1), C(m,k+2) with 0 <= k <= m-2 form an increasing arithmetic progression.
7, 1001, 490314, 927983760, 6973199770790, 209769429934732479, 25331521183260952835630, 12289694242827235919344118592, 23955991473971122736214778043009679, 187581456720371323313917970237305876898550, 5898404991626652623457605084827693331568853294440, 744569299056744628602691379013860201165514803170616390880
Offset: 1
Examples
For n = 1, row 7 of Pascal's triangle is 1, 7, 21, 35, 35, 21, 7, 1; C(7,1) = 7, C(7,2) = 21 and C(7,3) = 35 form an arithmetic progression with common difference = 14, hence a(3) = 7 = C(7,1). For n = 2, row 14 is 1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1; C(14,4) = 1001 , C(14,5) = 2002 and C(14,6) = 3003 form an arithmetic progression with common difference = 1001, hence a(4) = 1001 = C(14,4).
References
- G. L. Alexanderson, L. F. Klosinski and L. C. Larson, The William Lowell Putnam Mathematical Competition, Problems and Solutions 1965-1984, The Mathematical Association of America, 1985, page 17.
Links
Programs
-
Maple
Sequence = seq(binomial(n^2+4*n+2,(n^2+3*n-2)/2), n=1..16);
-
Mathematica
nterms=15; Table[Binomial[n^2+4n+2, (n^2+3n-2)/2], {n, nterms}] (* Paolo Xausa, Nov 29 2021 *) -
PARI
a(n) = binomial(n^2+4*n+2,(n^2+3*n-2)/2) \\ Andrew Howroyd, Oct 29 2023
Formula
a(n) ~ c*2^(n^2+4*n)/n, where c = 4*sqrt(2/(Pi*e)). - Stefano Spezia, Nov 29 2021
Extensions
Missing a(9) = 23955...79 inserted by Georg Fischer, Oct 29 2023
Comments