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Square array read by antidiagonals downwards: A(n, k) = (Sum_{i=1..n} i^k) - (n+1)^k for n >= 1, k >= 1.
1

%I #17 Sep 03 2017 22:06:58

%S -1,-3,0,-7,-4,2,-15,-18,-2,5,-31,-64,-28,5,9,-63,-210,-158,-25,19,14,

%T -127,-664,-748,-271,9,42,20,-255,-2058,-3302,-1825,-317,98,76,27,

%U -511,-6304,-14068,-10735,-3351,-126,272,123,35,-1023,-19170,-58718,-59425,-26141,-4606,580,567,185,44

%N Square array read by antidiagonals downwards: A(n, k) = (Sum_{i=1..n} i^k) - (n+1)^k for n >= 1, k >= 1.

%C Paul Erdős conjectured that A(n, k) = 0 only for (n, k) = (2, 1).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Moser_equation">Erdos-Moser equation</a>

%e Array starts

%e -1, -3, -7, -15, -31, -63, -127, -255

%e 0, -4, -18, -64, -210, -664, -2058, -6304

%e 2, -2, -28, -158, -748, -3302, -14068, -58718

%e 5, 5, -25, -271, -1825, -10735, -59425, -318271

%e 9, 19, 9, -317, -3351, -26141, -183111, -1216637

%e 14, 42, 98, -126, -4606, -50478, -446782, -3622206

%e 20, 76, 272, 580, -3760, -77324, -896848, -8869820

%e 27, 123, 567, 2211, 2727, -84477, -1485513, -18362109

%e 35, 185, 1025, 5333, 20825, -21595, -1919575, -32268667

%e 44, 264, 1694, 10692, 59774, 206844, -1406746, -46627548

%o (PARI) x(n, k) = sum(i=1, n, i^k)

%o y(n, k) = (n+1)^k

%o a(n, k) = x(n, k) - y(n, k)

%o array(rows, cols) = for(s=1, rows, for(t=1, cols, print1(a(s, t), ", ")); print(""))

%o array(10, 8) \\ print initial 10 rows and 8 columns of array

%Y Cf. A000096 (column 1), A126646 (row 1), A191686 (main diagonal).

%K sign,tabl

%O 1,2

%A _Felix Fröhlich_, Aug 12 2017