A290844 Square array read by antidiagonals downwards: A(n, k) = (Sum_{i=1..n} i^k) - (n+1)^k for n >= 1, k >= 1.

-1, -3, 0, -7, -4, 2, -15, -18, -2, 5, -31, -64, -28, 5, 9, -63, -210, -158, -25, 19, 14, -127, -664, -748, -271, 9, 42, 20, -255, -2058, -3302, -1825, -317, 98, 76, 27, -511, -6304, -14068, -10735, -3351, -126, 272, 123, 35, -1023, -19170, -58718, -59425, -26141, -4606, 580, 567, 185, 44

Offset: 1

Author

Felix Fröhlich, Aug 12 2017

Keywords

• sign
• tabl

Comments

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

Examples

			Array starts
  -1,  -3,   -7,   -15,   -31,    -63,     -127,      -255
   0,  -4,  -18,   -64,  -210,   -664,    -2058,     -6304
   2,  -2,  -28,  -158,  -748,  -3302,   -14068,    -58718
   5,   5,  -25,  -271, -1825, -10735,   -59425,   -318271
   9,  19,    9,  -317, -3351, -26141,  -183111,  -1216637
  14,  42,   98,  -126, -4606, -50478,  -446782,  -3622206
  20,  76,  272,   580, -3760, -77324,  -896848,  -8869820
  27, 123,  567,  2211,  2727, -84477, -1485513, -18362109
  35, 185, 1025,  5333, 20825, -21595, -1919575, -32268667
  44, 264, 1694, 10692, 59774, 206844, -1406746, -46627548
		

Links

• Wikipedia, Erdos-Moser equation

Crossrefs

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

Programs

PARI

x(n, k) = sum(i=1, n, i^k)
y(n, k) = (n+1)^k
a(n, k) = x(n, k) - y(n, k)
array(rows, cols) = for(s=1, rows, for(t=1, cols, print1(a(s, t), ", ")); print(""))
array(10, 8) \\ print initial 10 rows and 8 columns of array