cp's OEIS Frontend

A051128 Table T(n,k) = n^k read by upwards antidiagonals (n >= 1, k >= 1).

%I A051128 #44 Jul 02 2025 16:01:58
%S A051128 1,2,1,3,4,1,4,9,8,1,5,16,27,16,1,6,25,64,81,32,1,7,36,125,256,243,64,
%T A051128 1,8,49,216,625,1024,729,128,1,9,64,343,1296,3125,4096,2187,256,1,10,
%U A051128 81,512,2401,7776,15625,16384,6561,512,1,11,100,729,4096,16807,46656,78125,65536,19683,1024,1
%N A051128 Table T(n,k) = n^k read by upwards antidiagonals (n >= 1, k >= 1).
%C A051128 Sum of antidiagonals is A003101(n) for n>0. - _Alford Arnold_, Jan 14 2007
%H A051128 T. D. Noe, <a href="/A051128/b051128.txt">Rows n=1..50 of triangle, flattened</a>
%H A051128 G. Labelle, C. Lamathe and P. Leroux, <a href="https://arxiv.org/abs/math/0312424">Labeled and unlabeled enumeration of k-gonal 2-trees</a>, arXiv:math/0312424 [math.CO], 2003.
%F A051128 a(n) = A004736(n)^A002260(n) or ((t*t+3*t+4)/2-n)^(n-(t*(t+1)/2)), where t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Dec 14 2012
%e A051128 Table begins
%e A051128 1,    1,    1,    1,    1, ...
%e A051128 2,    4,    8,   16,   32, ...
%e A051128 3,    9,   27,   81,  243, ...
%e A051128 4,   16,   64,  256, 1024, ...
%p A051128 A051128 := proc(n)       # _Boris Putievskiy_'s formula
%p A051128 a := floor((sqrt(8*n-7)+1)/2);
%p A051128 b := (a+a^2)/2-n;
%p A051128 c := (a-a^2)/2+n;
%p A051128 (b+1)^c end:
%p A051128 seq(A051128(n), n=1..61);  # _Peter Luschny_, Dec 14 2012
%p A051128 # second Maple program:
%p A051128 T:= (n, k)-> n^k:
%p A051128 seq(seq(T(1+d-k, k), k=1..d), d=1..11);  # _Alois P. Heinz_, Apr 18 2020
%t A051128 Table[n^(k - n + 1), {k, 1, 11}, {n, k, 1, -1}] // Flatten (* _Jean-François Alcover_, Dec 14 2012 *)
%o A051128 (PARI) T(n,k) = n^k \\ _Charles R Greathouse IV_, Feb 09 2017
%Y A051128 Cf. A051129, A003992, A004248, A033918, A095891, A220415, A220416, A220417, A003101.
%K A051128 nonn,tabl,easy,nice
%O A051128 1,2
%A A051128 _N. J. A. Sloane_
%E A051128 More terms from _James Sellers_, Dec 11 1999