A033918 -id:A033918 - cp's OEIS frontend

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

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, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 11, 100, 729, 4096, 16807, 46656, 78125, 65536, 19683, 1024, 1

Offset: 1

N. J. A. Sloane

Sum of antidiagonals is A003101(n) for n>0. - Alford Arnold, Jan 14 2007

			Table begins
1,    1,    1,    1,    1, ...
2,    4,    8,   16,   32, ...
3,    9,   27,   81,  243, ...
4,   16,   64,  256, 1024, ...

T. D. Noe, Rows n=1..50 of triangle, flattened
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], 2003.

Cf. A051129, A003992, A004248, A033918, A095891, A220415, A220416, A220417, A003101.

A051128 := proc(n)       # Boris Putievskiy's formula
a := floor((sqrt(8*n-7)+1)/2);
b := (a+a^2)/2-n;
c := (a-a^2)/2+n;
(b+1)^c end:
seq(A051128(n), n=1..61);  # Peter Luschny, Dec 14 2012
# second Maple program:
T:= (n, k)-> n^k:
seq(seq(T(1+d-k, k), k=1..d), d=1..11);  # Alois P. Heinz, Apr 18 2020

Table[n^(k - n + 1), {k, 1, 11}, {n, k, 1, -1}] // Flatten (* Jean-François Alcover, Dec 14 2012 *)

T(n,k) = n^k \\ Charles R Greathouse IV, Feb 09 2017

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

More terms from James Sellers, Dec 11 1999

A108396 Triangle read by rows: T(n,k) = n*(1+n^k)/2, 0<=k<=n.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 3, 6, 15, 42, 4, 10, 34, 130, 514, 5, 15, 65, 315, 1565, 7815, 6, 21, 111, 651, 3891, 23331, 139971, 7, 28, 175, 1204, 8407, 58828, 411775, 2882404, 8, 36, 260, 2052, 16388, 131076, 1048580, 8388612, 67108868, 9, 45, 369, 3285, 29529, 265725

Offset: 0

Reinhard Zumkeller, Jun 02 2005

Row sums give A108397;
T(n,0) = A001477(n);
T(n,1) = A000217(n) for n>0;
T(n,2) = A006003(n) for n>1;
T(n,3) = A027441(n) for n>2;
T(n,4) = A021003(n) for n>3;
T(n,n) = A108398(n).

			.  0:  0
.  1:  1  1
.  2:  2  3   5
.  3:  3  6  15   42
.  4:  4 10  34  130   514
.  5:  5 15  65  315  1565   7815
.  6:  6 21 111  651  3891  23331  139971
.  7:  7 28 175 1204  8407  58828  411775  2882404
.  8:  8 36 260 2052 16388 131076 1048580  8388612  67108868
.  9:  9 45 369 3285 29529 265725 2391489 21523365 193710249 1743392205 .

Reinhard Zumkeller, Rows n = 0..124 of triangle, flattened

Cf. A079901, A000312, A033918, A001477, A000217, A006003, A027441, A021003, A108398, A108397 (row sums), A256512 (central terms).

a108396 n k = a108396_tabl !! n !! k
a108396_row n = a108396_tabl !! n
a108396_tabl = zipWith (\v ws -> map (flip div 2 . (* v) . (+ 1)) ws)
                       [0..] a079901_tabl
-- Reinhard Zumkeller, Mar 31 2015

Join[{0},Flatten[Table[n (1+n^k)/2,{n,10},{k,0,n}]]] (* Harvey P. Dale, Mar 19 2015 *)

Offset changed by Reinhard Zumkeller, Mar 31 2015

cp's OEIS Frontend

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

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