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%I #41 May 20 2021 23:04:57
%S 1,1,1,1,2,1,1,4,3,1,1,8,9,4,1,1,16,27,16,5,1,1,32,81,64,25,6,1,1,64,
%T 243,256,125,36,7,1,1,128,729,1024,625,216,49,8,1,1,256,2187,4096,
%U 3125,1296,343,64,9,1,1,512,6561,16384,15625,7776,2401,512,81,10,1
%N Triangle in which j-th entry in i-th row is (j+1)^(i-j).
%C Read as a square array this is the Hilbert transform of triangle A123125 (see A145905 for the definition of this term). For example, the fourth row of A123125 is (0,1,4,1) and the expansion (x + 4*x^2 + x^3)/(1-x)^4 = x + 8*x^2 + 27*x^3 + 64*x^4 + ... generates the entries in the fourth row of this array read as a square. - _Peter Bala_, Oct 28 2008
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 24.
%H T. D. Noe, <a href="/A009998/b009998.txt">Rows n=0..50 of triangle, flattened</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/FigurateNumber">Figurate number - a very short introduction</a>
%F T(n,n) = 1; T(n,k) = (k+1)*T(n-1,k) for k=0..n-1. - _Reinhard Zumkeller_, Feb 02 2014
%F T(n,m) = (m+1)*Sum_{k=0..n-m}((n+1)^(k-1)*(n-m)^(n-m-k)*(-1)^(n-m-k)*binomial(n-m-1,k-1)). - _Vladimir Kruchinin_, Sep 12 2015
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 4, 3, 1;
%e 1, 8, 9, 4, 1;
%e 1, 16, 27, 16, 5, 1;
%e 1, 32, 81, 64, 25, 6, 1;
%e ...
%e From _Gus Wiseman_, May 01 2021: (Start)
%e The rows of the triangle are obtained by reading antidiagonals upward in the following table of A(k,n) = n^k, with offset k = 0, n = 1:
%e n=1: n=2: n=3: n=4: n=5: n=6:
%e k=0: 1 1 1 1 1 1
%e k=1: 1 2 3 4 5 6
%e k=2: 1 4 9 16 25 36
%e k=3: 1 8 27 64 125 216
%e k=4: 1 16 81 256 625 1296
%e k=5: 1 32 243 1024 3125 7776
%e k=6: 1 64 729 4096 15625 46656
%e k=7: 1 128 2187 16384 78125 279936
%e k=8: 1 256 6561 65536 390625 1679616
%e k=9: 1 512 19683 262144 1953125 10077696
%e k=10: 1 1024 59049 1048576 9765625 60466176
%e (End)
%p E := (n,x) -> `if`(n=0,1,x*(1-x)*diff(E(n-1,x),x)+E(n-1,x)*(1+(n-1)*x));
%p G := (n,x) -> E(n,x)/(1-x)^(n+1);
%p A009998 := (n,k) -> coeff(series(G(n-k,x),x,18),x,k);
%p seq(print(seq(A009998(n,k),k=0..n)),n=0..6);
%p # _Peter Luschny_, Aug 02 2010
%t Flatten[Table[(j+1)^(i-j),{i,0,20},{j,0,i}]] (* _Harvey P. Dale_, Dec 25 2012 *)
%o (Haskell)
%o a009998 n k = (k + 1) ^ (n - k)
%o a009998_row n = a009998_tabl !! n
%o a009998_tabl = map reverse a009999_tabl
%o -- _Reinhard Zumkeller_, Feb 02 2014
%o (PARI) T(i,j)=(j+1)^(i-j) \\ _Charles R Greathouse IV_, Feb 06 2017
%Y Row sums give A026898.
%Y Cf. A088956, A123125, A179927.
%Y Column n = 2 of the array is A000079.
%Y Column n = 3 of the array is A000244.
%Y Row k = 2 of the array is A000290.
%Y Row k = 3 of the array is A000578.
%Y Diagonal n = k of the array is A000312.
%Y Diagonal n = k + 1 of the array is A000169.
%Y Diagonal n = k + 2 of the array is A000272.
%Y The transpose of the array is A009999.
%Y The numbers of divisors of the entries are A343656 (row sums: A343657).
%Y A007318 counts k-sets of elements of {1..n}.
%Y A059481 counts k-multisets of elements of {1..n}.
%Y Cf. A002109, A062319, A066959, A143773, A176029, A204688, A326358, A327527.
%K tabl,nonn,easy,nice
%O 0,5
%A _N. J. A. Sloane_
%E a(62) corrected to 512 by _T. D. Noe_, Dec 20 2007
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