A198062 - OEIS

%I #15 Mar 30 2017 06:48:05

%S 1,0,1,0,1,1,0,1,1,1,0,1,1,2,1,0,1,1,4,2,1,0,1,1,8,3,2,1,0,1,1,16,4,4,

%T 3,1,0,1,1,32,5,8,9,3,1,0,1,1,64,6,16,27,7,3,1,0,1,1,128,7,32,81,15,7,

%U 3,1,0,1,1,256,8,64,243,31,15,9,4,1,0,1,1,512,9

%N Array read by antidiagonals, m>=0, n>=0, k>=0, A(m, n, k) = sum{j=0..m} sum{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j).

%H Vincenzo Librandi, <a href="/A198062/b198062.txt">Table of n, a(n) for n = 0..350</a>

%F A007318(n,k) = A(0,n+1,k+1)*C(n,k)^1/(k+1)^0,

%F A103371(n,k) = A(1,n+1,k+1)*C(n,k)^2/(k+1)^1,

%F A194595(n,k) = A(2,n+1,k+1)*C(n,k)^3/(k+1)^2,

%F A197653(n,k) = A(3,n+1,k+1)*C(n,k)^4/(k+1)^3,

%F A197654(n,k) = A(4,n+1,k+1)*C(n,k)^5/(k+1)^4,

%F A197655(n,k) = A(5,n+1,k+1)*C(n,k)^6/(k+1)^5.

%e    [0] [1] [2]  [3] [4]  [5]  [6]  [7]  [8]  [9]

%e -------------------------------------------------

%e [0]  1   1   1    1   1    1    1    1    1    1    A000012

%e [1]  0   1   1    2   2    2    3    3    3    3    A003056

%e [2]  0   1   1    4   3    4    9    7    7    9    A073254

%e [3]  0   1   1    8   4    8   27   15   15   27    A198063

%e [4]  0   1   1   16   5   16   81   31   31   81    A198064

%e [5]  0   1   1   32   6   32  243   63   63  243    A198065

%p A198062_RowAsTriangle := proc(m) local pow; pow :=(a,b)->`if`(a=0 and b=0,1,a^b): proc(n, k) local i, j; add(add((-1)^(j + i)*binomial(i, j)*pow(n, j)* pow(k, m-j), i=0..m),j=0..m) end: end:

%p for m from 0 to 2 do seq(print(seq(A198062_RowAsTriangle(m)(n,k),k=0..n)),n=0..5) od;

%t max = 9; RowAsTriangle[m_][n_, k_] := Module[{pow}, pow[a_, b_] := If[a == 0 && b == 0, 1, a^b]; Module[{i, j}, Sum[Sum[(-1)^(j+i)*Binomial[i, j]*pow[n, j]*pow[k, m-j], {i, 0, m}], {j, 0, m}]]]; t = Flatten /@ Table[RowAsTriangle[m][n, k], {m, 0, max}, {n, 0, max}, {k, 0, n}]; Table[t[[n-k+1, k+1]], {n, 0, max}, {k, 0, n }] // Flatten (* _Jean-François Alcover_, Feb 25 2014, after Maple *)

%Y Cf. A198060, A198061.

%K nonn,tabl

%O 0,14

%A _Peter Luschny_, Nov 02 2011