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A371764 Triangle read by rows: Trace of the Akiyama-Tanigawa algorithm for powers x^3.

%I A371764 #17 Apr 21 2024 19:12:53
%S A371764 0,1,1,13,14,8,73,86,57,27,301,374,273,148,64,1081,1382,1065,628,305,
%T A371764 125,3613,4694,3729,2308,1205,546,216,11593,15206,12297,7828,4265,
%U A371764 2058,889,343,36301,47894,39153,25348,14165,7098,3241,1352,512
%N A371764 Triangle read by rows: Trace of the Akiyama-Tanigawa algorithm for powers x^3.
%C A371764 See the comments in A371763.
%F A371764 T(n, k) = n^3 if n=k, otherwise 2*binomial(k + 3, k)*3^(n - k + 1) - 6*binomial(k + 2, k)*2^(n - k + 1) + 7*(k + 1). - _Detlef Meya_, Apr 21 2024
%e A371764 Triangle starts:
%e A371764 0:                        0
%e A371764 1:                      1,  1
%e A371764 2:                   13,  14, 8
%e A371764 3:                 73, 86,  57, 27
%e A371764 4:             301, 374, 273, 148, 64
%e A371764 5:        1081, 1382, 1065, 628, 305, 125
%e A371764 6:     3613, 4694, 3729, 2308, 1205, 546, 216
%e A371764 7: 11593, 15206, 12297, 7828, 4265, 2058, 889, 343
%p A371764 # Using function ATPtriangle from A371763.
%p A371764 ATPtriangle(3, 9);
%p A371764 # Or, after _Detlef Meya_:
%p A371764 T := (n,k) -> (k+1)*(7-(k+2)*(3*2^(n-k+1)-(k+3)*3^(n-k)))-`if`(n=k,1,0):
%p A371764 seq(seq(T(n, k), k = 0..n), n = 0..8);  # _Peter Luschny_, Apr 21 2024
%t A371764 T[n_, k_] := If[n==k, n^3, 2*Binomial[k + 3, k]*3^(n - k + 1) - 6*Binomial[k + 2, k]*2^(n - k + 1) + 7*(k + 1)]; Flatten[Table[T[n, k],{n, 0, 8},{k, 0, n}]] (* _Detlef Meya_, Apr 21 2024 *)
%o A371764 (Python) # See function ATPowList in A371761.
%o A371764 (Julia)  # Using function ATPtriangle from A371763.
%o A371764 ATPtriangle(3, 8)
%Y A371764 Cf. A371761, A371763.
%Y A371764 Cf. A006230 (left edge).
%K A371764 nonn,tabl
%O A371764 0,4
%A A371764 _Peter Luschny_, Apr 15 2024