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A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).

%I A108084 #25 May 04 2025 16:09:18
%S A108084 1,2,1,8,6,1,64,56,14,1,1024,960,280,30,1,32768,31744,9920,1240,62,1,
%T A108084 2097152,2064384,666624,89280,5208,126,1,268435456,266338304,87392256,
%U A108084 12094464,755904,21336,254,1,68719476736,68451041280,22638755840,3183575040,205605888,6217920,86360,510,1
%N A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).
%C A108084 Triangle T(n,k), 0 <= k <= n, read by rows given by [2, 2, 8, 12, 32, 56, 128, 240, 512, ...] DELTA [1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, ...] = A014236 (first zero omitted) DELTA A077957 where DELTA is the operator defined in A084938. - _Philippe Deléham_, Aug 23 2006
%H A108084 G. C. Greubel, <a href="/A108084/b108084.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A108084 Sum_{k=0..n} T(n, k) = A028362(n).
%F A108084 T(n,0) = 2^(n*(n+1)/2) = A006125(n+1). - _Philippe Deléham_, Nov 05 2006
%F A108084 T(n,k) = 2^binomial(n+1-k,2) * A022166(n,k) for 0 <= k <= n. - _Werner Schulte_, Mar 25 2019
%e A108084 Triangle begins:
%e A108084       1;
%e A108084       2,     1;
%e A108084       8,     6,    1;
%e A108084      64,    56,   14,    1;
%e A108084    1024,   960,  280,   30,  1;
%e A108084   32768, 31744, 9920, 1240, 62, 1;
%t A108084 T[n_, k_, q_]:= T[n,k,q]= If[k<0 || k>n, 0, If[k==n, 1, q^n*T[n-1,k,q] +T[n-1,k-1,q] ]];
%t A108084 Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 20 2021 *)
%o A108084 (Sage)
%o A108084 def T(n, k, q):
%o A108084     if (k<0 or k>n): return 0
%o A108084     elif (k==n): return 1
%o A108084     else: return q^n*T(n-1,k,q) + T(n-1,k-1,q)
%o A108084 flatten([[T(n,k,2) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Feb 20 2021
%o A108084 (Magma)
%o A108084 function T(n,k,q)
%o A108084   if k lt 0 or k gt n then return 0;
%o A108084   elif k eq n then return 1;
%o A108084   else return q^n*T(n-1,k,q) + T(n-1,k-1,q);
%o A108084   end if; return T; end function;
%o A108084 [T(n,k,2): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Feb 20 2021
%Y A108084 Cf. A023531 (q=0), A007318 (q=1), this sequence (q=2), A173007 (q=3), A173008 (q=4).
%Y A108084 Cf. A006125, A022166, A028362.
%K A108084 nonn,tabl
%O A108084 0,2
%A A108084 _Gerald McGarvey_, Jun 05 2005