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A173008 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n, column 0<=k<=n, and q = 4.

%I A173008 #25 Feb 20 2021 23:12:23
%S A173008 1,4,1,64,20,1,4096,1344,84,1,1048576,348160,22848,340,1,1073741824,
%T A173008 357564416,23744512,371008,1364,1,4398046511104,1465657589760,
%U A173008 97615085568,1543393280,5957952,5460,1,72057594037927936,24017731997138944,1600791219535872,25384570585088,99158478848,95414592,21844,1
%N A173008 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n, column 0<=k<=n, and q = 4.
%C A173008 Row sums are 1, 5, 85, 5525, 1419925, 1455423125, 5962868543125, 97701601079103125, 6403069829921181503125, ... (partial products of A092896).
%C A173008 Triangle T(n,k), read by rows, given by [4,12,64,240,1024,4032,16384,...] DELTA [1,0,4,0,16,0,64,0,256,0,1024,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 01 2011
%H A173008 Robert Israel, <a href="/A173008/b173008.txt">Table of n, a(n) for n = 0..1430</a>
%F A173008 T(n,k) = 4^n*T(n-1,k) + T(n-1,k-1) with T(0,0)=1. - _Philippe Deléham_, Oct 01 2011
%F A173008 Sum_{k=0..n} T(n, k, 4) = A309327(n+1). - _G. C. Greubel_, Feb 20 2021
%e A173008 Triangle begins as:
%e A173008               1;
%e A173008               4,             1;
%e A173008              64,            20,           1;
%e A173008            4096,          1344,          84,          1;
%e A173008         1048576,        348160,       22848,        340,       1;
%e A173008      1073741824,     357564416,    23744512,     371008,    1364,    1;
%e A173008   4398046511104, 1465657589760, 97615085568, 1543393280, 5957952, 5460, 1;
%p A173008 P:= 1: A:= 1:
%p A173008 for n from 1 to 12 do
%p A173008   P:= expand(P*(x+4^n));
%p A173008   A:= A, seq(coeff(P,x,j),j=0..n)
%p A173008 od:
%p A173008 A; # _Robert Israel_, Aug 12 2015
%t A173008 (* First program *)
%t A173008 p[x_, n_, q_]= If[n==0, 1, Product[x + q^i, {i,n}]];
%t A173008 Table[CoefficientList[p[x, n, 4], x], {n, 0, 10}]//Flatten (* modified by _G. C. Greubel_, Feb 20 2021 *)
%t A173008 (* Second program *)
%t A173008 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 A173008 Table[T[n,k,4], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 20 2021 *)
%o A173008 (Sage)
%o A173008 def T(n, k, q):
%o A173008     if (k<0 or k>n): return 0
%o A173008     elif (k==n): return 1
%o A173008     else: return q^n*T(n-1,k,q) + T(n-1,k-1,q)
%o A173008 flatten([[T(n,k,4) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Feb 20 2021
%o A173008 (Magma)
%o A173008 function T(n,k,q)
%o A173008   if k lt 0 or k gt n then return 0;
%o A173008   elif k eq n then return 1;
%o A173008   else return q^n*T(n-1,k,q) + T(n-1,k-1,q);
%o A173008   end if; return T; end function;
%o A173008 [T(n,k,4): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Feb 20 2021
%Y A173008 Cf. A023531 (q=0), A007318 (q=1), A108084 (q=2), A173007 (q=3), this sequence (q=4).
%Y A173008 Cf. A092896, A108084, A309327.
%K A173008 nonn,tabl,easy
%O A173008 0,2
%A A173008 _Roger L. Bagula_, Feb 07 2010