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A173122 Irregular triangle T(n) = coefficients of Sum_{k=0..n} t(n,k,q) for powers of q, where t(n,k,q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with t(n,0,q) = t(n,n,q) = 1, read by rows.

%I A173122 #9 Apr 29 2021 03:50:43
%S A173122 1,2,4,1,8,2,16,4,32,8,2,64,16,4,128,32,8,2,256,64,16,4,512,128,32,8,
%T A173122 2,1024,256,64,16,4,2048,512,128,32,8,2,4096,1024,256,64,16,4,8192,
%U A173122 2048,512,128,32,8,16384,4096,1024,256,64,16,32768,8192,2048,512,128,32,65536,16384,4096,1024,256,64
%N A173122 Irregular triangle T(n) = coefficients of Sum_{k=0..n} t(n,k,q) for powers of q, where t(n,k,q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with t(n,0,q) = t(n,n,q) = 1, read by rows.
%H A173122 G. C. Greubel, <a href="/A173122/b173122.txt">Rows n = 0..100 of the irregular triangle, flattened</a>
%F A173122 T(n) = coefficients of Sum_{k=0..n} t(n,k,q) for powers of q, where t(n,k,q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with t(n,0,q) = t(n,n,q) = 1.
%e A173122 Irregular triangle begins as:
%e A173122      1;
%e A173122      2;
%e A173122      4,   1;
%e A173122      8,   2;
%e A173122     16,   4;
%e A173122     32,   8,  2;
%e A173122     64,  16,  4;
%e A173122    128,  32,  8,  2;
%e A173122    256,  64, 16,  4;
%e A173122    512, 128, 32,  8, 2;
%e A173122   1024, 256, 64, 16, 4;
%t A173122 t[n_, k_, q_]:= If[k==0 || k==n, 1, q*Boole[n==2] + Sum[q^j*Binomial[n-2*j, k-j] *Boole[n>2*j], {j, 0, 5}]];
%t A173122 T[n_]:= CoefficientList[Series[Sum[t[n,k,q], {k,0,n}], {q,0,n}], q];
%t A173122 Table[T[n], {n, 0, 12}]//Flatten (* modified by _G. C. Greubel_, Apr 29 2021 *)
%o A173122 (Sage)
%o A173122 @CachedFunction
%o A173122 def t(n, k, x): return 1 if (k==0 or k==n) else x*bool(n==2) + sum( x^j*binomial(n-2*j, k-j)*bool(n>2*j) for j in (0..5) )
%o A173122 def s(n,x): return sum( t(n,k,x) for k in (0..n) )
%o A173122 flatten([taylor(s(n,x), x, 0, n).list() for n in (0..12)]) # _G. C. Greubel_, Apr 29 2021
%Y A173122 Cf. A007318, A072405, A173117, A173118, A173119, A173120.
%K A173122 nonn,tabf,easy,less
%O A173122 0,2
%A A173122 _Roger L. Bagula_, Feb 10 2010
%E A173122 More terms and edited by _G. C. Greubel_, Apr 29 2021