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%I #9 Apr 10 2022 23:20:39
%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,
%T 1,1,2,2,2,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,3,2,2,1,2,1,1,1,1,
%U 1,1,1,1,1,1,1,1,2,2,3,2,3,2,3,2,3,3,3,3,2,3,2,2,1,2,1,1,1,1,1,1,1
%N Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = x^(n-1)*p(n-1, x) + p(n-2, x) with p(0, x) = 1 and p(1, x) = 1 + x, read by rows.
%C Rows have A089071(n) number of terms, i.e., 0 <= k <= A089071(n) - 1, for n >= 0. - _G. C. Greubel_, Apr 07 2022
%H G. C. Greubel, <a href="/A172098/b172098.txt">Rows n = 0..30 of the irregular triangle, flattened</a>
%F T(n, k) = [x^k]( p(n, x) ), where p(n, x) = x^(n-1)*p(n-1, x) + p(n-2, x) with p(0, x) = 1 and p(1, x) = 1 + x.
%F Sum_{k=0..A089071(n)-1} T(n, k) = Fibonacci(n+2) (row sums). - _G. C. Greubel_, Apr 07 2022
%e Irregular triangle begins as:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 1, 1, 1, 1;
%e 1, 1, 1, 1, 1, 1, 1, 1;
%e 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1;
%e 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1;
%e 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1;
%t p[n_, x_]:= p[n, x]= If[n<2, n*x+1, x^(n-1)*p[n-1, x] + p[n-2, x]];
%t Table[CoefficientList[p[n, x], x], {n,0,12}]//Flatten
%o (SageMath)
%o @CachedFunction
%o def p(n,x):
%o if (n<2): return n*x+1
%o else: return x^(n-1)*p(n-1, x) + p(n-2, x)
%o def T(n,k): return ( p(n,x) ).series(x, (n^2-n+4-2*0^n)/2+1).list()[k]
%o flatten([[T(n,k) for k in (0..(n^2-n+2-2*0^n)/2)] for n in (0..12)]) # _G. C. Greubel_, Apr 07 2022
%Y Cf. A000045, A089071, A172099.
%K nonn,tabf
%O 0,24
%A _Roger L. Bagula_, Jan 25 2010
%E Edited by _G. C. Greubel_, Apr 07 2022
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