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%I #18 Dec 01 2021 01:17:57
%S 1,1,0,1,2,0,2,0,3,6,0,6,0,15,0,7,24,0,24,0,84,0,52,0,37,120,0,120,0,
%T 540,0,380,0,485,0,141,720,0,720,0,3960,0,3000,0,5430,0,2406,0,1111,
%U 5040,0,5040,0,32760,0,26040,0,60690,0,32802,0,28147,0,5923
%N Expansion of e.g.f.: exp(t*x)/(1 - x/t - t^2 * x^2).
%H G. C. Greubel, <a href="/A158706/b158706.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%F T(n, k) = coefficients of the expansion : p(x,t) = exp(t*x)/(1 - x/t - t^2* x^2).
%F T(n, k) = coefficients of the series : Sum_{s=0..n} Sum_{j=0..floor(s/2)} (n!/(n-s)!)*A011973(s, j)*x^(4*j+2*n-2*s). - _G. C. Greubel_, Nov 30 2021
%e Irregular triangle begins as:
%e 1;
%e 1, 0, 1;
%e 2, 0, 2, 0, 3;
%e 6, 0, 6, 0, 15, 0, 7;
%e 24, 0, 24, 0, 84, 0, 52, 0, 37;
%e 120, 0, 120, 0, 540, 0, 380, 0, 485, 0, 141;
%e 720, 0, 720, 0, 3960, 0, 3000, 0, 5430, 0, 2406, 0, 1111;
%e 5040, 0, 5040, 0, 32760, 0, 26040, 0, 60690, 0, 32802, 0, 28147, 0, 5923;
%t (* First program *)
%t Table[CoefficientList[n!*t^n*SeriesCoefficient[Series[Exp[t*x]/(1 -x/t -t^2*x^2), {x,0,20}], n], t], {n,0,10}]//Flatten
%t (* Second program *)
%t Table[CoefficientList[Series[Sum[Sum[GegenbauerC[k, (s+1)/2 -k, 1]*x^(4*k+2*n - 2*s)*(n!/(n-s)!), {k,0,Floor[s/2]}], {s,0,n}], {x,0,20}], x], {n,0,10}] (* _G. C. Greubel_, Nov 30 2021 *)
%o (Sage)
%o @CachedFunction
%o def A011973(n,k): return 0 if (k<0 or k>(n//2)) else binomial(n-k, k)
%o def f(n,x): return sum( sum( (A011973(s,j)/factorial(n-s))*x^(4*j+2*n-2*s) for j in (0..(s//2)) ) for s in (0..n) )
%o def A158706(n,k): return factorial(n)*( f(n,x) ).series(x,2*n+1).list()[k]
%o flatten([[A158706(n,k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, Nov 30 2021
%Y Cf. A011973, A110313 (row sums).
%K nonn,tabf
%O 0,5
%A _Roger L. Bagula_, Mar 24 2009
%E Edited by _G. C. Greubel_, Nov 30 2021
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