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Catalan numbers as convergents of power series.

MatsGranvik Nov 25th, 2012 (edited) 260 Never
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  1. Mathematica:
  2.  
  3. (*program start*)
  4. (*coefficients (coeff) in power series can be changed*)
  5. Clear[t, n, k, i, nn, x];
  6. coeff = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
  7.   1}; mp[m_, e_] :=
  8.  If[e == 0, IdentityMatrix@Length@m, MatrixPower[m, e]]; nn =
  9.  Length[coeff]; cc = Range[nn]*0 + 1; Monitor[
  10.  Do[Clear[t]; t[n_, 1] := t[n, 1] = cc[[n]];
  11.   t[n_, k_] :=
  12.    t[n, k] =
  13.     If[n >= k,
  14.      Sum[t[n - i, k - 1], {i, 1, 2 - 1}] -
  15.       0*Sum[t[n - i, k], {i, 1, k - 1}], 0];
  16.   A4 = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
  17.   A5 = A4[[1 ;; nn - 1]]; A5 = Prepend[A5, ConstantArray[0, nn]];
  18.   cc = Total[
  19.     Table[coeff[[n]]*mp[A5, n - 1][[All, 1]], {n, 1, nn}]];, {i, 1,
  20.    nn}], i]; cc
  21. (*Mats Granvik,Jul 11 2015*)
  22. (*program end*)
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