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A119725 Triangular array read by rows: T(n,1) = T(n,n) = 1, T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k).

%I A119725 #18 Sep 08 2022 08:45:25
%S A119725 1,1,1,1,5,1,1,13,17,1,1,29,73,53,1,1,61,233,325,161,1,1,125,649,1349,
%T A119725 1297,485,1,1,253,1673,4645,6641,4861,1457,1,1,509,4105,14309,27217,
%U A119725 29645,17497,4373,1,1,1021,9737,40933,97361,140941,123929,61237,13121,1
%N A119725 Triangular array read by rows: T(n,1) = T(n,n) = 1, T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k).
%C A119725 Second column is like A036563.
%C A119725 Second diagonal is A048473.
%H A119725 G. C. Greubel, <a href="/A119725/b119725.txt">Rows n = 1..100 of triangle, flattened</a>
%H A119725 Termeszet Vilaga A XI. Természet-Tudomány Diákpályázat díjnyertesei 133.EVF. 6.SZ. jun. 2002. Vegh Lea (and Vegh Erika): <a href="http://www.termeszetvilaga.hu/tv2002/tv0206/tartalom.html">Pascal-tipusu haromszogek</a>
%e A119725 Triangle begins:
%e A119725   1;
%e A119725   1,    1;
%e A119725   1,    5,    1;
%e A119725   1,   13,   17,     1;
%e A119725   1,   29,   73,    53,     1;
%e A119725   1,   61,  233,   325,   161,      1;
%e A119725   1,  125,  649,  1349,  1297,    485,      1;
%e A119725   1,  253, 1673,  4645,  6641,   4861,   1457,     1;
%e A119725   1,  509, 4105, 14309, 27217,  29645,  17497,  4373,     1;
%e A119725   1, 1021, 9737, 40933, 97361, 140941, 123929, 61237, 13121, 1;
%p A119725 T:= proc(n, k) option remember;
%p A119725       if k=1 and k=n then 1
%p A119725     else 3*T(n-1, k-1) + 2*T(n-1, k)
%p A119725       fi
%p A119725     end:
%p A119725 seq(seq(T(n, k), k=1..n), n=1..12); # _G. C. Greubel_, Nov 18 2019
%t A119725 T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, 3*T[n-1, k-1] + 2*T[n-1, k]]; Table[T[n,k], {n,10}, {k,n}]//Flatten (* _G. C. Greubel_, Nov 18 2019 *)
%o A119725 (PARI) T(n,k) = if(k==1 || k==n, 1, 3*T(n-1,k-1) + 2*T(n-1,k)); \\ _G. C. Greubel_, Nov 18 2019
%o A119725 (Magma)
%o A119725 function T(n,k)
%o A119725   if k eq 1 or k eq n then return 1;
%o A119725   else return 3*T(n-1,k-1) + 2*T(n-1,k);
%o A119725   end if;
%o A119725   return T;
%o A119725 end function;
%o A119725 [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Nov 18 2019
%o A119725 (Sage)
%o A119725 @CachedFunction
%o A119725 def T(n, k):
%o A119725     if (k==1 or k==n): return 1
%o A119725     else: return 3*T(n-1, k-1) + 2*T(n-1, k)
%o A119725 [[T(n, k) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Nov 18 2019
%Y A119725 Cf. A007318, A036563, A048473, A119726, A119727.
%K A119725 easy,nonn,tabl
%O A119725 1,5
%A A119725 _Zerinvary Lajos_, Jun 14 2006
%E A119725 Edited by _Don Reble_, Jul 24 2006