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A119727 Triangular array: T(n,k) = T(n,n) = 1, T(n,k) = 5*T(n-1, k-1) + 2*T(n-1, k), read by rows.

%I A119727 #7 Sep 08 2022 08:45:25
%S A119727 1,1,1,1,7,1,1,19,37,1,1,43,169,187,1,1,91,553,1219,937,1,1,187,1561,
%T A119727 5203,7969,4687,1,1,379,4057,18211,41953,49219,23437,1,1,763,10009,
%U A119727 56707,174961,308203,292969,117187,1,1,1531,23833,163459,633457,1491211,2126953,1699219,585937,1
%N A119727 Triangular array: T(n,k) = T(n,n) = 1, T(n,k) = 5*T(n-1, k-1) + 2*T(n-1, k), read by rows.
%C A119727 Second column is A048488. Second diagonal is A057651.
%D A119727 TERMESZET VILAGA XI.TERMESZET-TUDOMANY DIAKPALYAZAT 133.EVF. 6.SZ. jun. 2002. Vegh Lea (and Vegh Erika): "Pascal-tipusu haromszogek" http://www.kfki.hu/chemonet/TermVil/tv2002/tv0206/tartalom.html
%H A119727 G. C. Greubel, <a href="/A119727/b119727.txt">Rows n = 1..100 of triangle, flattened</a>
%e A119727 Triangle begins as:
%e A119727   1;
%e A119727   1,    1;
%e A119727   1,    7,     1;
%e A119727   1,   19,    37,      1;
%e A119727   1,   43,   169,    187,      1;
%e A119727   1,   91,   553,   1219,    937,       1;
%e A119727   1,  187,  1561,   5203,   7969,    4687,       1;
%e A119727   1,  379,  4057,  18211,  41953,   49219,   23437,       1;
%e A119727   1,  763, 10009,  56707, 174961,  308203,  292969,  117187,      1;
%e A119727   1, 1531, 23833, 163459, 633457, 1491211, 2126953, 1699219, 585937, 1;
%p A119727 T:= proc(n, k) option remember;
%p A119727       if k=1 and k=n then 1
%p A119727     else 5*T(n-1, k-1) + 2*T(n-1, k)
%p A119727       fi
%p A119727 end: seq(seq(T(n, k), k=1..n), n=1..12); # _G. C. Greubel_, Nov 18 2019
%t A119727 T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, 5*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 A119727 (PARI) T(n,k) = if(k==1 || k==n, 1, 5*T(n-1,k-1) + 2*T(n-1,k)); \\ _G. C. Greubel_, Nov 18 2019
%o A119727 (Magma)
%o A119727 function T(n,k)
%o A119727   if k eq 1 or k eq n then return 1;
%o A119727   else return 5*T(n-1,k-1) + 2*T(n-1,k);
%o A119727   end if;
%o A119727   return T;
%o A119727 end function;
%o A119727 [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Nov 18 2019
%o A119727 (Sage)
%o A119727 @CachedFunction
%o A119727 def T(n, k):
%o A119727     if (k==1 or k==n): return 1
%o A119727     else: return 5*T(n-1, k-1) + 2*T(n-1, k)
%o A119727 [[T(n, k) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Nov 18 2019
%Y A119727 Cf. A007318, A048488, A057651, A119725, A119726.
%K A119727 easy,nonn,tabl
%O A119727 1,5
%A A119727 _Zerinvary Lajos_, Jun 14 2006
%E A119727 Edited by _Don Reble_, Jul 24 2006