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A335436 Triangle read by rows: T(n,k) = 2*n+1 for k = 0 and otherwise T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j).

%I A335436 #92 Dec 21 2024 02:35:24
%S A335436 1,3,4,5,8,21,7,12,35,96,9,16,49,144,410,11,20,63,192,574,1680,13,24,
%T A335436 77,240,738,2240,6692,15,28,91,288,902,2800,8604,26112,17,32,105,336,
%U A335436 1066,3360,10516,32640,100296,19,36,119,384,1230,3920,12428,39168,122584,380480
%N A335436 Triangle read by rows: T(n,k) = 2*n+1 for k = 0 and otherwise T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j).
%H A335436 Oboifeng Dira, <a href="/A335436/b335436.txt">Table of n, a(n) for n = 0..54</a>
%F A335436 T(n,0) = 2*n+1 for k=0;
%F A335436 T(n,k) = ((2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1))*(2*n-k+1)/(4*sqrt(2)) for 1<=k<=n.
%e A335436 Triangle begins:
%e A335436   1;
%e A335436   3,  4;
%e A335436   5,  8, 21;
%e A335436   7, 12, 35,  96;
%e A335436   9, 16, 49, 144, 410;
%e A335436   ...
%e A335436 T(3,2) = ((2+sqrt(2))^3-(2-sqrt(2))^3)*(6-2+1)/(4*sqrt(2)) = (28*sqrt(2))*(5)/(4*sqrt(2)) = 35.
%p A335436 T := proc (n, k) if k = 0 and 0 <= n then 2*n+1 elif 1 <= k and k <= n then round((((2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1))*(2*n-k+1)/(4*sqrt(2)))) else 0 end if end proc:seq(print(seq(T(n, k), k=0..n)), n=0..9);
%o A335436 (PARI) T(n,k) = if (k==0, 2*n+1, if (k<=n, sum(i=n-k, n, sum(j=0, i-n+k, if ((i==n) && (j==k), 0, T(i,j)), 0))));
%o A335436 matrix(10, 10, n, k, T(n-1, k-1)) \\ _Michel Marcus_, Sep 08 2020
%o A335436 (PARI) T(n, k) = if (k==0, 2*n+1, if (k>n, 0, my(w=quadgen(8, 'w)); ((2+w)^(k+1)-(2-w)^(k+1))*(2*n-k+1)/(4*w)));
%o A335436 matrix(10, 10, n, k, T(n-1, k-1)) \\ _Michel Marcus_, Sep 10 2020
%Y A335436 Cf. A053199, A294285, A064339.
%K A335436 nonn,tabl
%O A335436 0,2
%A A335436 _Oboifeng Dira_, Jul 14 2020