A201972 - OEIS

%I #20 Feb 18 2020 01:18:41

%S 1,2,2,5,8,3,12,28,20,4,29,88,94,40,5,70,262,372,244,70,6,169,752,

%T 1333,1184,539,112,7,408,2104,4472,5016,3144,1064,168,8,985,5776,

%U 14316,19408,15526,7344,1932,240,9

%N Triangle T(n,k), read by rows, given by (2,1/2,-1/2,0,0,0,0,0,0,0,...) DELTA (2,-1/2,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

%C Diagonal sums: A201967(n), row sums: A000302(n) (powers of 4).

%H G. C. Greubel, <a href="/A201972/b201972.txt">Rows n = 0..100 of triangle, flattened</a>

%F G.f.: 1/(1-2*(y+1)*x+(y+1)*(y-1)*x^2).

%F Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000129(n+1), A000302(n), A138395(n), A057084(n) for x = -1, 0, 1, 2, 3, respectively.

%F Sum_{k=0..n} T(n,k)*x^(n-k) = A000027(n), A000302(n), A090018(n), A057091(n) for x = 0, 1, 2, 3, respectively.

%F T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.

%e Triangle begins:

%e     1;

%e     2,   2;

%e     5,   8,    3;

%e    12,  28,   20,    4;

%e    29,  88,   94,   40,   5;

%e    70, 262,  372,  244,  70,   6;

%e   169, 752, 1333, 1184, 539, 112, 7;

%p T:= proc(n, k) option remember;

%p       if k=0 and n=0 then 1

%p     elif k<0 or  k>n  then 0

%p     else 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2)

%p       fi; end:

%p seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Feb 17 2020

%t With[{m = 8}, CoefficientList[CoefficientList[Series[1/(1-2*(y+1)*x+(y+1)*(y-1)*x^2), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* _Georg Fischer_, Feb 17 2020 *)

%o (PARI) T(n,k) = if(n<k,0,if(k<0,0,if((n==0)&&(k==0),1,2*T(n-1,k)+2*T(n-1,k-1)+T(n-2,k)-T(n-2,k-2))));

%o matrix(10, 10, n, k, T(n-1,k-1)) \\ to see the triangle \\ _Michel Marcus_, Feb 17 2020

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o     if (k<0 or k>n): return 0

%o     elif (k==0 and n==0): return 1

%o     else: return 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2)

%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Feb 17 2020

%Y Cf. A000027, A000129, A006645, A007290.

%K nonn,tabl

%O 0,2

%A _Philippe Deléham_, Dec 07 2011

%E a(40) corrected by _Georg Fischer_, Feb 17 2020