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A164981 A triangle with Pell numbers in the first column.

%I A164981 #30 Aug 06 2023 05:12:05
%S A164981 1,2,1,5,3,1,12,10,4,1,29,30,16,5,1,70,87,56,23,6,1,169,245,185,91,31,
%T A164981 7,1,408,676,584,334,136,40,8,1,985,1836,1784,1158,546,192,50,9,1,
%U A164981 2378,4925,5312,3850,2052,834,260,61,10,1,5741,13079,15497,12386,7342,3366,1212,341,73,11,1
%N A164981 A triangle with Pell numbers in the first column.
%C A164981 Rows sum up to A000244 (powers of 3), diagonals to A001654 (golden rectangles).
%C A164981 Up to reflection at the vertical axis, the triangle of numbers given here coincides with the triangle given in A210557, i.e. the numbers are the same just read row-wise in the opposite direction. [_Christine Bessenrodt_, Jul 20 2012]
%C A164981 Subtriangle of (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 10 2013
%F A164981 From _R. J. Mathar_, Jan 27 2011: (Start)
%F A164981 T(1,1) =1. T(n,k)=0 if n<1 or k<1 or k>n. T(n,k) = 2*T(n-1,k)+T(n-1,k-1)+T(n-2,k)-T(n-2,k-1) otherwise.
%F A164981 T(n,1) = A000129(n).
%F A164981 T(n,n-1) = n.
%F A164981 T(n,n-2) = A052905(n-2).
%F A164981 T(n,2) = A026937(n-2). (End)
%F A164981 G.f. x*y/(1-2*x-x^2+x^2*y-x*y). - _R. J. Mathar_, Aug 11 2015
%e A164981 Triangle begins
%e A164981   1
%e A164981   2,1
%e A164981   5,3,1
%e A164981   12,10,4,1
%e A164981   29,30,16,5,1
%e A164981   70,87,56,23,6,1
%e A164981   169,245,185,91,31,7,1
%e A164981   ...
%e A164981 From _Philippe Deléham_, Oct 10 2013: (Start)
%e A164981 Triangle (0, 2, 1/2, -1/2, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, ...):
%e A164981   1
%e A164981   0, 1
%e A164981   0, 2, 1
%e A164981   0, 5, 3, 1
%e A164981   0, 12, 10, 4, 1
%e A164981   0, 29, 30, 16, 5, 1
%e A164981   0, 70, 87, 56, 23, 6, 1
%e A164981   0, 169, 245, 185, 91, 31, 7, 1
%e A164981   ... (End)
%p A164981 A164981 := proc(n,k) option remember; if n <1 or k<1 or k>n then 0; elif n = 1 then 1; else 2*procname(n-1,k)+procname(n-1,k-1)+procname(n-2,k)-procname(n-2,k-1) ; end if; end proc:
%t A164981 T[n_, k_] := T[n, k] = Which[n < 1 || k < 1 || k > n, 0, n == 1, 1, True, 2*T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1]];
%t A164981 Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 06 2023 *)
%o A164981 (PARI) T(n,k) = if ((n==1) && (k==1), return(1)); if ((n<=0) || (k<=0) || (n<k), return(0)); 2*T(n-1,k)+T(n-1,k-1)+T(n-2,k)-T(n-2,k-1); \\ _Michel Marcus_, Feb 01 2023
%Y A164981 Cf. A000244, A001654, A164975, A164976, A210557.
%K A164981 nonn,tabl
%O A164981 1,2
%A A164981 _Mark Dols_, Sep 03 2009
%E A164981 Rows 10-11 from _Michel Marcus_, Feb 01 2023