This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A111049 #25 Sep 06 2024 05:07:32
%S A111049 1,1,1,1,3,2,1,6,9,4,1,11,27,25,8,1,20,70,100,65,16,1,37,170,330,325,
%T A111049 161,32,1,70,399,980,1295,966,385,64,1,135,917,2723,4515,4501,2695,
%U A111049 897,128,1,264,2076,7224,14406,17976,14364,7176,2049,256
%N A111049 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
%F A111049 T(n, k) = 2^(n-1)binomial(n-1, k-1) + binomial(n-1, k).
%F A111049 Sum_{k=0..n} T(n, k) = 2^(n-1)*(1+2^(n-1)) = A063376(n-1) for n >= 1.
%F A111049 From _Peter Bala_, Mar 20 2013: (Start)
%F A111049 O.g.f.: (1 - 2*t + x*t*(t-2) + x^2*t^2)/((1 - t*(1+x))*(1 - 2*t*(1+x))) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2 + ....
%F A111049 E.g.f.: (x + 2*exp((1+x)*t) + x*exp(2*t*(1+x)))/(2*(1+x)) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2/2! + ....
%F A111049 Recurrence equation: for n >= 1, T(n+1,k) = 2*T(n,k) + 2*T(n,k-1) - binomial(n,k). (End)
%F A111049 From _Philippe Deléham_, Oct 18 2013: (Start)
%F A111049 G.f.: (1 - 2*x - 2*x*y + x^2*y + x^2*y^2)/(1 - 3*x - 3*x*y + 2*x^2 + 4*x^2*y + 2*x^2*y^2).
%F A111049 T(n,k) = 3*T(n-1,k) + 3*T(n-1,k-1) - 2*T(n-2,k) - 4*T(n-2,k-1) - 2*T(n-2,k-2), T(0,0) = T(1,1) = T(1,0) = T(2,0) = 1, T(2,1) = 3, T(2,2) = 2, T(n,k) = 0 if k > n or if k < 0. (End)
%e A111049 Rows begin:
%e A111049 1;
%e A111049 1, 1;
%e A111049 1, 3, 2;
%e A111049 1, 6, 9, 4;
%e A111049 1, 11, 27, 25, 8;
%e A111049 1, 20, 70, 100, 65, 16;
%e A111049 1, 37, 170, 330, 325, 161, 32;
%e A111049 1, 70, 399, 980, 1295, 966, 385, 64;
%e A111049 1, 135, 917, 2723, 4515, 4501, 2695, 897, 128;
%e A111049 1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049, 256;
%t A111049 With[{m = 9}, CoefficientList[CoefficientList[Series[(1 - 2*x - 2*x*y + x^2 *y + x^2*y^2)/(1 - 3*x - 3*x*y + 2*x^2 + 4*x^2*y + 2*x^2*y^2), {x, 0 , m}, {y, 0, m} ], x], y]] // Flatten (* _Georg Fischer_, Feb 17 2020 *)
%o A111049 (PARI) T(n, k) = if (k<=n, 2^(n-1)*binomial(n-1, k-1)+binomial(n-1, k));
%o A111049 matrix(10, 10, n, k, T(n-1, k-1)) \\ to see the triangle \\ _Michel Marcus_, Feb 17 2020
%Y A111049 Cf. A002064, A006127.
%K A111049 easy,nonn,tabl
%O A111049 0,5
%A A111049 _Philippe Deléham_, Oct 07 2005
%E A111049 Wrong a(42) removed by _Georg Fischer_, Feb 17 2020