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 A323211 #18 Sep 26 2024 08:56:55
%S A323211 1,1,1,1,2,1,1,2,2,1,1,2,3,2,1,1,2,4,4,2,1,1,2,5,7,5,2,1,1,2,6,11,11,
%T A323211 6,2,1,1,2,7,16,21,16,7,2,1,1,2,8,22,36,36,22,8,2,1,1,2,9,29,57,71,57,
%U A323211 29,9,2,1,1,2,10,37,85,127,127,85,37,10,2,1
%N A323211 Level 1 of Pascal's pyramid. T(n, k) triangle read by rows for n >= 0 and 0 <= k <= n.
%C A323211 Pascal's pyramid is defined by recurrence. P(0) is Pascal's triangle. Now assume P(n-1) already constructed. Then P(n) is found by the steps: (1) Add 1 to each term of P(n-1). (2) Add at the left and at the right side a diagonal consisting all of 1s and complement the top with the rows 1 and 1, 1. A similar construction starting from the Pascal's triangle and subtracting 1 from all terms leads to A014473.
%H A323211 G. C. Greubel, <a href="/A323211/b323211.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A323211 T(n, k) = binomial(n-2, k-1) + 1 if n != 1 else 1.
%F A323211 G.f.: (1 + 3*y + y^2 + x^4*y^2*(1 + y)^2 + x^2*y*(2 + 5*y + 2*y^2) - x^3*y*(1 + 4*y + 4*y^2 + y^3) - x*(1 + 5*y + 5*y^2 + y^3)/((1 - x)*(1 + y)^2*(1 - x*y)*(1 - x - x*y)). - _Stefano Spezia_, Sep 26 2024
%F A323211 From _G. C. Greubel_, Sep 26 2024: (Start)
%F A323211 T(n, n-k) = T(n, k) (symmetry).
%F A323211 T(2*n, n) = A323230(n).
%F A323211 Sum_{k=0..n} (-1)^k*T(n, k) = (n+1 mod 2) - [n=2].
%F A323211 Sum_{k=0..floor(n/2)} T(n-k, k) = Fibonacci(n-2) + (1/4)*(2*n + 3 + (-1)^n) +[n=0] - [n=1]. (End)
%e A323211 Triangle starts:
%e A323211 1
%e A323211 1, 1
%e A323211 1, 2, 1
%e A323211 1, 2, 2, 1
%e A323211 1, 2, 3, 2, 1
%e A323211 1, 2, 4, 4, 2, 1
%e A323211 1, 2, 5, 7, 5, 2, 1
%e A323211 1, 2, 6, 11, 11, 6, 2, 1
%e A323211 1, 2, 7, 16, 21, 16, 7, 2, 1
%e A323211 1, 2, 8, 22, 36, 36, 22, 8, 2, 1
%e A323211 1, 2, 9, 29, 57, 71, 57, 29, 9, 2, 1
%p A323211 T := (n, k) -> `if`(n=1, 1, binomial(n-2, k-1) + 1):
%p A323211 seq(seq(T(n, k), k=0..n), n=0..10);
%p A323211 # Alternative:
%p A323211 T := proc(n, k) option remember;
%p A323211 if k = n then return 1 fi; if k < 2 then return k+1 fi;
%p A323211 T(n-1, k-1) + T(n-1, k) - 1 end:
%p A323211 seq(seq(T(n, k), k=0..n), n=0..10);
%t A323211 A323211[n_, k_]:= If[n<2, 1, Binomial[n-2, k-1] +1];
%t A323211 Table[A323211[n,k], {n,0,13}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 26 2024 *)
%o A323211 (Magma)
%o A323211 A323211:= func< n,k | n le 1 select 1 else 1 + Binomial(n-2,k-1) >;
%o A323211 [A323211(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Sep 26 2024
%o A323211 (SageMath)
%o A323211 def A323211(n,k): return 1 if (n<2) else 1 + binomial(n-2,k-1)
%o A323211 flatten([[A323211(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Sep 26 2024
%Y A323211 Differs from A323231 only in the second term.
%Y A323211 Row sums are A323227.
%Y A323211 Cf. A000045, A014473, A323230.
%K A323211 nonn,tabl,easy
%O A323211 0,5
%A A323211 _Peter Luschny_, Feb 11 2019