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A072405 Triangle T(n, k) = C(n,k) - C(n-2,k-1) for n >= 3 and T(n, k) = 1 otherwise, read by rows.

%I A072405 #28 Jan 12 2024 22:43:22
%S A072405 1,1,1,1,1,1,1,2,2,1,1,3,4,3,1,1,4,7,7,4,1,1,5,11,14,11,5,1,1,6,16,25,
%T A072405 25,16,6,1,1,7,22,41,50,41,22,7,1,1,8,29,63,91,91,63,29,8,1,1,9,37,92,
%U A072405 154,182,154,92,37,9,1,1,10,46,129,246,336,336,246,129,46,10,1,1,11,56,175,375,582,672,582,375,175,56,11,1
%N A072405 Triangle T(n, k) = C(n,k) - C(n-2,k-1) for n >= 3 and T(n, k) = 1 otherwise, read by rows.
%C A072405 Starting 1,0,1,1,1,... this is the Riordan array ((1-x+x^2)/(1-x), x/(1-x)). Its diagonal sums are A006355. Its inverse is A106509. - _Paul Barry_, May 04 2005
%H A072405 G. C. Greubel, <a href="/A072405/b072405.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A072405 Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%F A072405 T(n, k) = C(n,k) - C(n-2,k-1) for n >= 3 and T(n, k) = 1 otherwise.
%F A072405 T(n, k) = T(n-1, k-1) + T(n-1, k) starting with T(2, 0) = T(2, 1) = T(2, 2) = 1 and T(n, 0) = T(n, n) = 1.
%F A072405 G.f.: (1-x^2*y) / (1 - x*(1+y)). - _Ralf Stephan_, Jan 31 2005
%F A072405 From _G. C. Greubel_, Apr 28 2021: (Start)
%F A072405 Sum_{k=0..n} T(n, k) = (n+1)*[n<3] + 3*2^(n-2)*[n>=3].
%F A072405 T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = -1. (End)
%e A072405 Rows start as:
%e A072405   1;
%e A072405   1, 1;
%e A072405   1, 1,  1; (key row for starting the recurrence)
%e A072405   1, 2,  2,  1;
%e A072405   1, 3,  4,  3,  1;
%e A072405   1, 4,  7,  7,  4, 1;
%e A072405   1, 5, 11, 14, 11, 5, 1;
%t A072405 t[2, 1] = 1; t[n_, n_] = t[_, 0] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + t[n-1, k]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 28 2013, after _Ralf Stephan_ *)
%o A072405 (Magma)
%o A072405 T:= func< n,k | n lt 3 select 1 else Binomial(n,k) - Binomial(n-2,k-1) >;
%o A072405 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 28 2021
%o A072405 (Sage)
%o A072405 def T(n,k): return 1 if n<3 else binomial(n,k) - binomial(n-2,k-1)
%o A072405 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 28 2021
%o A072405 (PARI) A072405(n, k) = if(n>2, binomial(n, k)-binomial(n-2, k-1), 1) \\ _M. F. Hasler_, Jan 06 2024
%Y A072405 Row sums give essentially A003945, A007283, or A042950.
%Y A072405 Cf. A072406 for number of odd terms in each row.
%Y A072405 Cf. A051597, A096646, A122218 (identical for n > 1).
%Y A072405 Cf. A007318 (q=0), A072405 (q= -1), A173117 (q=1), A173118 (q=2), A173119 (q=3), A173120 (q=-4).
%K A072405 easy,nonn,tabl
%O A072405 0,8
%A A072405 _Henry Bottomley_, Jun 16 2002