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A131830 Triangle read by rows: T(n,0) = T(n,n) = n + 1 for n >= 0, and T(n,k) = binomial(n,k) for 1 <= k <= n - 1, n >= 2.

%I A131830 #25 Feb 18 2020 09:15:17
%S A131830 1,2,2,3,2,3,4,3,3,4,5,4,6,4,5,6,5,10,10,5,6,7,6,15,20,15,6,7,8,7,21,
%T A131830 35,35,21,7,8,9,8,28,56,70,56,28,8,9,10,9,36,84,126,126,84,36,9,10,11,
%U A131830 10,45,120,210,252,210,120,45,10,11,12,11,55,165,330,462,462,330,165,55,11,12
%N A131830 Triangle read by rows: T(n,0) = T(n,n) = n + 1 for n >= 0, and T(n,k) = binomial(n,k) for 1 <= k <= n - 1, n >= 2.
%C A131830 Given Pascal's triangle, replace the two (1, 1, 1, ...) borders with (1, 2, 3, ...).
%H A131830 Georg Fischer, <a href="/A131830/b131830.txt">Table of n, a(n) for n = 0..10152</a> [Rows 0..141] (older version by B. D. Swan)
%F A131830 T(n,k) = A131821(n,k) + A007318(n,k) - 1.
%F A131830 From _Franck Maminirina Ramaharo_, Dec 19 2018: (Start)
%F A131830 G.f.: (1 - (1 + x)*y - 2*x*y^2 + (3*x + 3*x^2)*y^3 - (x + x^2 + x^3)*y^4)/((1 - y)^2*(1 - x*y)^2*(1 - y - x*y)).
%F A131830 E.g.f.: y*exp(y) + (x*y + exp(y))*exp(x*y). (End)
%e A131830 First few rows of the triangle are:
%e A131830   1;
%e A131830   2, 2;
%e A131830   3, 2,  3;
%e A131830   4, 3,  3,  4;
%e A131830   5, 4,  6,  4,  5;
%e A131830   6, 5, 10, 10,  5, 6;
%e A131830   7, 6, 15, 20, 15, 6, 7;
%e A131830   ...
%t A131830 Flatten[Table[If[Or[k==n,k==0], n+1, Binomial[n, k]], {n, 0, 11}, {k, 0, n}]] (* _Georg Fischer_, Feb 18 2020 *)
%o A131830 (Maxima) T(n, k) := if k = 0 or k = n then n + 1 else binomial(n, k)$
%o A131830 create_list(T(n, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Dec 19 2018 */
%Y A131830 Row sums: A100314.
%Y A131830 Cf. A007318, A131821, A131821.
%K A131830 nonn,tabl,easy
%O A131830 0,2
%A A131830 _Gary W. Adamson_, Jul 20 2007
%E A131830 Edited by _Franck Maminirina Ramaharo_, Dec 19 2018
%E A131830 B-file corrected from a(1678) onwards by _Georg Fischer_, Feb 18 2020