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A176992 Triangle T(n,m) = binomial(2n-k+1, n+1) read by rows, 0 <= k <= n.

%I A176992 #23 Sep 08 2022 08:45:53
%S A176992 1,3,1,10,4,1,35,15,5,1,126,56,21,6,1,462,210,84,28,7,1,1716,792,330,
%T A176992 120,36,8,1,6435,3003,1287,495,165,45,9,1,24310,11440,5005,2002,715,
%U A176992 220,55,10,1,92378,43758,19448,8008,3003,1001,286,66,11,1,352716,167960,75582,31824,12376,4368,1365,364,78,12,1
%N A176992 Triangle T(n,m) = binomial(2n-k+1, n+1) read by rows, 0 <= k <= n.
%C A176992 Row sums are A001791.
%C A176992 Obtained from A059481 by removal of the last two terms in each row, followed by row reversal.
%C A176992 Riordan array (c(x)/sqrt(1 - 4*x), x*c(x)) where c(x) is the g.f. of A000108. - _Philippe Deléham_, Jul 12 2015
%F A176992 n-th row of the triangle = top row of M^n, where M is the following infinite square production matrix:
%F A176992   3, 1, 0, 0, 0, ...
%F A176992   1, 1, 1, 0, 0, ...
%F A176992   1, 1, 1, 1, 0, ...
%F A176992   1, 1, 1, 1, 1, ...
%F A176992   ... - _Philippe Deléham_, Jul 12 2015
%e A176992 Triangle begins:
%e A176992        1;
%e A176992        3,      1;
%e A176992       10,      4,     1;
%e A176992       35,     15,     5,     1;
%e A176992      126,     56,    21,     6,    1;
%e A176992      462,    210,    84,    28,    7,     1;
%e A176992     1716,    792,   330,   120,   36,     8,    1;
%e A176992     6435,   3003,  1287,   495,   165,   45,    9,   1;
%e A176992    24310,  11440,  5005,  2002,   715,  220,   55,  10,  1;
%e A176992    92378,  43758, 19448,  8008,  3003, 1001,  286,  66, 11,  1;
%e A176992   352716, 167960, 75582, 31824, 12376, 4368, 1365, 364, 78, 12, 1;
%p A176992 A176992 := proc(n,k) binomial(1+2*n-k,n+1) ; end proc: # _R. J. Mathar_, Dec 09 2010
%t A176992 p[t_, j_] = ((-1)^(j + 1)/2)*Sum[Binomial[k - j - 1, j + 1]*t^k, {k, 0, Infinity}];
%t A176992 Flatten[Table[CoefficientList[ExpandAll[p[t, j]], t], {j, 0, 10}]]
%o A176992 (Magma) /* As triangle */ [[Binomial(2*n-k+1,n+1): k in [0..n]]: n in [0.. 10]]; // _Vincenzo Librandi_, Jul 12 2015
%Y A176992 Cf. A092392, A001791, A078812.
%Y A176992 Cf. Similar triangle: A033184, A054445.
%Y A176992 Cf. A178300 (reversal).
%K A176992 nonn,tabl,easy
%O A176992 0,2
%A A176992 _Roger L. Bagula_, Dec 08 2010