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

%I #23 Sep 08 2022 08:45:53

%S 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 120,36,8,1,6435,3003,1287,495,165,45,9,1,24310,11440,5005,2002,715,

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

%C Row sums are A001791.

%C Obtained from A059481 by removal of the last two terms in each row, followed by row reversal.

%C 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 n-th row of the triangle = top row of M^n, where M is the following infinite square production matrix:

%F 3, 1, 0, 0, 0, ...

%F 1, 1, 1, 0, 0, ...

%F 1, 1, 1, 1, 0, ...

%F 1, 1, 1, 1, 1, ...

%F ... - _Philippe Deléham_, Jul 12 2015

%e Triangle begins:

%e 1;

%e 3, 1;

%e 10, 4, 1;

%e 35, 15, 5, 1;

%e 126, 56, 21, 6, 1;

%e 462, 210, 84, 28, 7, 1;

%e 1716, 792, 330, 120, 36, 8, 1;

%e 6435, 3003, 1287, 495, 165, 45, 9, 1;

%e 24310, 11440, 5005, 2002, 715, 220, 55, 10, 1;

%e 92378, 43758, 19448, 8008, 3003, 1001, 286, 66, 11, 1;

%e 352716, 167960, 75582, 31824, 12376, 4368, 1365, 364, 78, 12, 1;

%p A176992 := proc(n,k) binomial(1+2*n-k,n+1) ; end proc: # _R. J. Mathar_, Dec 09 2010

%t p[t_, j_] = ((-1)^(j + 1)/2)*Sum[Binomial[k - j - 1, j + 1]*t^k, {k, 0, Infinity}];

%t Flatten[Table[CoefficientList[ExpandAll[p[t, j]], t], {j, 0, 10}]]

%o (Magma) /* As triangle */ [[Binomial(2*n-k+1,n+1): k in [0..n]]: n in [0.. 10]]; // _Vincenzo Librandi_, Jul 12 2015

%Y Cf. A092392, A001791, A078812.

%Y Cf. Similar triangle: A033184, A054445.

%Y Cf. A178300 (reversal).

%K nonn,tabl,easy

%O 0,2

%A _Roger L. Bagula_, Dec 08 2010