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A371401 Triangle read by rows: T(n, k) = [x^k] (n*x + 1)*Hypergeometric([-n, -n + 1], [1], x).

%I A371401 #5 Mar 22 2024 17:04:11
%S A371401 1,1,1,1,4,4,1,9,21,9,1,16,66,76,16,1,25,160,340,205,25,1,36,330,1100,
%T A371401 1275,456,36,1,49,609,2905,5425,3801,889,49,1,64,1036,6664,18130,
%U A371401 20776,9604,1576,64,1,81,1656,13776,51156,86436,65856,21456,2601,81
%N A371401 Triangle read by rows: T(n, k) = [x^k] (n*x + 1)*Hypergeometric([-n, -n + 1], [1], x).
%F A371401 Sum_{k=0..n} a(n) = (n + 1)*binomial(2*n - 1, n).
%e A371401 Triangle starts:
%e A371401 [0] 1;
%e A371401 [1] 1,  1;
%e A371401 [2] 1,  4,    4;
%e A371401 [3] 1,  9,   21,    9;
%e A371401 [4] 1, 16,   66,   76,    16;
%e A371401 [5] 1, 25,  160,  340,   205,    25;
%e A371401 [6] 1, 36,  330, 1100,  1275,   456,   36;
%e A371401 [7] 1, 49,  609, 2905,  5425,  3801,  889,   49;
%e A371401 [8] 1, 64, 1036, 6664, 18130, 20776, 9604, 1576, 64;
%p A371401 P := (n, x) -> (n*x + 1)*hypergeom([-n, -n + 1], [1], x):
%p A371401 T := (n, k) -> coeff(simplify(P(n, x)), x, k):
%p A371401 seq(seq(T(n, k), k = 0..n), n = 0..9);
%Y A371401 Cf. A371400, A097070 (row sums, shifted).
%K A371401 nonn,tabl,easy
%O A371401 0,5
%A A371401 _Peter Luschny_, Mar 22 2024