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%I A371400 #19 Mar 22 2024 18:06:31
%S A371400 1,2,2,6,9,6,20,40,40,20,70,175,225,175,70,252,756,1176,1176,756,252,
%T A371400 924,3234,5880,7056,5880,3234,924,3432,13728,28512,39600,39600,28512,
%U A371400 13728,3432,12870,57915,135135,212355,245025,212355,135135,57915,12870
%N A371400 Triangle read by rows: T(n, k) = binomial(k + n, k)*binomial(2*n - k, n).
%C A371400 The main diagonal and column 0 of the triangle are the central binomial coefficients, which are the sums of the squares of Pascal's triangle entries. This sum representation can be generalized, and all terms can be seen as sums of coefficients of some polynomials. (See the Example section.)
%C A371400 To see this, consider T(n, k) as the value of the polynomials P(n, k)(x) at x = 1, where P(n, k)(x) = H([-n, -k], [1], x)*H([-n, -n + k], [1], x) and H denotes the hypergeometric sum 2F1. For instance column 0 is given by the row sums of A008459, and column 1 by the row sums of A371401.
%F A371400 T(n, k) = A046899(n, k) * A092392(n, k).
%F A371400 T(n, k) = A046899(n, k) * A046899(n, n - k).
%F A371400 T(n, k) = A092392(n, k) * A092392(n, n - k).
%F A371400 T(n, k) = A371395(n, k) * (n + 1).
%F A371400 T(n, k) = hypergeom([-n, -k], [1], 1) * hypergeom([-n, -n + k], [1], 1).
%F A371400 2^n*Sum_{k=0..n} T(n, k)*(1/2)^k = A244038(n).
%F A371400 2^n*Sum_{k=0..n} T(n, k)*(-1/2)^k = A371399(n).
%e A371400 Triangle starts:
%e A371400 [0]    1;
%e A371400 [1]    2,     2;
%e A371400 [2]    6,     9,     6;
%e A371400 [3]   20,    40,    40,    20;
%e A371400 [4]   70,   175,   225,   175,    70;
%e A371400 [5]  252,   756,  1176,  1176,   756,   252;
%e A371400 [6]  924,  3234,  5880,  7056,  5880,  3234,   924;
%e A371400 [7] 3432, 13728, 28512, 39600, 39600, 28512, 13728, 3432;
%e A371400 .
%e A371400 Because of the symmetry, only the sum representation of terms with k <= n/2 are shown.
%e A371400 0:                 [1]
%e A371400 1:               [1+1]
%e A371400 2:             [1+4+1],               [1+4+4]
%e A371400 3:           [1+9+9+1],            [1+9+21+9]
%e A371400 4:      [1+16+36+16+1],       [1+16+66+76+16],        [1+16+76+96+36]
%e A371400 5: [1+25+100+100+25+1], [1+25+160+340+205+25], [1+25+190+460+400+100]
%p A371400 T := (n, k) -> binomial(k + n, k) * binomial(2*n - k, n):
%p A371400 seq(print(seq(T(n, k), k = 0..n)), n = 0..8);
%t A371400 T[n_, k_] := Hypergeometric2F1[-n, -k, 1, 1] Hypergeometric2F1[-n, -n +k, 1, 1];
%t A371400 Table[T[n, k], {n, 0, 7}, {k, 0, n}]
%Y A371400 Column 0 and main diagonal are A000984.
%Y A371400 Column 1 and subdiagonal are A097070.
%Y A371400 Row sums are A045721.
%Y A371400 The even bisection of the alternating row sums is A005809.
%Y A371400 The central terms are A188662.
%Y A371400 Cf. A046899, A092392, A371395, A244038, A371399.
%Y A371400 Cf. A008459, A371401.
%K A371400 nonn,tabl,easy
%O A371400 0,2
%A A371400 _Peter Luschny_, Mar 21 2024

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