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Triangle read by rows, T(n, k) is the coefficient of x^n in expansion of ((-1 - x + sqrt(1 + 2*x + 5*x^2)) /2)^k.
1

%I #21 May 19 2021 05:15:35

%S 1,-1,1,0,-2,1,2,1,-3,1,-3,4,3,-4,1,-1,-10,5,6,-5,1,11,4,-21,4,10,-6,

%T 1,-15,28,21,-35,0,15,-7,1,-13,-64,42,56,-50,-8,21,-8,1,77,9,-162,36,

%U 114,-63,-21,28,-9,1,-86,230,135,-312,-15,198,-70,-40,36,-10,1

%N Triangle read by rows, T(n, k) is the coefficient of x^n in expansion of ((-1 - x + sqrt(1 + 2*x + 5*x^2)) /2)^k.

%F T(n, k) = (k/n) * Sum_{j=0..n} (-1)^(j-k) * binomial(n,j) * binomial(j,-n-k+2*j).

%F T(n, k) = binomial(n, k)*hypergeom([(k - n)/2, (k - n + 1)/2], [k + 2], -4)*(-1)^(n - k), assuming offset = 0. - _Peter Luschny_, May 19 2021

%e 1;

%e -1, 1;

%e 0, -2, 1;

%e 2, 1, -3, 1;

%e -3, 4, 3, -4, 1;

%e -1, -10, 5, 6, -5, 1;

%e 11, 4, -21, 4, 10, -6, 1;

%p # Assuming offset = 0.

%p T := (n,k) -> (-1)^(n-k)*binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], -4): for n from 0 to 9 do seq(simplify(T(n, k)), k=0..n) od; # _Peter Luschny_, May 19 2021

%o (Maxima)

%o T(n,k):=(k*sum(binomial(j,-n-k+2*j)*(-1)^(j-k)*binomial(n,j),j,0,n))/n;

%Y Cf. A007440 (1st column), A108624 (row sums).

%K sign,tabl

%O 1,5

%A _Vladimir Kruchinin_, Dec 17 2011

%E More terms from _Sean A. Irvine_, Mar 03 2021