A084610 - OEIS

%I #23 Jul 17 2024 16:56:17

%S 1,1,1,-1,1,2,-1,-2,1,1,3,0,-5,0,3,-1,1,4,2,-8,-5,8,2,-4,1,1,5,5,-10,

%T -15,11,15,-10,-5,5,-1,1,6,9,-10,-30,6,41,-6,-30,10,9,-6,1,1,7,14,-7,

%U -49,-14,77,29,-77,-14,49,-7,-14,7,-1,1,8,20,0,-70,-56,112,120,-125,-120,112,56,-70,0,20,-8,1

%N Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x-x^2)^n.

%H Paul D. Hanna, <a href="/A084610/b084610.txt">Rows n=0..34 of triangle, flattened</a>

%H Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, <a href="https://arxiv.org/abs/2407.04409">The Fibonacci-Fubini and Lucas-Fubini numbers</a>, arXiv:2407.04409 [math.CO], 2024. See p. 13.

%F G.f.: G(0)/2 , where G(k)= 1 + 1/( 1 - (1+x-x^2)*x^(2*k+1)/((1+x-x^2)*x^(2*k+1) + 1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 06 2013

%F From _G. C. Greubel_, Mar 26 2023: (Start)

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

%F T(n, 2*n) = (-1)^n.

%F T(n, 2*n-1) = (-1)^(n-1)*n, n >= 1.

%F Sum_{k=0..2*n} T(n, k) = 1.

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

%F Sum_{k=0..n} T(n-k, k) = floor((n+2)/2).

%F Sum_{k=0..n} (-1)^k*T(n-k, k) = (-1)^n*A057597(n+2). (End)

%e Rows:

%e   1;

%e   1, 1, -1;

%e   1, 2, -1,  -2,   1;

%e   1, 3,  0,  -5,   0,   3, -1;

%e   1, 4,  2,  -8,  -5,   8,  2,  -4,   1;

%e   1, 5,  5, -10, -15,  11, 15, -10,  -5,   5, -1;

%e   1, 6,  9, -10, -30,   6, 41,  -6, -30,  10,  9, -6,   1;

%e   1, 7, 14,  -7, -49, -14, 77,  29, -77, -14, 49, -7, -14, 7, -1;

%t T[n_, k_]:= Sum[Binomial[n,k-j]*Binomial[k-j,j]*(-1)^j, {j,0,k}];

%t Table[T[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Mar 26 2023 *)

%o (PARI) for(n=0,12, for(k=0,2*n,t=polcoeff((1+x-x^2)^n,k,x); print1(t",")); print(" "))

%o (Magma)

%o A084610:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-1)^j: j in [0..k]]) >;

%o [A084610(n,k): k in [0..2*n], n in [0..13]]; // _G. C. Greubel_, Mar 26 2023

%o (SageMath)

%o def A084610(n,k): return sum(binomial(n,k-j)*binomial(k-j,j)*(-1)^j for j in range(k+1))

%o flatten([[A084610(n,k) for k in range(2*n+1)] for n in range(14)]) # _G. C. Greubel_, Mar 26 2023

%Y Cf. A002426, A057597, A084600 - A084609, A084611 - A084615.

%K sign,tabf

%O 0,6

%A _Paul D. Hanna_, Jun 01 2003