A084610 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x-x^2)^n.
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, -15, 11, 15, -10, -5, 5, -1, 1, 6, 9, -10, -30, 6, 41, -6, -30, 10, 9, -6, 1, 1, 7, 14, -7, -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
Offset: 0
Examples
Rows: 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, -15, 11, 15, -10, -5, 5, -1; 1, 6, 9, -10, -30, 6, 41, -6, -30, 10, 9, -6, 1; 1, 7, 14, -7, -49, -14, 77, 29, -77, -14, 49, -7, -14, 7, -1;
Links
- Paul D. Hanna, Rows n=0..34 of triangle, flattened
- Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, The Fibonacci-Fubini and Lucas-Fubini numbers, arXiv:2407.04409 [math.CO], 2024. See p. 13.
Programs
-
Magma
A084610:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-1)^j: j in [0..k]]) >; [A084610(n,k): k in [0..2*n], n in [0..13]]; // G. C. Greubel, Mar 26 2023
-
Mathematica
T[n_, k_]:= Sum[Binomial[n,k-j]*Binomial[k-j,j]*(-1)^j, {j,0,k}]; Table[T[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, Mar 26 2023 *) -
PARI
for(n=0,12, for(k=0,2*n,t=polcoeff((1+x-x^2)^n,k,x); print1(t",")); print(" ")) -
SageMath
def A084610(n,k): return sum(binomial(n,k-j)*binomial(k-j,j)*(-1)^j for j in range(k+1)) flatten([[A084610(n,k) for k in range(2*n+1)] for n in range(14)]) # G. C. Greubel, Mar 26 2023
Formula
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
From G. C. Greubel, Mar 26 2023: (Start)
T(n, k) = Sum_{j=0..k} binomial(n, k-j)*binomial(k-j, j)*(-1)^j.
T(n, 2*n) = (-1)^n.
T(n, 2*n-1) = (-1)^(n-1)*n, n >= 1.
Sum_{k=0..2*n} T(n, k) = 1.
Sum_{k=0..2*n} (-1)^k*T(n, k) = (-1)^n.
Sum_{k=0..n} T(n-k, k) = floor((n+2)/2).
Sum_{k=0..n} (-1)^k*T(n-k, k) = (-1)^n*A057597(n+2). (End)