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

0,6

Links

Paul D. Hanna, Rows n=0..34 of triangle, flattened

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)

Example

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;

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 *)

Prog

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

Crossrefs

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

Sequence in context: A269329 A117166 A078996 * A303336 A361462 A129479

Adjacent sequences: A084607 A084608 A084609 * A084611 A084612 A084613

Keyword

sign,tabf

Author

Paul D. Hanna, Jun 01 2003

Status

approved