A084614 Triangle, read by rows, where the n-th row lists the (2*n+1) coefficients of (1 + x - 3*x^2)^n.

1, 1, 1, -3, 1, 2, -5, -6, 9, 1, 3, -6, -17, 18, 27, -27, 1, 4, -6, -32, 19, 96, -54, -108, 81, 1, 5, -5, -50, 5, 211, -15, -450, 135, 405, -243, 1, 6, -3, -70, -30, 366, 181, -1098, -270, 1890, -243, -1458, 729, 1, 7, 0, -91, -91, 546, 637, -2015, -1911, 4914, 2457, -7371, 0, 5103, -2187, 1, 8, 4, -112, -182, 728, 1456

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

From G. C. Greubel, Mar 25 2023: (Start)

T(n, k) = Sum_{j=0..k} binomial(n, k-j)*binomial(k-j, j)*(-3)^j, for 0 <= k <= 2*n.

T(n, 2*n) = (-3)^n.

T(n, 2*n-1) = (-1)^(n-1)*A027471(n+1), n >= 1.

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

Sum_{k=0..2*n} (-1)^k*T(n, k) = (-3)^n. (End)

Example

Rows:
  1;
  1, 1, -3;
  1, 2, -5,  -6,   9;
  1, 3, -6, -17,  18,  27, -27;
  1, 4, -6, -32,  19,  96, -54,  -108,   81;
  1, 5, -5, -50,   5, 211, -15,  -450,  135,  405, -243;
  1, 6, -3, -70, -30, 366, 181, -1098, -270, 1890, -243, -1458, 729;

Mathematica

With[{eq= (1+x-3*x^2)}, Flatten[Table[CoefficientList[Expand[eq^n], x], {n, 0, 13}]]] (* G. C. Greubel, Mar 02 2017 *)

Prog

(PARI) for(n=0, 12, for(k=0, 2*n, t=polcoeff((1+x-3*x^2)^n, k, x); print1(t", ")); print(" "))
(Magma)
A084614:= func< n, k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-3)^j: j in [0..k]]) >;
[A084614(n, k): k in [0..2*n], n in [0..15]]; // G. C. Greubel, Mar 25 2023
(SageMath)
def A084614(n, k): return ( (1+x-3*x^2)^n ).series(x, 30).list()[k]
flatten([[A084614(n, k) for k in range(2*n+1)] for n in range(13)]) # G. C. Greubel, Mar 25 2023

Crossrefs

Cf. A002426, A027471, A084600 - A084613, A084615.

Sequence in context: A183386 A231818 A280633 * A050058 A144204 A048226

Adjacent sequences: A084611 A084612 A084613 * A084615 A084616 A084617

Keyword

sign,tabf

Author

Paul D. Hanna, Jun 01 2003

Status

approved