A084608 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2*x+3*x^2)^n.

1, 1, 2, 3, 1, 4, 10, 12, 9, 1, 6, 21, 44, 63, 54, 27, 1, 8, 36, 104, 214, 312, 324, 216, 81, 1, 10, 55, 200, 530, 1052, 1590, 1800, 1485, 810, 243, 1, 12, 78, 340, 1095, 2712, 5284, 8136, 9855, 9180, 6318, 2916, 729, 1, 14, 105, 532, 2009, 5922, 13993, 26840, 41979

Offset

0,3

Links

Alois P. Heinz, Rows n = 0..100, flattened

Formula

From G. C. Greubel, Mar 27 2023: (Start)

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

T(n, n) = A084609(n).

T(n, 2*n-1) = A212697(n), n >= 1.

T(n, 2*n) = A000244(n).

Sum_{j=0..2*n} T(n, k) = A000400(n).

Sum_{k=0..2*n} (-1)^k*T(n, k) = A000079(n).

Sum_{k=0..n} T(n-k, k) = A101822(n). (End)

Example

Triangle begins:
  1;
  1,  2,  3;
  1,  4, 10,  12,   9;
  1,  6, 21,  44,  63,   54,   27;
  1,  8, 36, 104, 214,  312,  324,  216,   81;
  1, 10, 55, 200, 530, 1052, 1590, 1800, 1485, 810, 243;

Maple

f:= proc(n) option remember; expand((1+2*x+3*x^2)^n) end:
T:= (n, k)-> coeff(f(n), x, k):
seq(seq(T(n, k), k=0..2*n), n=0..10);  # Alois P. Heinz, Apr 03 2011

Mathematica

row[n_] := (1+2x+3x^2)^n + O[x]^(2n+1) // CoefficientList[#, x]&; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 01 2017 *)

Prog

(PARI) for(n=0, 10, for(k=0, 2*n, t=polcoeff((1+2*x+3*x^2)^n, k, x); print1(t", ")); print(" "))
(Haskell)
a084608 n = a084608_list !! n
a084608_list = concat $ iterate ([1, 2, 3] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
   _ * _               = []
-- Reinhard Zumkeller, Apr 02 2011
(Magma)
A084608:= func< n, k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*2^(k-2*j)*3^j: j in [0..k]]) >;
[A084608(n, k): k in [0..2*n], n in [0..13]]; // G. C. Greubel, Mar 27 2023
(SageMath)
def A084608(n, k): return sum(binomial(n, j)*binomial(n-j, k-2*j)*2^(k-2*j)*3^j for j in range(k//2+1))
flatten([[A084608(n, k) for k in range(2*n+1)] for n in range(14)]) # G. C. Greubel, Mar 27 2023

Crossrefs

Cf. A000079, A000244, A000400, A002426, A084600 - A084606, A006139, A084609 - A084615, A101822, A212697.

Sequence in context: A076732 A130152 A211233 * A078990 A176566 A079639

Adjacent sequences: A084605 A084606 A084607 * A084609 A084610 A084611

Keyword

nonn,tabf

Author

Paul D. Hanna, Jun 01 2003

Status

approved