A261357 - OEIS

%I #57 Oct 31 2015 15:31:54

%S 1,1,2,2,1,4,4,4,8,4,1,6,6,12,24,12,8,24,24,8,1,8,8,24,48,24,32,96,96,

%T 32,16,64,96,64,16,1,10,10,40,80,40,80,240,240,80,80,320,480,320,80,

%U 32,160,320,320,160,32

%N Pyramid of coefficients in expansion of (1 + 2*x + 2*y)^n.

%C T(n,j,k) is the number of lattice paths from (0,0,0) to (n,j,k) with steps (1,0,0), two kinds of steps (1,1,0) and two kinds of steps (1,1,1).

%C The sum of the numbers in each slice of the pyramid is 5^n.

%C The terms of the j-th row of the n-th slice of this pyramid are the sum of the terms in each antidiagonal of the j-th triangle of the n-th slice of A261358. - _Dimitri Boscainos_, Aug 21 2015

%H Alois P. Heinz, <a href="/A261357/b261357.txt">Rows n = 0..38, flattened</a>

%F T(i+1,j,k) = 2*T(i,j-1,k-1)+ 2*T(i,j-1,k) + T(i,j,k); T(i,j,-1) = 0, ...; T(0,0,0) = 1.

%F T(n,j,k) = 2^j*binomial(n,j)*binomial(j,k). - _Dimitri Boscainos_, Aug 21 2015

%e Here is the fourth (n=3) slice of the pyramid:

%e         1

%e       6   6

%e    12  24  12

%e   8  24  24   8

%p p:= proc(i, j, k) option remember;

%p       if k<0 or i<0 or i>k or j<0 or j>i then 0

%p     elif {i, j, k}={0} then 1

%p     else p(i, j, k-1) +2*p(i-1, j, k-1) +2*p(i-1, j-1, k-1)

%p       fi

%p     end:

%p seq(seq(seq(p(i, j, k), j=0..i), i=0..k), k=0..5);

%p # Adapted from _Alois P. Heinz_'s Maple program for A261356

%o (PARI) tabf(nn) = {for (n=0, nn, for (j=0, n, for (k=0, j, print1(2^j*binomial(n,j)*binomial(j,k), ", ")); print();); print(););} \\ _Michel Marcus_, Oct 07 2015

%Y Cf. A046816, A261356, A261358.

%K nonn,tabf,walk

%O 0,3

%A _Dimitri Boscainos_, Aug 16 2015