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A261357
Pyramid of coefficients in expansion of (1 + 2*x + 2*y)^n.
3
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, 32, 16, 64, 96, 64, 16, 1, 10, 10, 40, 80, 40, 80, 240, 240, 80, 80, 320, 480, 320, 80, 32, 160, 320, 320, 160, 32
OFFSET
0,3
COMMENTS
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).
The sum of the numbers in each slice of the pyramid is 5^n.
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
LINKS
FORMULA
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.
T(n,j,k) = 2^j*binomial(n,j)*binomial(j,k). - Dimitri Boscainos, Aug 21 2015
EXAMPLE
Here is the fourth (n=3) slice of the pyramid:
1
6 6
12 24 12
8 24 24 8
MAPLE
p:= proc(i, j, k) option remember;
if k<0 or i<0 or i>k or j<0 or j>i then 0
elif {i, j, k}={0} then 1
else p(i, j, k-1) +2*p(i-1, j, k-1) +2*p(i-1, j-1, k-1)
fi
end:
seq(seq(seq(p(i, j, k), j=0..i), i=0..k), k=0..5);
# Adapted from Alois P. Heinz's Maple program for A261356
PROG
(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
CROSSREFS
KEYWORD
nonn,tabf,walk
AUTHOR
Dimitri Boscainos, Aug 16 2015
STATUS
approved