cp's OEIS Frontend

A261357 Pyramid of coefficients in expansion of (1 + 2*x + 2*y)^n.

%I A261357 #63 Mar 17 2025 09:02:05
%S A261357 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 A261357 32,16,64,96,64,16,1,10,10,40,80,40,80,240,240,80,80,320,480,320,80,
%U A261357 32,160,320,320,160,32
%N A261357 Pyramid of coefficients in expansion of (1 + 2*x + 2*y)^n.
%C A261357 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 A261357 The sum of the numbers in each slice of the pyramid is 5^n.
%C A261357 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 A261357 Alois P. Heinz, <a href="/A261357/b261357.txt">Rows n = 0..38, flattened</a>
%F A261357 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 A261357 T(n,j,k) = 2^j*binomial(n,j)*binomial(j,k). - _Dimitri Boscainos_, Aug 21 2015
%e A261357 Here is the fourth (n=3) slice of the pyramid:
%e A261357         1
%e A261357       6   6
%e A261357    12  24  12
%e A261357   8  24  24   8
%p A261357 p:= proc(i, j, k) option remember;
%p A261357       if k<0 or i<0 or i>k or j<0 or j>i then 0
%p A261357     elif {i, j, k}={0} then 1
%p A261357     else p(i, j, k-1) +2*p(i-1, j, k-1) +2*p(i-1, j-1, k-1)
%p A261357       fi
%p A261357     end:
%p A261357 seq(seq(seq(p(i, j, k), j=0..i), i=0..k), k=0..5);
%p A261357 # Adapted from _Alois P. Heinz_'s Maple program for A261356
%t A261357 p[i_, j_, k_] := p[i, j, k] = If[k < 0 || i < 0 || i > k || j < 0 || j > i, 0, If[Union@{i, j, k} == {0}, 1, p[i, j, k - 1] + 2*p[i - 1, j, k - 1] + 2*p[i - 1, j - 1, k - 1]]];
%t A261357 Table[Table[Table[p[i, j, k], {j, 0, i}], {i, 0, k}], {k, 0, 5}] // Flatten (* _Jean-François Alcover_, Mar 17 2025, after _Alois P. Heinz_ *)
%o A261357 (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 A261357 Cf. A000079, A046816, A059304, A261356, A261358.
%K A261357 nonn,tabf,walk
%O A261357 0,3
%A A261357 _Dimitri Boscainos_, Aug 16 2015