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A133826 Triangle whose rows are sequences of increasing and decreasing tetrahedral numbers: 1; 1,4,1; 1,4,10,4,1; ... .

%I A133826 #39 Sep 17 2022 14:09:58
%S A133826 1,1,4,1,1,4,10,4,1,1,4,10,20,10,4,1,1,4,10,20,35,20,10,4,1,1,4,10,20,
%T A133826 35,56,35,20,10,4,1,1,4,10,20,35,56,84,56,35,20,10,4,1,1,4,10,20,35,
%U A133826 56,84,120,84,56,35,20,10,4,1,1,4,10,20,35,56,84,120,165,120,84,56,35,20,10,4,1
%N A133826 Triangle whose rows are sequences of increasing and decreasing tetrahedral numbers: 1; 1,4,1; 1,4,10,4,1; ... .
%C A133826 Reading the triangle by rows produces the sequence 1,1,4,1,1,4,10,4,1,..., analogous to A004737.
%C A133826 T(n,k) = min(n*(n+1)*(n+2)/6, k*(k+1)*(k+2)/6) n, k > 0. The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - _Boris Putievskiy_, Jan 13 2013
%H A133826 Harvey P. Dale, <a href="/A133826/b133826.txt">Table of n, a(n) for n = 0..1000</a>
%H A133826 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F A133826 O.g.f.: (1+q*x)/((1-x)*(1-q*x)^3*(1-q^2x)) = 1 + x*(1 + 4*q + q^2) + x^2*(1 + 4*q + 10*q^2 + 4*q^3 + q^4) + ... .
%F A133826 From _Boris Putievskiy_, Jan 13 2013: (Start)
%F A133826 a(n) = A004737(n)*(A004737(n)+1)*(A004737(n)+2)/2.
%F A133826 a(n) = z*(z+1)*(z+2)/6, where z = floor(sqrt(n-1)) - |n - floor(sqrt(n-1))^2 - floor(sqrt(n-1)) - 1| + 1. (End)
%e A133826 Triangle T(n,k) starts:
%e A133826   1;
%e A133826   1,  4,  1;
%e A133826   1,  4, 10,  4,  1;
%e A133826   1,  4, 10, 20, 10,  4,  1;
%e A133826   1,  4, 10, 20, 35, 20, 10,  4,  1;
%e A133826   1,  4, 10, 20, 35, 56, 35, 20, 10,  4,  1;
%e A133826   1,  4, 10, 20, 35, 56, 84, 56, 35, 20, 10,  4,  1;
%e A133826   ...
%p A133826 T:= n-> (f-> (f(i)$i=1..n, f(n-i)$i=1..n-1))(t-> t*(t+1)*(t+2)/6):
%p A133826 seq(T(n), n=1..10);  # _Alois P. Heinz_, Feb 17 2022
%t A133826 Module[{nn=10,tet},tet=Table[(n(n+1)(n+2))/6,{n,nn}];Table[Join[Take[ tet,k], Reverse[ Take[tet,k-1]]],{k,nn}]]//Flatten (* _Harvey P. Dale_, Oct 22 2017 *)
%t A133826 Table[Series[(1-h^(2*N+4))^2/(1-h^2)^4-((2+N)^2 *h^(2N+2))/(1-h^2)^2, {h, 0, 4*N}], {N,0,5}] // Normal (* _Sergii Voloshyn_, Sep 09 2022 *)
%Y A133826 Cf. A000292, A002415 (row sums), A004737, A124258, A133825.
%K A133826 easy,nonn,tabf
%O A133826 0,3
%A A133826 _Peter Bala_, Sep 25 2007