A156917 - OEIS

%I #6 Sep 08 2022 08:45:41

%S 1,1,1,1,40,1,1,1210,1210,1,1,33880,1024870,33880,1,1,925771,

%T 784128037,784128037,925771,1,1,25095280,580812061522,16262737722616,

%U 580812061522,25095280,1,1,678468820,425659125229240,325671796712891524

%N General q-Narayana triangle sequence: T(n, k) = Product_{j=0..2} ( q_binomial(n+j, j+k, 3)/q_binomial(n+j-k, j, 3) ).

%C Row sums are: {1, 2, 42, 2422, 1092632, 1570107618, 17424412036222, 652194913033179170, 189060566695044668933610, ...}.

%H G. C. Greubel, <a href="/A156917/b156917.txt">Rows n = 0..50 of triangle, flattened</a>

%F T(n, k) = Product_{j=0..2} ( q_binomial(n+j, j+k, 3)/q_binomial(n+j-k, j, 3) ). - _G. C. Greubel_, May 22 2019

%e Triangle begins as:

%e   1;

%e   1,        1;

%e   1,       40,            1;

%e   1,     1210,         1210,              1;

%e   1,    33880,      1024870,          33880,            1;

%e   1,   925771,    784128037,      784128037,       925771,        1;

%e   1, 25095280, 580812061522, 16262737722616, 580812061522, 25095280, 1;

%t (* First Program *)t[n_, m_]:= If[m==0, n!, Product[Sum[(m+1)^i, {i,0,k-1}], {k,1,n}]];

%t b[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];

%t c[n_, l_, m_]:= Product[b[n+k, l+k, 2]/b[n-l+k, k, 2], {k, 0, m}];

%t Table[c[n, k, 2], {n, 0, 10}, {k, 0, n}]//Flatten(* Second Program *)

%t T[n_, k_]:= Product[QBinomial[n+j,j+k,3]/QBinomial[n+j-k,j,3], {j,0,2}];

%t Table[T[n, k], {n, 0, 5}, {k, 0, n}]//Flatten (* _G. C. Greubel_, May 22 2019 *)

%o (PARI)

%o b(n,k,q) = prod(j=1, k, (1-q^(n-j+1))/(1-q^j));

%o T(n, k) = prod(j=0, 2, b(n+j, j+k, 3)/b(n-k+j, j, 3));

%o for(n=0, 12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 22 2019

%o (Magma)

%o B:= func< n,k,q | (&*[(1-q^(n-j+1))/(1-q^j): j in [1..k]]) >;

%o T:= func< n,k | k eq 0 select 1 else B(n,k,3)*(&*[B(n+j, j+k, 3)/B(n-k+j,j,3): j in [1..2]]) >;

%o [[T(n,k) : k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, May 22 2019

%o (Sage)

%o def T(n, k): return product((q_binomial(n+j, j+k, 3)/q_binomial(n+j-k, j, 3)) for j in (0..2))

%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, May 22 2019

%Y Cf. A001263, A156916, this sequence, A156939.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 18 2009

%E Edited by _G. C. Greubel_, May 22 2019