A156725 - OEIS

%I #5 Feb 26 2021 20:14:23

%S 1,1,1,1,-2,1,1,22,22,1,1,-440,4840,-440,1,1,12760,2807200,2807200,

%T 12760,1,1,-484880,3093534400,-61870688000,3093534400,-484880,1,1,

%U 22789360,5525052438400,3204530414272000,3204530414272000,5525052438400,22789360,1

%N Triangle T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(3*i-2) ) and m = 2, read by rows.

%C Row sums are: {1, 2, 0, 46, 3962, 5639922, -55684588958, 6420110978999522, 8653645559546848833282, 120959123027642635275104364802, ...}.

%H G. C. Greubel, <a href="/A156725/b156725.txt">Rows n = 0..30 of the triangle, flattened</a>

%F T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(3*i-2) ) and m = 2.

%F T(n, k, m, p, q) = (-p*(m+1))^(k*(n-k)) * (f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q))) where Product_{j=1..n} Pochhammer( (q*(m+1) -1)/(p*(m+1)), j) for (m, p, q) = (2, 3, -2). - _G. C. Greubel_, Feb 26 2021

%e Triangle begins as:

%e   1;

%e   1,       1;

%e   1,      -2,          1;

%e   1,      22,         22,            1;

%e   1,    -440,       4840,         -440,          1;

%e   1,   12760,    2807200,      2807200,      12760,       1;

%e   1, -484880, 3093534400, -61870688000, 3093534400, -484880, 1;

%t (* First program *)

%t t[n_, k_]:= If[k==0, n!, Product[1 -(3*i-2)*(k+1), {j,n}, {i,0,j-1}] ];

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

%t Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 26 2021 *)

%t (* Second program *)

%t f[n_, m_, p_, q_]:= Product[Pochhammer[(q*(m+1) -1)/(p*(m+1)), j], {j,n}];

%t T[n_, k_, m_, p_, q_]:= (-p*(m+1))^(k*(n-k))*(f[n,m,p,q]/(f[k,m,p,q]*f[n-k,m,p,q]));

%t Table[T[n,k,2,3,-2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 26 2021 *)

%o (Sage)

%o @CachedFunction

%o def f(n, m, p, q): return product( rising_factorial( (q*(m+1)-1)/(p*(m+1)), j) for j in (1..n))

%o def T(n, k, m, p, q): return (-p*(m+1))^(k*(n-k))*(f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q)))

%o flatten([[T(n,k,2,3,-2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 26 2021

%o (Magma)

%o f:= func< n,m,p,q | n eq 0 select 1 else m eq 0 select Factorial(n) else (&*[ 1 -(p*i+q)*(m+1): i in [0..j], j in [0..n-1]]) >;

%o T:= func< n,k,m,p,q | f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q)) >;

%o [T(n,k,2,3,-2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 26 2021

%Y Cf. A007318 (m=0), A156722 (m=1), this sequence (m=2), A156727 (m=3).

%Y Cf. A156691, A156697, A156730.

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 14 2009

%E Edited by _G. C. Greubel_, Feb 26 2021