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Triangle T(n, k, m) = t(n,m)/( t(k,nm) * 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)*(2*i-1) ) and m = 2, read by rows.
T(n, k, m) = t(n,m)/( t(k,nm) * 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)*(2*i-1) ) and m = 2.
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General qp odd weighted combinations as : m=2;q=3: t(n,m)=If[m == 0, n!, Product[Product[1 - (2*i - 1)*( m + 1), {i, 0, k - 1}], {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].
Triangle T(n, k, m) = t(n,m)/( t(k,n) * 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)*(2*i-1) ) and m = 2, read by rows.
1, 1, 1, 1, -2, 1, 1, 16, 16, 1, 1, -224, 1792, -224, 1, 1, 4480, 501760, 501760, 4480, 1, 1, -116480, 260915200, -3652812800, 260915200, -116480, 1, 1, 3727360, 217081446400, 60782804992000, 60782804992000, 217081446400, 3727360, 1, 1
Row sums are: {1, 2, 0, 34, 1346, 1012482, -3131215358, 121999780331522,
{1, 2, 0, 34, 1346, 1012482, -3131215358, 121999780331522,
34591292869081661442, 107137531255480378706493442, ...}.
-3432787564907030237721525583871998,...}.
G. C. Greubel, <a href="/A156697/b156697.txt">Rows n = 0..30 of the triangle, flattened</a>
T(n, k, m) = t(n,m)/( t(k,n) * 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)*(2;q*i-1) ) and m =3: 2.
tT(n, k, m, p, q) =If[ (-p*(m+1))^(k*(n-k)) * (f(n,m,p,q)/(f(k,m == 0, ,p,q)*f(n!, Product[-k,m,p,q))) where Product[_{j=1 - ..n} Pochhammer( (2q*i (m+1) - 1)/(p*( m + 1), {i, 0, k ), j) for (m, p, q) = (2, 2, - 1}], {k, 1, n}]];). - _G. C. Greubel_, Feb 25 2021
b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].
{1},
Triangle begins as:
1;
{ 1, 1},;
{ 1, -2, 1},;
{ 1, 16, 16, 1},;
{ 1, -224, 1792, -224, 1},;
{ 1, 4480, 501760, 501760, 4480, 1},;
{ 1, -116480, 260915200, -3652812800, 260915200, -116480, 1},;
{1, 3727360, 217081446400, 60782804992000, 60782804992000, 217081446400, 3727360, 1},
{1, -141639680, 263971038822400, -1921709162627072000, 38434183252541440000, -1921709162627072000, 263971038822400, -141639680, 1},
{1, 6232145920, 441359576911052800, 102819127037198860288000, 53465946059343407349760000, 53465946059343407349760000, 102819127037198860288000, 441359576911052800, 6232145920, 1},
{1, -311607296000, 970991069204316160000, -8595679020309824720076800000, 143032098897955483342077952000000, -3718834571346842566894026752000000, 143032098897955483342077952000000, -8595679020309824720076800000, 970991069204316160000, -311607296000, 1}
(* First program *)
t[n_, m_k_] := If[m k== 0, n!, Product[Product[1 - (2*i - 1)*( m k+ 1), {j, n}, {i, 0, k j- 1}], {k, 1, n}] ];
bT[n_, k_, m_] := If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]
Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 25 2021 *)
(* Second program *)
f[n_, m_, p_, q_]:= Product[Pochhammer[(q*(m+1) -1)/(p*(m+1)), j], {j, n}];
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]));
Table[T[n, k, 2, 2, -1], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 25 2021 *)
(Sage)
@CachedFunction
def f(n, m, p, q): return product( rising_factorial( (q*(m+1)-1)/(p*(m+1)), j) for j in (1..n))
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)))
flatten([[T(n, k, 2, 2, -1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 25 2021
(Magma)
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]]) >;
T:= func< n, k, m, p, q | f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)) >;
[T(n, k, 2, 2, -1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 25 2021
sign,tabl,uned
Edited by G. C. Greubel, Feb 25 2021
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t[n_, m_] = If[m == 0, n!, Product[Product[1 - (2*i - 1)*( m + 1), {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]
t[n_, m_] = If[m == 0, n!, Product[Product[1 - (2*i - 1)*( m + 1), {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]
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_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Feb 13 2009