|
%I #9 Dec 26 2023 12:07:42
%S 1,1,1,1,0,1,1,-78,-78,1,1,1070,1200,1070,1,1,-16530,-14665,-14665,
%T -16530,1,1,240667,179242,163044,179242,240667,1,1,-2572332,-726012,
%U -638358,-638358,-726012,-2572332,1,1,-29453058,-82571646,-81432978,-79275240,-81432978,-82571646,-29453058,1
%N Triangle, read by rows, T(n,k) = f(n,k) - f(n,0) + 1, where f(n, k) = Sum_{j=0..k} (-1)^j*(n+k)!/((n-j)!*(k-j)!*j!) + Sum_{j=0..n-k} (-1)^j*(2*n - k)!/((n-j)!*(n-k-j)!*j!).
%C Row sums are: {1, 2, 2, -154, 3342, -62388, 1002864, -7873402, -466190602, 41337748316, -2470134563444, ...}.
%H G. C. Greubel, <a href="/A176094/b176094.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n,k) = f(n,k) - f(n,0) + 1, where f(n, k) = Sum_{j=0..k} (-1)^j*(n+k)!/((n-j)!*(k-j)!*j!) + Sum_{j=0..n-k} (-1)^j*(2*n - k)!/((n-j)!*(n-k-j)!*j!).
%F f(n,k) = binomial(n+k, n)*2F0(-n, -k; -; -1) + binomial(2*n-k, n)*2F0(-n, k-n; -; -1), where 2F0 is a generalized hypergeometric function. - _G. C. Greubel_, Nov 28 2019
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 0, 1;
%e 1, -78, -78, 1;
%e 1, 1070, 1200, 1070, 1;
%e 1, -16530, -14665, -14665, -16530, 1;
%e 1, 240667, 179242, 163044, 179242, 240667, 1;
%p b:=binomial; f(n,k):=b(n+k,n)*add((-1)^j*j!*b(n,j)*b(k,j), j=0..k) + b(2*n-k,n)*add((-1)^j*j!*b(n,j)*b(n-k,j), j=0..n-k); seq(seq(f(n,k)-f(n,0)+1, k=0..n), n=0..10); # _G. C. Greubel_, Nov 27 2019
%t (* First program *)
%t f[n_, k_]:= Sum[(-1)^j*(n+k)!/((n-j)!*(k-j)!*j!), {j,0,k}] + Sum[(-1)^j*(2*n-k)! /((n-j)!*(n-k-j)!*j!), {j,0,(n-m)}]; Table[f[n, m] -f[n, 0] +1, {n, 0, 10}, {k, 0, n}]//Flatten
%t (* Second program *)
%t f[n_, k_]:= Binomial[n+k, n]*HypergeometricPFQ[{-n, -k}, {}, -1] + Binomial[2*n-k, n]*HypergeometricPFQ[{-n, k - n}, {}, -1]; Table[f[n, k] - f[n, 0] + 1, {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 27 2019 *)
%o (PARI) b=binomial; f(n,k) = b(n+k,n)*sum(j=0,k, (-1)^j*j!*b(n,j)*b(k,j)) + b(2*n-k,n)* sum(j=0,n-k, (-1)^j*j!*b(n,j)*b(n-k,j));
%o T(n,k) = f(n,k) - f(n,0) + 1; \\ _G. C. Greubel_, Nov 27 2019
%o (Magma)
%o function f(n,k)
%o B:=Binomial;
%o return B(n+k,n)*(&+[(-1)^j*Factorial(j)*B(n,j)*B(k,j): j in [0..k]]) + B(2*n-k,n)* (&+[(-1)^j*Factorial(j)*B(n,j)*B(n-k,j): j in [0..n-k]]); end function;
%o [f(n,k) -f(n,0) +1: k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 27 2019
%o (Sage)
%o def f(n, k):
%o b=binomial;
%o return b(n+k,n)*sum((-1)^j*factorial(j)*b(n,j)*b(k,j) for j in (0..k)) + b(2*n-k, n)*sum((-1)^j*factorial(j)*b(n,j)*b(n-k,j) for j in (0..n-k))
%o [[f(n, k) -f(n,0) +1 for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 27 2019
%o (GAP)
%o B:=Binomial;;
%o f:= function(n,k) return B(n+k,n)*Sum([0..k], j-> (-1)^j*Factorial(j)*B(n,j)* B(k,j)) + B(2*n-k,n)*Sum([0..n-k], j-> (-1)^j*Factorial(j)*B(n,j)*B(n-k,j )); end;
%o Flat(List([0..10], n-> List([0..n], k-> f(n,k)-f(n,0)+1 ))); # _G. C. Greubel_, Nov 27 2019
%K sign,tabl
%O 0,8
%A _Roger L. Bagula_, Apr 08 2010
|
