A176022 - OEIS

%I #7 Feb 15 2021 02:01:12

%S -2,3,3,-7,-18,-7,25,96,96,25,-121,-650,-800,-650,-121,721,5490,7500,

%T 7500,5490,721,-5041,-53067,-92610,-73500,-92610,-53067,-5041,40321,

%U 564704,1328096,987840,987840,1328096,564704,40321,-362881,-6532164,-20345472,-18373824,-10668672,-18373824,-20345472,-6532164,-362881

%N Triangle T(n, k) = A176013(n, k) + A176013(n, n-k+1), read by rows.

%C Row sums are: -2, 6, -32, 242, -2342, 27422, -374936, 5841922, -101897354, 1962916022, ...

%H G. C. Greubel, <a href="/A176022/b176022.txt">Rows n = 1..100 of the triangle, flattened</a>

%F T(n, k) = ((-1)^n*n!/(k*k!))*binomial(n-1, k-1)*binomial(n, k-1) + ((-1)^n*n!)/((n-k+1)*(n-k+1)!)*binomial(n-1, n-k)*binomial(n, n-k).

%F From _G. C. Greubel_, Feb 15 2021: (Start)

%F T(n, k) = A176013(n, k) + A176013(n, n-k+1), where A176013(n, k) = (-1)^n*(n!/(k*k!))*binomial(n-1, k-1)*binomial(n, k-1).

%F Sum_{k=1..n} T(n, k) = 2*(-1)^n * n! * Hypergeometric2F2(-n, -(n-1); 2, 2; 1). (End)

%e Triangle begins as:

%e      -2;

%e       3,      3;

%e      -7,    -18,      -7;

%e      25,     96,      96,     25;

%e    -121,   -650,    -800,   -650,   -121;

%e     721,   5490,    7500,   7500,   5490,     721;

%e   -5041, -53067,  -92610, -73500, -92610,  -53067,  -5041;

%e   40321, 564704, 1328096, 987840, 987840, 1328096, 564704, 40321;

%t (* First program *)

%t T[n_, m_]:= ((-1)^n*n!/(m*m!))*Binomial[n-1, m-1]*Binomial[n, m-1] + ((-1)^n*n!)/((n-m+1)*(n-m+1)!)*Binomial[n-1, n-m] Binomial[n, n-m];

%t Table[T[n, m], {m,n}], {n,10}]//Flatten

%t (* Second program *)

%t A176013[n_, k_] := (-1)^n*(n!/(k*k!))*Binomial[n-1, k-1]*Binomial[n, k-1];

%t T[n_, k_]:= A176013[n, k] + A176013[n, n-k+1];

%t Table[T[n, k], {n, 10}, {k, n}]//Flatten (* _G. C. Greubel_, Feb 15 2021 *)

%o (Sage)

%o def A176013(n, k): return (-1)^n*(factorial(n)/(k*factorial(k)))*binomial(n-1, k-1)*binomial(n, k-1)

%o flatten([[A176013(n, k) + A176013(n, n-k+1) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 15 2021

%o (Magma)

%o A176013:= func< n, k | (-1)^n*(Factorial(n)/(k*Factorial(k)))*Binomial(n-1, k-1)*Binomial(n, k-1) >;

%o [A176013(n, k) + A176013(n, n-k+1) : k in [1..n], n in [1..12]]; // _G. C. Greubel_, Feb 15 2021

%Y Cf. A008297, A176013, A176021.

%K sign,tabl

%O 1,1

%A _Roger L. Bagula_, Apr 06 2010

%E Edited by _G. C. Greubel_, Feb 15 2021