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 A169658 Triangle, read by rows, defined by T(n, k) = b(n, k) + b(n, n-k+1) - (b(n,1) + b(n,n)) + 1, where b(n, k) = (-1)^n*(n!/k!)^2 *binomial(n-1, k-1). 1

%I #8 Sep 08 2022 08:45:49

%S 1,1,1,1,2,1,1,-96,-96,1,1,-98,9602,-98,1,1,129780,-365400,-365400,

%T 129780,1,1,-12701092,14791142,23637602,14791142,-12701092,1,1,

%U 1219277248,-677310144,-1522967040,-1522967040,-677310144,1219277248,1

%N Triangle, read by rows, defined by T(n, k) = b(n, k) + b(n, n-k+1) - (b(n,1) + b(n,n)) + 1, where b(n, k) = (-1)^n*(n!/k!)^2 *binomial(n-1, k-1).

%C Row sums are: {1, 2, 4, -190, 9408, -471238, 27817704, -1961999870, 163293385984, -15674630045398, ...}.

%H G. C. Greubel, <a href="/A169658/b169658.txt">Rows n = 1..100 of triangle, flattened</a>

%F T(n, k) = b(n, k) + b(n, n-k+1) - b(n, n) - b(n, 1) + 1, where b(n, k) = (-1)^n*(n!/m!)^2 *binomial(n-1, k-1), where 1 <= k <= n, n >= 1.

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, -96, -96, 1;

%e 1, -98, 9602, -98, 1;

%e 1, 129780, -365400, -365400, 129780, 1;

%e 1, -12701092, 14791142, 23637602, 14791142, -12701092, 1;

%t L[n_, m_] = (-1)^n*(n!/m!)^2*Binomial[n-1, m-1];

%t t[n_, m_] = L[n, m] + L[n, n-m+1];

%t Table[t[n, m] - t[n, 1] + 1, {n, 1, 10}, {m, 1, n}]//Flatten

%o (PARI) b(n, k) = (-1)^n*(n!/k!)^2 *binomial(n-1, k-1);

%o t(n, k) = b(n, k) + b(n, n-k+1);

%o for(n=1, 10, for(k=1, n, print1(t(n,k) - t(n,1) + 1, ", "))) \\ _G. C. Greubel_, May 20 2019

%o (Magma)

%o b:= func< n,k | (-1)^n*(Factorial(n)/Factorial(k))^2*Binomial(n-1, k-1) >;

%o [[b(n, k) +b(n, n-k+1) -b(n,1) -b(n,n) +1: k in [1..n]]: n in [1..10]]; // _G. C. Greubel_, May 20 2019

%o (Sage)

%o def b(n, k): return (-1)^n*factorial(n-k)^2*binomial(n,k)^2*binomial(n-1, k-1)

%o def t(n, k): return b(n, k) + b(n, n-k+1)

%o [[t(n,k) - t(n,1) + 1 for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, May 20 2019

%Y Cf. A008297.

%K sign,tabl

%O 1,5

%A _Roger L. Bagula_, Apr 05 2010

%E Edited by _G. C. Greubel_, May 20 2019

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Last modified August 9 20:04 EDT 2024. Contains 375044 sequences. (Running on oeis4.)