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 A169656 Triangle, read by rows, T(n, k) = (-1)^n*(n!/k!)^2*binomial(n-1, k-1). 1

%I #9 Sep 08 2022 08:45:49

%S -1,4,1,-36,-18,-1,576,432,48,1,-14400,-14400,-2400,-100,-1,518400,

%T 648000,144000,9000,180,1,-25401600,-38102400,-10584000,-882000,

%U -26460,-294,-1,1625702400,2844979200,948326400,98784000,3951360,65856,448,1

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

%C Row sums are: {-1, 5, -55, 1057, -31301, 1319581, -74996755, 5521809665, -510921831817, 58003632177301, ...}.

%H G. C. Greubel, <a href="/A169656/b169656.txt">Rows n = 1..100 of triangle, flattened</a>

%F T(n, k) = (-1)^n * (n!/k!)^2 * binomial(n-1, k-1).

%e Triangle begins as:

%e -1;

%e 4, 1;

%e -36, -18, -1;

%e 576, 432, 48, 1;

%e -14400, -14400, -2400, -100, -1;

%e 518400, 648000, 144000, 9000, 180, 1;

%e -25401600, -38102400, -10584000, -882000, -26460, -294, -1;

%p seq(seq( (-1)^n*(n!/k!)^2*binomial(n-1, k-1), k=1..n), n=1..10); # _G. C. Greubel_, Nov 28 2019

%t T[n_, k_]:= (-1)^n*(n!/k!)^2*Binomial[n-1, k-1]; Table[T[n, k], {n,10}, {k,n}]//Flatten

%o (PARI) T(n,k) = (-1)^n*(n!/k!)^2*binomial(n-1, k-1); \\ _G. C. Greubel_, Nov 28 2019

%o (Magma) F:=Factorial; [(-1)^n*(F(n)/F(k))^2*Binomial(n-1, k-1): k in [1..n], n in [1..10]]; // _G. C. Greubel_, Nov 28 2019

%o (Sage) f=factorial; [[(-1)^n*(f(n)/f(k))^2*binomial(n-1, k-1) for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, Nov 28 2019

%o (GAP) F:=Factorial;; Flat(List([1..10], n-> List([1..n], k-> (-1)^n*(F(n)/F(k) )^2*Binomial(n-1, k-1) ))); # _G. C. Greubel_, Nov 28 2019

%Y Cf. A008297.

%K sign,tabl

%O 1,2

%A _Roger L. Bagula_, Apr 05 2010

%E Edited by _G. C. Greubel_, Nov 28 2019

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Last modified July 21 15:14 EDT 2024. Contains 374474 sequences. (Running on oeis4.)