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%I #8 Sep 08 2022 08:45:52
%S 1,1,1,1,6,1,1,15,15,1,1,37,60,37,1,1,125,212,212,125,1,1,641,904,
%T 1058,904,641,1,1,4375,5310,5990,5990,5310,4375,1,1,35351,40270,42546,
%U 43800,42546,40270,35351,1,1,322649,358918,367194,373320,373320,367194,358918,322649,1
%N Triangle, read by rows, T(n, k) = 2*binomial(n, k)*binomial(n+1, k)/(k+1) - (k! - n! + (n-k)!).
%C Row sums are: {1, 2, 8, 32, 136, 676, 4150, 31352, 280136, 2844164, 31958544, ...}.
%H G. C. Greubel, <a href="/A176152/b176152.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n, k) = 2*binomial(n, k)*binomial(n+1, k)/(k+1) - (k! - n! + (n-k)!).
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 6, 1;
%e 1, 15, 15, 1;
%e 1, 37, 60, 37, 1;
%e 1, 125, 212, 212, 125, 1;
%e 1, 641, 904, 1058, 904, 641, 1;
%e 1, 4375, 5310, 5990, 5990, 5310, 4375, 1;
%e 1, 35351, 40270, 42546, 43800, 42546, 40270, 35351, 1;
%e 1, 322649, 358918, 367194, 373320, 373320, 367194, 358918, 322649, 1;
%p T:= 2*binomial(n, k)*binomial(n+1, k)/(k+1) -(k! -n! +(n-k)!); seq(seq(T(n,k), k=0..n), n=0..10); # _G. C. Greubel_, Nov 23 2019
%t T[n_, k_] = 2*Binomial[n, k]*Binomial[n+1, k]/(k+1) -(k! -n! +(n-k)!); Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Nov 23 2019 *)
%o (PARI) T(n, k) = 2*binomial(n, k)*binomial(n+1, k)/(k+1) -(k!-n!+(n-k)!); \\ _G. C. Greubel_, Nov 23 2019
%o (Magma) F:=Factorial; [2*Binomial(n, k)*Binomial(n+1, k)/(k+1) - (F(k) - F(n) + F(n-k)): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 23 2019
%o (Sage) f=factorial; b=binomial; [[2*b(n, k)*b(n+1, k)/(k+1) -f(k) +f(n) - f(n-k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 23 2019
%o (GAP) F:=Factorial;; B:=Binomial;; Flat(List([0..10], n-> List([0..n], k-> 2*B(n, k)*B(n+1, k)/(k+1) -F(k) +F(n) -F(n-k) ))); # _G. C. Greubel_, Nov 23 2019
%Y Cf. A155170.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Apr 10 2010
%E Edited by _G. C. Greubel_, Nov 23 2019