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%I #8 Sep 08 2022 08:45:41
%S 1,1,1,1,4,1,1,11,11,1,1,40,48,40,1,1,197,236,236,197,1,1,1206,1405,
%T 1438,1405,1206,1,1,8647,9859,10057,10057,9859,8647,1,1,70568,79226,
%U 80446,80616,80446,79226,70568,1,1,645129,715714,724394,725600,725600,724394,715714,645129,1
%N Triangle read by rows: T(n, k) = binomial(n, k) + 2*(1 + n! - k! - (n-k)!).
%C Row sum are: {1, 2, 6, 24, 130, 868, 6662, 57128, 541098, 5621676, 63682990, ...}.
%H G. C. Greubel, <a href="/A156049/b156049.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n, k) = binomial(n, k) + 2*(1 + n! - k! - (n-k)!).
%F Sum_{k=0..n} T(n, k) = 2^n + 2*(n+1) - 2*(n+1)! - 4* !(n+1), where !n =
%F A003422(n). - _G. C. Greubel_, Dec 01 2019
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 11, 11, 1;
%e 1, 40, 48, 40, 1;
%e 1, 197, 236, 236, 197, 1;
%e 1, 1206, 1405, 1438, 1405, 1206, 1;
%e 1, 8647, 9859, 10057, 10057, 9859, 8647, 1;
%e 1, 70568, 79226, 80446, 80616, 80446, 79226, 70568, 1;
%p seq(seq( binomial(n,k) + 2*(1+n!-k!-(n-k)!), k=0..n), n=0..10); # _G. C. Greubel_, Dec 01 2019
%t Table[Binomial[n, k] +2*(1 +n! -k! -(n-k)!), {n,0,10}, {k,0,n}]//Flatten
%o (PARI) T(n,k) = binomial(n,k) + 2*(1+n!-k!-(n-k)!); \\ _G. C. Greubel_, Dec 01 2019
%o (Magma) F:=Factorial; [Binomial(n,k) +2*(1 +F(n) -F(k) -F(n-k)): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 01 2019
%o (Sage) f=factorial; [[binomial(n,k) +2*(1 +f(n) -f(k) -f(n-k)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 01 2019
%o (GAP) F:=Factorial;; Flat(List([0..10], n-> List([0..n], k-> Binomial(n,k) +2*(1 +F(n) -F(k) -F(n-k)) ))); # _G. C. Greubel_, Dec 01 2019
%Y Cf. A003422.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 02 2009
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