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%I #14 Sep 08 2022 08:45:24
%S 1,1,3,1,6,5,1,9,15,7,1,12,30,28,9,1,15,50,70,45,11,1,18,75,140,135,
%T 66,13,1,21,105,245,315,231,91,15,1,24,140,392,630,616,364,120,17,1,
%U 27,180,588,1134,1386,1092,540,153,19,1,30,225,840,1890,2772
%N Triangle, row sums = number of edges in n-dimensional hypercubes.
%C Terms in the array rows tend to A001787, number of edges in n-dimensional hypercubes: 1, 4, 12, 32, 80, 192, 448... Row sums of the sequence also = A001787.
%H Reinhard Zumkeller, <a href="/A116666/b116666.txt">Rows n = 1..125 of table, flattened</a>
%F From an array, rows = binomial transforms of (1,0,0,0...); (1,3,0,0,0...); (1,3,5,0,0,0...); difference rows of the columns become rows of the triangle.
%F T(n,k) = binomial(n,k-1) * (2*k - 1), 1 <= k <= n. - _Reinhard Zumkeller_, Nov 02 2013
%e First few rows of the array are:
%e 1 1 1 1 1...
%e 1 4 7 10 13...
%e 1 4 12 25 43...
%e 1 4 12 32 71...
%e 1 4 12 32 80...
%e ...
%e Then take differences of columns which become rows of the triangle:
%e 1;
%e 1, 3;
%e 1, 6, 5;
%e 1, 9, 15, 7;
%e 1, 12, 30, 28, 9;
%e 1, 15, 50, 70, 45, 11;
%e 1, 18, 75, 140, 135, 66, 13;
%e 1, 21, 105, 245, 315, 231, 91, 15;
%e ...
%p seq(seq(binomial(n,k-1)*(2*k-1), k=1..n+1),n=0..100); # _Muniru A Asiru_, Jan 30 2018
%t Table[Binomial[n,k]*(2*k+1), {n,0,10}, {k,0,n}] (* _G. C. Greubel_, Jan 29 2018 *)
%o (Haskell)
%o a116666 n k = a116666_tabl !! (n-1) !! (k-1)
%o a116666_row n = a116666_tabl !! (n-1)
%o a116666_tabl = zipWith (zipWith (*)) a007318_tabl a158405_tabl
%o -- _Reinhard Zumkeller_, Nov 02 2013
%o (PARI) for(n=0,10, for(k=0,n, print1(binomial(n,k)*(2*k+1), ", "))) \\ _G. C. Greubel_, Jan 29 2018
%o (Magma) /* As triangle */ [[(2*k+1)*Binomial(n,k): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Jan 29 2018
%o (GAP) Flat(List([0..100],n->List([1..n+1],k->Binomial(n,k-1)*(2*k-1)))); # _Muniru A Asiru_, Jan 30 2018
%Y Cf. A001787.
%Y Cf. A007318, A005408, A002457 (central terms).
%K nonn,tabl
%O 1,3
%A _Gary W. Adamson_, Feb 22 2006
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