A116666 - OEIS

%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