cp's OEIS Frontend

A116666 Triangle, row sums = number of edges in n-dimensional hypercubes.

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