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

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, 66, 13, 1, 21, 105, 245, 315, 231, 91, 15, 1, 24, 140, 392, 630, 616, 364, 120, 17, 1, 27, 180, 588, 1134, 1386, 1092, 540, 153, 19, 1, 30, 225, 840, 1890, 2772

Offset: 1

Author

Gary W. Adamson, Feb 22 2006

Keywords

• nonn
• tabl

Comments

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.

Examples

			First few rows of the array are:
1 1 1 1 1...
1 4 7 10 13...
1 4 12 25 43...
1 4 12 32 71...
1 4 12 32 80...
...
Then take differences of columns which become rows of the triangle:
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, 66, 13;
1, 21, 105, 245, 315, 231, 91, 15;
...
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of table, flattened

Crossrefs

Cf. A001787.

Cf. A007318, A005408, A002457 (central terms).

Programs

GAP

Flat(List([0..100],n->List([1..n+1],k->Binomial(n,k-1)*(2*k-1)))); # Muniru A Asiru, Jan 30 2018

Haskell

a116666 n k = a116666_tabl !! (n-1) !! (k-1)
a116666_row n = a116666_tabl !! (n-1)
a116666_tabl = zipWith (zipWith (*)) a007318_tabl a158405_tabl
-- Reinhard Zumkeller, Nov 02 2013

Magma

/* As triangle */ [[(2*k+1)*Binomial(n,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jan 29 2018

Maple

seq(seq(binomial(n,k-1)*(2*k-1), k=1..n+1),n=0..100); # Muniru A Asiru, Jan 30 2018

Mathematica

Table[Binomial[n,k]*(2*k+1), {n,0,10}, {k,0,n}] (* G. C. Greubel, Jan 29 2018 *)

PARI

for(n=0,10, for(k=0,n, print1(binomial(n,k)*(2*k+1), ", "))) \\ G. C. Greubel, Jan 29 2018

Formula

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.

T(n,k) = binomial(n,k-1) * (2*k - 1), 1 <= k <= n. - Reinhard Zumkeller, Nov 02 2013