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

1,3

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.

Links

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

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

Example

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;
...

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 *)

Prog

(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
(PARI) for(n=0, 10, for(k=0, n, print1(binomial(n, k)*(2*k+1), ", "))) \\ G. C. Greubel, Jan 29 2018
(Magma) /* As triangle */ [[(2*k+1)*Binomial(n, k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jan 29 2018
(GAP) Flat(List([0..100], n->List([1..n+1], k->Binomial(n, k-1)*(2*k-1)))); # Muniru A Asiru, Jan 30 2018

Crossrefs

Cf. A001787.

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

Sequence in context: A016575 A308948 A225246 * A208331 A061702 A112351

Adjacent sequences: A116663 A116664 A116665 * A116667 A116668 A116669

Keyword

nonn,tabl

Author

Gary W. Adamson, Feb 22 2006

Status

approved