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A116666 Triangle, row sums = number of edges in n-dimensional hypercubes. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Feb 22 2006
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)