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A199478
Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.
1
1, 1, 1, 3, 1, 5, 2, 1, 7, 8, 1, 9, 18, 4, 1, 11, 32, 20, 1, 13, 50, 56, 8, 1, 15, 72, 120, 48, 1, 17, 98, 220, 160, 16, 1, 19, 128, 364, 400, 112, 1, 21, 162, 560, 840, 432, 32, 1, 23, 200, 816, 1568, 1232, 256, 1, 25, 242, 1140, 2688, 2912, 1120, 64, 1, 27
OFFSET
1,4
COMMENTS
Row sums are 2^n. T(n,k) is the number of n-letter words from alphabet {a,b,c} not starting with a, having no consecutive letters and exactly k b's. - David Scambler, Dec 10 2011
LINKS
Eddie Cheng, Ke Qiu and Zhizhang Shen, On the surface area of the augmented cubes, J. of Supercomputing, DOI: 10.1007/s11227-011-0641-1
EXAMPLE
Irregular triangle begins :
1, 1
1, 3
1, 5, 2
1, 7, 8
1, 9, 18, 4
1, 11, 32, 20
1, 13, 50, 56, 8
1, 15, 72, 120, 48
1, 17, 98, 220, 160, 16
1, 19, 128, 364, 400, 112
For n=2 one word has no b: ca, and three words have one b: ba, bc, cb.
MAPLE
scan3:=proc(a, M1) local lis, n, k;
lis:=[];
for n from 1 to M1 do for k from 0 to floor((n+1)/2) do
lis:=[op(lis), a(n, k)];
od: od:
lis;
end:
a:=(n, i)-> if i=0 then 1 else 2^(i-1)*(binomial(n-i+1, i)+binomial(n-i, i)); fi;
scan3(a, 15);
MATHEMATICA
T[ n_, k_] := Sum[ Binomial[ n - k, i - k] Binomial[ n - i + 1, 2 k - i], {i, 0, n}]; (* Michael Somos, Sep 20 2014 *)
PROG
(PARI) {T(n, k) = sum(i=0, n, binomial( n-k, i-k) * binomial( n-i+1, 2*k-i))}; /* Michael Somos, Sep 20 2014 */
CROSSREFS
Sequence in context: A130303 A100898 A101350 * A134867 A102573 A233940
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Nov 06 2011
STATUS
approved