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A201641
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Triangle read by rows, T(n,k) for 0<=k<=n, generalizes the colored Motzkin paths of A129400.
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3
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1, 2, 1, 8, 4, 1, 32, 20, 6, 1, 144, 96, 36, 8, 1, 672, 480, 200, 56, 10, 1, 3264, 2432, 1104, 352, 80, 12, 1, 16256, 12544, 6048, 2128, 560, 108, 14, 1, 82688, 65536, 33152, 12544, 3680, 832, 140, 16, 1, 427520, 346368, 182016, 72960, 23232, 5904, 1176, 176, 18, 1
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OFFSET
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0,2
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LINKS
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FORMULA
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Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n>=1 T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + 4*T(n-1,k+1).
T(n,k) = ((k+1)/(n+1))*2^(n-k)*Sum_{j=0..floor((n-k)/3)} (-1)^j*C(n+1,j) *C(2*n-k-3*j,n-k-3*j). - Vladimir Kruchinin, Apr 06 2019
T(n,k) = 2^n*Sum_{j=0..n}(C(n,j)*(C(n-j, j+k) - C(n-j, j+k+2))*2^(-k). - Peter Luschny, Dec 31 2019
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EXAMPLE
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Triangle begins as:
[0] [1]
[1] [2, 1]
[2] [8, 4, 1]
[3] [32, 20, 6, 1]
[4] [144, 96, 36, 8, 1]
[5] [672, 480, 200, 56, 10, 1]
[6] [3264, 2432, 1104, 352, 80, 12, 1]
[7] [16256, 12544, 6048, 2128, 560, 108, 14, 1]
[8] [82688, 65536, 33152, 12544, 3680, 832, 140, 16, 1]
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MAPLE
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T := (n, k) -> 2^n*add(binomial(n, j)*(binomial(n-j, j+k) - binomial(n-j, j+k+2)) *2^(-k), j=0..n); seq(seq(T(n, k), k=0..n), n=0..8); # Peter Luschny, Dec 31 2019
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MATHEMATICA
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T[n_, k_]:= If[k==n, 1, 2^(n-k)*((k+1)/(n+1))*Sum[(-1)^j*Binomial[n+1, j]* Binomial[2*n-k-3*j, n-k-3*j], {j, 0, Floor[(n-k)/3]}]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Apr 04 2019 *)
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PROG
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(Sage)
M = matrix(ZZ, dim, dim)
for n in range(dim): M[n, n] = 1
for n in (1..dim-1):
for k in (0..n-1):
M[n, k] = M[n-1, k-1]+2*M[n-1, k]+4*M[n-1, k+1]
return M
(Sage)
@CachedFunction
def T(n, k):
if k==n: return 1
else: return 2^(n-k)*((k+1)/(n+1))*sum((-1)^j*binomial(n+1, j)* binomial(2*n-k-3*j, n-k-3*j) for j in (0..floor((n-k)/3)))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 05 2019
(Maxima)
T(n, k):=(k+1)/(n+1)*2^(n-k)*sum((-1)^j*binomial(n+1, j)*binomial(2*n-k-3*j, n-k-3*j), j, 0, floor((n-k)/3)); /* Vladimir Kruchinin, Apr 06 2019 */
(PARI) {T(n, k) = if(k==n, 1, 2^(n-k)*((k+1)/(n+1))*sum(j=0, floor((n-k)/3), (-1)^j*binomial(n+1, j)*binomial(2*n-k-3*j, n-k-3*j)))};
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Apr 04 2019
(Magma) [[k eq n select 1 else 2^(n-k)*((k+1)/(n+1))*(&+[(-1)^j* Binomial(n+1, j)*Binomial(2*n-k-3*j, n-k-3*j): j in [0..Floor((n-k)/3)]]) :k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 05 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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