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A113413
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A Riordan array of coordination sequences.
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17
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1, 2, 1, 2, 4, 1, 2, 8, 6, 1, 2, 12, 18, 8, 1, 2, 16, 38, 32, 10, 1, 2, 20, 66, 88, 50, 12, 1, 2, 24, 102, 192, 170, 72, 14, 1, 2, 28, 146, 360, 450, 292, 98, 16, 1, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1, 2, 36, 258, 952, 1970, 2364, 1666, 688, 162, 20, 1, 2, 40, 326
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OFFSET
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0,2
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COMMENTS
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T(n,k) is the number of length n words on alphabet {0,1,2} with no two consecutive 1's and no two consecutive 2's and having exactly k 0's. - Geoffrey Critzer, Jun 11 2015
Triangle of coefficients (from low to high degree) of x^-n * vertex cover polynomial of the n-ladder graph P_2 \square p_n:
Psi_{L_1}: x*(2 + x) -> {2, 1}
Psi_{L_2}: x^2*(2 + 4 x + x^2) -> {2, 4, 1}
Psi_{L_3}: x^3*(2 + 8 x + 6 x^2 + x^3) -> {2, 8, 6, 1}
(End)
Let c(n, k), n > 0, be multiplicative sequences for some fixed integer k >= 0 with c(p^e, k) = T(e+k, k) for prime p and e >= 0. Then we have Dirichlet g.f.: Sum_{n>0} c(n, k) / n^s = zeta(s)^(2*k+2) / zeta(2*s)^(k+1). Examples: For k = 0 see A034444 and for k = 1 see A322328. Dirichlet convolution of c(n, k) and lambda(n) is Dirichlet inverse of c(n, k). - Werner Schulte, Oct 31 2022
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LINKS
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FORMULA
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Riordan array ((1+x)/(1-x), x(1+x)/(1-x)).
T(n, k) = Sum_{i=0..n-k} C(k+1, i)*C(n-i, k).
T(n, k) = Sum_{j=0..n-k} C(k+j, j)*C(k+1, n-k-j).
T(n, k) = D(n, k) + D(n-1, k) where D(n, k) = Sum_{j=0..n-k} C(n-k, j)*C(k, j)*2^j = A008288(n, k).
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k-1).
T(n, k) = Sum_{j=0..n} C(floor((n+j)/2), k)*C(k, floor((n-j)/2)). (End)
T(n, k) = C(n, k)*hypergeometric([-k-1, k-n], [-n], -1). - Peter Luschny, Sep 17 2014
T(n, k) = (Sum_{i=2..k+2} A137513(k+2, i) * (n-k)^(i-2)) / (k!) for 0 <= k < n (conjectured). - Werner Schulte, Oct 31 2022
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EXAMPLE
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Triangle begins
1;
2, 1;
2, 4, 1;
2, 8, 6, 1;
2, 12, 18, 8, 1;
2, 16, 38, 32, 10, 1;
2, 20, 66, 88, 50, 12, 1;
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MATHEMATICA
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nn = 10; Map[Select[#, # > 0 &] &, CoefficientList[Series[1/(1 - 2 x/(1 + x) - y x), {x, 0, nn}], {x, y}]] // Grid (* Geoffrey Critzer, Jun 11 2015 *)
CoefficientList[CoefficientList[Series[1/(1 - 2 x/(1 + x) - y x), {x, 0, 10}, {y, 0, 10}], x], y] (* Eric W. Weisstein, Feb 17 2016 *)
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PROG
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(Sage)
T = lambda n, k : binomial(n, k)*hypergeometric([-k-1, k-n], [-n], -1).simplify_hypergeometric()
A113413 = lambda n, k : 1 if n==0 and k==0 else T(n, k)
for n in (0..12): print([A113413(n, k) for k in (0..n)]) # Peter Luschny, Sep 17 2014 and Mar 16 2016
(Sage)
# Alternatively:
@cached_function
def prec(n, k):
if k==n: return 1
if k==0: return 0
return prec(n-1, k-1)+2*sum(prec(n-i, k-1) for i in (2..n-k+1))
return [prec(n, k) for k in (1..n)]
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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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