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A073387
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Convolution triangle of A002605(n) (generalized (2,2)-Fibonacci), n>=0.
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16
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1, 2, 1, 6, 4, 1, 16, 16, 6, 1, 44, 56, 30, 8, 1, 120, 188, 128, 48, 10, 1, 328, 608, 504, 240, 70, 12, 1, 896, 1920, 1872, 1080, 400, 96, 14, 1, 2448, 5952, 6672, 4512, 2020, 616, 126, 16, 1, 6688, 18192, 23040, 17856, 9352, 3444, 896, 160, 18, 1
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OFFSET
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0,2
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COMMENTS
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The g.f. for the row polynomials P(n,x) = Sum_{m=0..n} T(n,m)*x^m is 1/(1-(2+x+2*z)*z). See Shapiro et al. reference and comment under A053121 for such convolution triangles.
T(n, k) is the number of words of length n over {0,1,2,3} having k letters 3 and avoiding runs of odd length for the letters 0,1. - Milan Janjic, Jan 14 2017
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LINKS
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FORMULA
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T(n, k) = 2*(p(k-1, n-k)*(n-k+1)*T(n-k+1) + q(k-1, n-k)*(n-k+2)*T(n-k))/(k!*12^k), n >= k >= 1, with T(n) = T(n, k=0) = A002605(n), else 0; p(m, n) = Sum_{j=0..m} A(m, j)*n^(m-j) and q(m, n) = Sum_{j=0..m} B(m, j)*n^(m-j) with the number triangles A(k, m) = A073403(k, m) and B(k, m) = A073404(k, m).
T(n, k) = Sum_{j=0..floor((n-k)/2)} 2^(n-k-j)*binomial(n-j, k)*binomial(n-k-j, j) if n > k, else 0.
T(n, k) = ((n-k+1)*T(n, k-1) + 2*(n+k)*T(n-1, k-1))/(6*k), n >= k >= 1, T(n, 0) = A002605(n+1), else 0.
G.f. for column m (without leading zeros): 1/(1-2*x*(1+x))^(m+1), m>=0.
T(n,k) = 2^(n-k)*binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -2) for n>=1. - Peter Luschny, Apr 25 2016
T(n, n-2) = 2*A005563(n-1), n >= 2.
T(n, n-3) = 4*A159920(n-1), n >= 2.
Sum_{k=0..n} (-1)^k*T(n, k) = A001045(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A015518(n+1). (End)
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EXAMPLE
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Lower triangular matrix, T(n,k), n >= k >= 0, else 0:
1;
2, 1;
6, 4, 1;
16, 16, 6, 1;
44, 56, 30, 8, 1;
120, 188, 128, 48, 10, 1;
328, 608, 504, 240, 70, 12, 1;
896, 1920, 1872, 1080, 400, 96, 14, 1;
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MAPLE
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T := (n, k) -> `if`(n=0, 1, 2^(n-k)*binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -2)): seq(seq(simplify(T(n, k)), k=0..n), n=0..10); # Peter Luschny, Apr 25 2016
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MATHEMATICA
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T[n_, k_]:=T[n, k]=Sum[2^(n-k-j)*Binomial[n-j, k]*Binomial[n-k-j, j], {j, 0, (n-k)/2}];
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PROG
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(Magma)
A073387:= func< n, k | (&+[2^(n-k-j)*Binomial(n-j, k)*Binomial(n-k-j, j): j in [0..Floor((n-k)/2)]]) >;
(SageMath)
def A073387(n, k): return sum(2^(n-k-j)*binomial(n-j, k)*binomial(n-k-j, j) for j in range(((n-k+2)//2)))
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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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