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A073387 Convolution triangle of A002605(n) (generalized (2,2)-Fibonacci), n>=0. 16

%I

%S 1,2,1,6,4,1,16,16,6,1,44,56,30,8,1,120,188,128,48,10,1,328,608,504,

%T 240,70,12,1,896,1920,1872,1080,400,96,14,1,2448,5952,6672,4512,2020,

%U 616,126,16,1,6688,18192,23040,17856,9352,3444,896,160,18,1

%N Convolution triangle of A002605(n) (generalized (2,2)-Fibonacci), n>=0.

%C The g.f. for the row polynomials P(n,x) := Sum_{m=0..n} a(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.

%C The column sequences (without leading zeros) give A002605, A073388-94, A073397-8 for m=0..9. Row sums give A007482.

%C 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

%H W. Lang, <a href="http://www.itp.kit.edu/~wl/EISpub/A073387.text">First 10 rows</a>.

%F a(n, m) = 2*(p(m-1, n-m)*(n-m+1)*a(n-m+1) + q(m-1, n-m)*(n-m+2)*a(n-m))/(m!*12^m), n>=m>=1, with a(n)=a(n, m=0) := A002605(n), else 0; p(k, n) := Sum_{l=0..k} A(k, l)*n^(k-l) and q(k, n) := Sum_{l=0..k} B(k, l)*n^(k-l) with the number triangles A(k, m) := A073403(k, m) and B(k, m) := A073404(k, m).

%F a(n, m) = Sum_{k=0..floor((n-m)/2)} ((2^(n-m))*binomial(n-k, m)*binomial(n-m-k, k)*(1/2)^k) if n>m, else 0.

%F a(n, m) = ((n-m+1)*a(n, m-1) + 2*(n+m)*a(n-1, m-1))/(6*m), n >= m >= 1, a(n, 0) = A002605(n+1), else 0.

%F G.f. for column m (without leading zeros): 1/(1-2*x*(1+x))^(m+1), m>=0.

%F 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

%e Lower triangular matrix a(n,m), n >= m >= 0, else 0:

%e {1},

%e {2, 1},

%e {6, 4, 1},

%e {16, 16, 6, 1},

%e {44, 56, 30, 8, 1},

%e {120, 188, 128, 48, 10, 1},

%e {328, 608, 504, 240, 70, 12, 1},

%e ...

%p 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

%t a[n_, m_] := Sum[2^(n-m)*Binomial[n-k, m]*Binomial[n-m-k, k]*(1/2)^k, {k, 0, (n-m)/2}];

%t Table[a[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* _Jean-Fran├žois Alcover_, Jun 04 2019 *)

%Y Cf. A002605, A053121, A073403, A073404.

%K nonn,easy,tabl

%O 0,2

%A _Wolfdieter Lang_, Aug 02 2002

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Last modified August 19 08:17 EDT 2019. Contains 326115 sequences. (Running on oeis4.)