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A073387 Convolution triangle of A002605(n) (generalized (2,2)-Fibonacci), n>=0. 16
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

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

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

LINKS

Table of n, a(n) for n=0..54.

W. Lang, First 10 rows.

FORMULA

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).

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.

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.

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

EXAMPLE

Lower triangular matrix a(n,m), n >= m >= 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},

  ...

MAPLE

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

MATHEMATICA

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

Table[a[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 04 2019 *)

CROSSREFS

Cf. A002605, A053121, A073403, A073404.

Sequence in context: A269505 A269479 A118040 * A259099 A125693 A094527

Adjacent sequences:  A073384 A073385 A073386 * A073388 A073389 A073390

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Aug 02 2002

STATUS

approved

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Last modified July 22 10:03 EDT 2019. Contains 325219 sequences. (Running on oeis4.)