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 A037027 Skew Fibonacci-Pascal triangle read by rows. 37

%I

%S 1,1,1,2,2,1,3,5,3,1,5,10,9,4,1,8,20,22,14,5,1,13,38,51,40,20,6,1,21,

%T 71,111,105,65,27,7,1,34,130,233,256,190,98,35,8,1,55,235,474,594,511,

%U 315,140,44,9,1,89,420,942,1324,1295,924,490,192,54,10,1,144,744,1836

%N Skew Fibonacci-Pascal triangle read by rows.

%C T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (0,1), (1,0), (2,0). [Joerg Arndt, Jun 30 2011]

%C T(n,k) is the number of lattice paths of length n, starting from the origin and ending at (n,k), using horizontal steps H=(1,0), up steps U=(1,1) and down steps D=(1,-1), never containing UUU, DD, HD. For instance, for n=4 and k=2, we have the paths; HHUU, HUHU, HUUH, UHHU, UHUH, UUHH, UUDU, UDUU, UUUD. [Emanuele Munarini, Mar 15 2011]

%C Row sums form Pell numbers A000129, T(n,0) forms Fibonacci numbers A000045, T(n,1) forms A001629. T(n+k,n-k) is polynomial sequence of degree k.

%C T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, etc.).

%C As a Riordan array, this is (1/(1-x-x^2),x/(1-x-x^2)). An interesting factorization is (1/(1-x^2),x/(1-x^2))*(1/(1-x),x/(1-x)) [abs(A049310) times A007318]. Diagonal sums are the Jacobsthal numbers A001045(n+1). - Paul Barry, Jul 28 2005

%C T(n,k) = T'(n+1,k+1), T' given by [0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM, Nov 19 2005

%C Equals A049310 * A007318 as infinite lower triangular matrices. - Gary W. Adamson, Oct 28 2007

%C This triangle may also be obtained from the coefficients of the Morgan-Voyce polynomials defined by: Mv(x, n) = (x + 1)*Mv(x, n - 1) + Mv(x, n - 2). - Roger L. Bagula, Apr 09 2008

%C Row sums are A000129. - Roger L. Bagula, Apr 09 2008

%C Absolute value of coefficients of the characteristic polynomial of tridiagonal matrices with 1's along the main diagonal, and i's along the superdiagonal and the subdiagonal (where i=sqrt(-1), see Mathematica program). [From John M. Campbell, Aug 23 2011]

%C A037027 is jointly generated with A122075 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=u(n-1,x)+x*v(n-1,x). See the Mathematica section at A122075. [From Clark Kimberling, Mar 5 2012]

%D Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart., Nov. 2005, pp. 359-370.

%H _Reinhard Zumkeller_, <a href="/A037027/b037027.txt">Rows n = 0..150 of triangle, flattened</a>

%H T. Mansour, <a href="http://arXiv.org/abs/math.CO/0301157">Generalization of some identities involving the Fibonacci numbers</a>

%H P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved Fibonacci numbers</a>

%H Eric W. Weisstein, from MathWorld: <a href="http://mathworld.wolfram.com/Morgan-VoycePolynomials.html">Morgan-Voyce Polynomials.</a>

%F T(n, m) = T'(n-1, m)+T'(n-2, m)+T'(n-1, m-1), where T'(n, m) = T(n, m) for n >= 0 and 0< = m< = n and T'(n, m) = 0 otherwise.

%F G.f.: 1/(1 - y - y*z - y^2).

%F G.f. for k-th column: x/(1-x-x^2)^k.

%F T(n, m)= sum(binomial(m+k, m)*binomial(k, n-k-m), k=0..n-m), n>=m>=0, else 0. Wolfdieter Lang, Jun 17 2002

%F T(n, m) = ((n-m+1)*T(n, m-1) + 2*(n+m)*T(n-1, m-1))/(5*m), n >= m >= 1; T(n, 0)= A000045(n+1); T(n, m)= 0 if n<m. Wolfdieter Lang, Apr 12 2000

%F Chebyshev coefficient triangle (abs(A049310)) times Pascal's triangle (A007318) as product of lower triangular matrices. T(n, k)=sum{k=0..n, C((n+j)/2, j)(1+(-1)^(n+j))C(j, k)/2}. - Paul Barry, Dec 22 2004

%F Let R(n) = n-th row polynomial in x, with R(0)=1, then R(n+1)/R(n) equals the continued fraction [1+x;1+x, ...(1+x) occurring (n+1) times..., 1+x] for n>=0. - Paul D. Hanna, Feb 27 2004

%F T(n,k)=sum{j=0..n, C(n-j,j)*C(n-2*j,k)}; in Egorychev notation, T(n,k)=res_w(1-w-w^2)^(-k-1)*w^(-n+k+1). - Paul Barry, Sep 13 2006

%F sum(k=0..n, T(n,k)*x^k ) = A000045(n+1), A000129(n+1), A006190(n+1), A001076(n+1), A052918(n), A005668(n+1), A054413(n), A041025(n), A099371(n+1), A041041(n), A049666(n+1), A041061(n), A140455(n+1), A041085(n), A154597(n+1), A041113(n) for n = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 respectively. [From Philippe DELEHAM, Nov 29 2009]

%F T((m+1)*n+r-1,m*n+r-1)*r/(m*n+r)=sum(k=1..n, k/n*T((m+1)*n-k-1,m*n-1)*(r+k,r)), n>=m>1.

%F T(n-1,m-1)=m/n*sum(k=1..n-m+1,k*A000045(k)*T(n-k-1,m-2),k,1,n-m+1), n>=m>1; [From Vladimir Kruchinin, Mar 17 2011]

%e 1; 1,1; 2,2,1; 3,5,3,1; 5,10,9,4,1; etc.

%e Ratio of row polynomials R(3)/R(2) = (3+5*x+3*x^2+x^3)/(2+2*x+x^2) = [1+x;1+x,1+x].

%e Triangle begins:

%e ..................{1},

%e .................{1,1},

%e ................{2,2,1},

%e ...............{3,5,3,1},

%e ..............{5,10,9,4,1},

%e ............{8,20,22,14,5,1},

%e ..........{13,38,51,40,20,6,1},

%e ........{21,71,111,105,65,27,7,1},

%e ......{34,130,233,256,190,98,35,8,1},

%e ....{55,235,474,594,511,315,140,44,9,1},

%e {89,420,942,1324,1295,924,490,192,54,10,1}

%t Mv[x, -1] = 0; Mv[x, 0] = 1; Mv[x, 1] = 1 + x; Mv[x_, n_] := Mv[x, n] = ExpandAll[(x + 1)*Mv[x, n - 1] + Mv[x, n - 2]]; Table[ CoefficientList[ Mv[x, n], x], {n, 0, 10}] // Flatten - Roger L. Bagula, Apr 09 2008

%t Abs[Flatten[Table[CoefficientList[CharacteristicPolynomial[Array[KroneckerDelta[#1,#2]+KroneckerDelta[#1,#2+1]*I+KroneckerDelta[#1,#2-1]*I&,{n,n}],x],x],{n,1,20}]]] (* John M. Campbell, Aug 23 2011 *)

%o (PARI) T(n,k)=if(k<0|k>n,0,if(n==0&k==0,1,T(n-1,k)+T(n-1,k-1)+T(n-2,k))) (from Michael Somos)

%o (PARI) T(n,k)=if(n<k|k<0,0,polcoeff(contfracpnqn(vector(n,i,1+x))[1,1],k,x)) (from Paul D. Hanna)

%o a037027 n k = a037027_tabl !! n !! k

%o a037027_row n = a037027_tabl !! n

%o a037027_tabl = [1] : [1,1] : f [1] [1,1] where

%o f xs ys = ys' : f ys ys' where

%o ys' = zipWith3 (\u v w -> u + v + w) (ys ++ [0]) (xs ++ [0,0]) ([0] ++ ys)

%o -- _Reinhard Zumkeller_, Jul 07 2012

%Y A038112(n)=T(2n, n). A038137 is reflected version. Maximal row entries: A038149.

%Y Diagonal differences are in A055830. Vertical sums are in A091186.

%Y Cf. A049310.

%Y Cf. A000129.

%Y Cf. A155161, A122542, A059283.

%K easy,nonn,tabl

%O 0,4

%A Floor van Lamoen (fvlamoen(AT)hotmail.com), Jan 01 1999

%E Examples from _Paul D. Hanna_, Feb 27 2004

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