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A037027
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Skew Fibonacci-Pascal triangle read by rows.
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31
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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, 71, 111, 105, 65, 27, 7, 1, 34, 130, 233, 256, 190, 98, 35, 8, 1, 55, 235, 474, 594, 511, 315, 140, 44, 9, 1, 89, 420, 942, 1324, 1295, 924, 490, 192, 54, 10, 1, 144, 744, 1836
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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]
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]
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.
T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, etc.).
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
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
Equals A049310 * A007318 as infinite lower triangular matrices. - Gary W. Adamson, Oct 28 2007
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
Row sums are A000129. - Roger L. Bagula, Apr 09 2008
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]
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REFERENCES
| Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart., Nov. 2005, pp. 359-370.
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LINKS
| H. J. Brothers, Pascal's Prism: Supplementary Material, PDF version.
T. Mansour, Generalization of some identities involving the Fibonacci numbers
P. Moree, Convoluted convolved Fibonacci numbers
Eric W. Weisstein, from MathWorld: Morgan-Voyce Polynomials.
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FORMULA
| 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.
G.f.: 1/(1 - y - y*z - y^2).
G.f. for k-th column: x/(1-x-x^2)^k.
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
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
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
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
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
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]
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.
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]
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EXAMPLE
| 1; 1,1; 2,2,1; 3,5,3,1; 5,10,9,4,1; etc.
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].
Triangle begins:
..................{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},
........{21,71,111,105,65,27,7,1},
......{34,130,233,256,190,98,35,8,1},
....{55,235,474,594,511,315,140,44,9,1},
{89,420,942,1324,1295,924,490,192,54,10,1}
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MATHEMATICA
| Clear[Mv, a] Mv[x, -1] = 0; Mv[x, 0] = 1; Mv[x, 1] = 1 + x; Mv[x_, n_] := Mv[x, n] = (x + 1)*Mv[x, n - 1] + Mv[x, n - 2]; Table[ExpandAll[Mv[x, n]], {n, 0, 10}]; a = Table[CoefficientList[Mv[x, n], x], {n, 0, 10}]; Flatten[a] - Roger L. Bagula, Apr 09 2008
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 *)
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PROG
| (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)
(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)
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CROSSREFS
| A038112(n)=T(2n, n). A038137 is reflected version. Maximal row entries: A038149.
Diagonal differences are in A055830. Vertical sums are in A091186.
Cf. A049310.
Cf. A000129.
Sequence in context: A081572 A144287 A106196 * A182810 A139375 A106198
Adjacent sequences: A037024 A037025 A037026 * A037028 A037029 A037030
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Floor van Lamoen (fvlamoen(AT)hotmail.com), Jan 01 1999
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EXTENSIONS
| Examples from Paul D. Hanna (pauldhanna(AT)juno.com), Feb 27 2004
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