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A037027 Skew Fibonacci-Pascal triangle read by rows. 52

%I #142 Sep 22 2023 05:17:35

%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 Deléham_, 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). - _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. - _Clark Kimberling_, Mar 05 2012

%C For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - _Boris Putievskiy_, Aug 18 2013

%C For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 09 2013

%C Row n, for n>=0, shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^n which is the denominator of the n-th convergent of the continued fraction [x+1, x+1, x+1, ...]; see A230000. - _Clark Kimberling_, Nov 13 2013

%C T(n,k) is the number of ternary words of length n having k letters 2 and avoiding a runs of odd length for the letter 0. - _Milan Janjic_, Jan 14 2017

%C Let T(m, n, k) be an m-bonacci Pascal's triangle, where T(m, n, 0) gives the values of F(m, n), the n-th m-bonacci number, and T(m, n, k) gives the values for the k-th convolution of F(m, n). Then the classic Pascal triangle is T(1, n, k) and this sequence is T(2, n, k). T(m, n, k) is the number of compositions of n using only the positive integers 1, 1' and 2 through m, with the part 1' used exactly k times. G.f. for k-th column of T(m, n, k): x/(1 - x - x^2 - ... - x^m)^k. The row sum for T(m, n, k) is the number of compositions of n using only the positive integers 1, 1' and 2 through m. G.f. for row sum of T(m, n, k): 1/(1 - 2x - x^2 - ... - x^m). - _Gregory L. Simay_, Jul 24 2021

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

%H Harlan J. Brothers, <a href="http://www.brotherstechnology.com/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>

%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), #18.1.4.

%H T. Mansour, <a href="https://arxiv.org/abs/math/0301157">Generalization of some identities involving the Fibonacci numbers</a>, arXiv:math/0301157 [math.CO], 2003.

%H P. Moree, <a href="https://arxiv.org/abs/math/0311205">Convoluted convolved Fibonacci numbers</a>, arXiv:math/0311205 [math.CO], 2003.

%H Yidong Sun, <a href="http://www.fq.math.ca/Papers1/43-4/paper43-4-10b.pdf">Numerical Triangles and Several Classical Sequences</a>, Fib. Quart. 43, no. 4, Nov. 2005, pp. 359-370.

%H Eric Weisstein's World of Mathematics, <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_{k=0..n-m} binomial(m+k, m)*binomial(k, n-k-m), n >= m >= 0, otherwise 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_{j=0..n} binomial((n+j)/2, j)*(1+(-1)^(n+j))*binomial(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} binomial(n-j,j)*binomial(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 x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 respectively. - _Philippe Deléham_, 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. - _Vladimir Kruchinin_, Mar 17 2011

%F T(n,k) = binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -4) for n >= 1. - _Peter Luschny_, Apr 25 2016

%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;

%p T := (n,k) -> `if`(n=0,1,binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -4)): seq(seq(simplify(T(n,k)), k=0..n), n=0..10); # _Peter Luschny_, Apr 25 2016

%p # Uses function PMatrix from A357368. Adds a row above and a column to the left.

%p PMatrix(10, n -> combinat:-fibonacci(n)); # _Peter Luschny_, Oct 07 2022

%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 *)

%t T[n_, k_] := Binomial[n, k] Hypergeometric2F1[(k-n)/2, (k-n+1)/2, -n, -4];

%t Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 16 2019, after _Peter Luschny_ *)

%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)))}; /* _Michael Somos_, Sep 29 2003 */

%o (PARI) T(n,k)=if(n<k||k<0,0,polcoeff(contfracpnqn(vector(n,i,1+x))[1,1],k,x)) \\ _Paul D. Hanna_, Feb 27 2004

%o (Haskell)

%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. A007318, A049310, A000129, A155161, A122542, A059283, A228196, A228576.

%Y Some other Fibonacci-Pascal triangles: A027926, A036355, A074829, A105809, A109906, A111006, A114197, A162741, A228074.

%K easy,nonn,tabl

%O 0,4

%A _Floor van Lamoen_, Jan 01 1999

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

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)