A123965 -id:A123965 - cp's OEIS frontend

A126124 Triangle, matrix inverse of A124733, companion to A123965.

1, -2, 1, 5, -5, 1, -13, 19, -8, 1, 34, -65, 42, -11, 1, -89, 210, -183, 74, -14, 1, 233, -654, 717, -394, 115, -17, 1, -610, 1985, -2622, 1825, -725, 165, -20, 1, 1597, -5911, 9134, -7703, 3885, -1203, 224, -23, 1

Offset: 1

Gary W. Adamson, Dec 17 2006

Left border (unsigned) = odd-indexed Fibonacci numbers. Left border (unsigned) of A123965 = even-indexed Fibonacci numbers.
Subtriangle of the triangle T(n,k) given by [0,-2,-1/2,-1/2,0,0,0,0,...] DELTA [1,0,1/2,-1/2,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 02 2007
Equals A129818*A130595 as lower triangular matrices. - Philippe Deléham, Oct 26 2007
Reversals = bisection of triangle A152063: (1; 1,2; 1,5,5; ...) having the following property: Product_{k=1..floor((n-1)/2)} (1 + 4*cos^2 k*2Pi/n) = the odd-indexed Fibonacci numbers. Example: x^3 - 8x^2 - 19x + 13 relates to the heptagon, and with k=1,2,3,..., the product = 13. - Gary W. Adamson, Aug 15 2010
Apart from signs, equals A123971.
Matrix inverse of A124733.

			First few rows of the triangle are:
    1;
   -2,    1;
    5,   -5,    1;
  -13,   19,   -8,    1;
   34,  -65,   42,  -11,    1;
  -89,  210, -183,   74,  -14,    1;
  ...
Triangle (n >= 0 and 0 <= k <= n) [0,-2,-1/2,-1/2,0,0,0,0,0,...] DELTA [1,0,1/2,-1/2,0,0,0,0,0,...] begins:
  1;
  0,    1;
  0,   -2,    1;
  0,    5,   -5,    1;
  0,  -13,   19,   -8,    1;
  0,   34,  -65,   42,  -11,    1;
  0,  -89,  210, -183,   74,  -14,    1;
  0,  233, -654,  717, -394,  115,  -17,    1;

Cf. A123965, A124733, A123971, A152063.

Sum_{k=1..n} (-1)^(n-k)*T(n,k) = A001835(n). - Philippe Deléham, Jul 14 2007
T(n,k) = T(n-1,k-1) - 3*T(n-1,k) - T(n-2,k). - Philippe Deléham, Dec 13 2011
T(n,k) = (-1)^(n+k)*Sum_{m=k..n} binomial(m,k)*binomial(m+n,2*m). - Wadim Zudilin, Jan 11 2012
G.f.: (1+x)*x*y/(1+3*x+x^2-x*y). - R. J. Mathar, Aug 11 2015

Corrected by Philippe Deléham, Jul 14 2007

More terms from Philippe Deléham, Dec 13 2011

A124025 Duplicate of A123965.

Original entry on oeis.org

1, 3, -1, 8, -6, 1, 21, -25, 9, -1, 55, -90, 51, -12, 1, 144, -300, 234, -86, 15, -1, 377, -954, 951, -480, 130, -18, 1, 987, -2939, 3573, -2305, 855, -183, 21, -1, 2584, -8850, 12707, -10008, 4740, -1386, 245, -24, 1, 6765, -26195, 43398, -40426, 23373, -8715, 2100, -316, 27, -1, 17711, -76500, 143682

Offset: 1

Roger L. Bagula, Oct 31 2006

dead

Joanne Dombrowski, Tridiagonal matrix representations of cyclic selfadjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334

Cf. A101950, A104562, A123965.

b[k_] = 3; a[k_] = -1; p[0, x] = 1; p[1, x] = (x - b[1])/a[1]; p[k_, x_] := p[k, x] = ((x - b[k - 1])*p[k - 1, x] - a[k - 2]*p[k - 2, x])/a[k - 1]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]

Recursive polynomial from a tridiagonal matrix version of A123965 ( first number different): p(k, x) = ((x - b(k - 1))*p(k - 1, x) - a(k - 2)*p(k - 2, x))/a(n - 1); a(n)=-1;b(n)=3;

A091965 Triangle read by rows: T(n,k) = number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and three types of steps H=(1,0) (left factors of 3-Motzkin steps).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 36, 29, 9, 1, 137, 132, 57, 12, 1, 543, 590, 315, 94, 15, 1, 2219, 2628, 1629, 612, 140, 18, 1, 9285, 11732, 8127, 3605, 1050, 195, 21, 1, 39587, 52608, 39718, 19992, 6950, 1656, 259, 24, 1, 171369, 237129, 191754, 106644, 42498, 12177, 2457

Offset: 0

Emeric Deutsch, Mar 13 2004

T(n,0) = A002212(n+1), T(n,1) = A045445(n+1); row sums give A026378.
The inverse is A207815. - Gary W. Adamson, Dec 17 2006 [corrected by Philippe Deléham, Feb 22 2012]
Reversal of A084536. - Philippe Deléham, Mar 23 2007
Triangle T(n,k), 0 <= k <= n, read by rows given by T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Mar 27 2007
This triangle belongs to the family of triangles defined by T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
5^n = (n-th row terms) dot (first n+1 terms in (1,2,3,...)). Example for row 4: 5^4 = 625 = (137, 132, 57, 12, 1) dot (1, 2, 3, 4, 5) = (137 + 264 + 171 + 48 + 5) = 625. - Gary W. Adamson, Jun 15 2011
Riordan array ((1-3*x-sqrt(1-6*x+5*x^2))/(2*x^2), (1-3*x-sqrt(1-6*x+5*x^2))/(2*x)). - Philippe Deléham, Feb 19 2012

			Triangle begins:
     1;
     3,    1;
    10,    6,    1;
    36,   29,    9,    1;
   137,  132,   57,   12,    1;
   543,  590,  315,   94,   15,    1;
  2219, 2628, 1629,  612,  140,   18,    1;
T(3,1)=29 because we have UDU, UUD, 9 HHU paths, 9 HUH paths and 9 UHH paths.
Production matrix begins
  3, 1;
  1, 3, 1;
  0, 1, 3, 1;
  0, 0, 1, 3, 1;
  0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 1;
- _Philippe Deléham_, Nov 07 2011

A. Nkwanta, Lattice paths and RNA secondary structures, DIMACS Series in Discrete Math. and Theoretical Computer Science, 34, 1997, 137-147.

Vincenzo Librandi, Rows n = 0..100, flattened
Shu-Chiuan Chang and Robert Shrock, Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips, J. Stat. Physics 137 (2009) 667, table 5.
Helmut Prodinger, The amplitude of Motzkin paths, arXiv:2104.07596 [math.CO], 2021. Mentions this sequence.
Helmut Prodinger, Multi-edge trees and 3-coloured Motzkin paths: bijective issues, arXiv:2105.03350 [math.CO], 2021.
Helmut Prodinger, Weighted unary-binary trees, Hex-trees, marked ordered trees, and related structures, arXiv:2106.14782 [math.CO], 2021.

Cf. A002212, A045445, A026378.

Cf. A123965.

nmax = 9; t[n_, k_] := ((k+1)*n!*Hypergeometric2F1[k+3/2, k-n, 2k+3, -4]) / ((k+1)!*(n-k)!); Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)
T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,
T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 3, 3], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

T(n,k):=(k+1)*sum((binomial(2*(m+1),m-k)*binomial(n,m))/(m+1),m,k,n); /* Vladimir Kruchinin, Oct 08 2011 */

@CachedFunction
def A091965(n,k):
    if n==0 and k==0: return 1
    if k<0 or k>n: return 0
    if k==0: return 3*A091965(n-1,0)+A091965(n-1,1)
    return A091965(n-1,k-1)+3*A091965(n-1,k)+A091965(n-1,k+1)
for n in (0..7):
    [A091965(n,k) for k in (0..n)] # Peter Luschny, Nov 05 2012

G.f.: G = 2/(1 - 3*z - 2*t*z + sqrt(1-6*z+5*z^2)). Alternatively, G = M/(1 - t*z*M), where M = 1 + 3*z*M + z^2*M^2.
Sum_{k>=0} T(m, k)*T(n, k) = T(m+n, 0) = A002212(m+n+1). - Philippe Deléham, Sep 14 2005
The triangle may also be generated from M^n * [1,0,0,0,...], where M = an infinite tridiagonal matrix with 1's in the super and subdiagonals and [3,3,3,...] in the main diagonal. - Gary W. Adamson, Dec 17 2006
Sum_{k=0..n} T(n,k)*(k+1) = 5^n. - Philippe Deléham, Mar 27 2007
Sum_{k=0..n} T(n,k)*x^k = A117641(n), A033321(n), A007317(n), A002212(n+1), A026378(n+1) for x = -3, -2, -1, 0, 1 respectively. - Philippe Deléham, Nov 28 2009
T(n,k) = (k+1)*Sum_{m=k..n} binomial(2*(m+1),m-k)*binomial(n,m)/(m+1). - Vladimir Kruchinin, Oct 08 2011
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x^2)*(1 + 3*x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022

A125662 A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).

Original entry on oeis.org

1, 3, 1, 8, 6, 1, 21, 25, 9, 1, 55, 90, 51, 12, 1, 144, 300, 234, 86, 15, 1, 377, 954, 951, 480, 130, 18, 1, 987, 2939, 3573, 2305, 855, 183, 21, 1, 2584, 8850, 12707, 10008, 4740, 1386, 245, 24, 1, 6765, 26195, 43398, 40426, 23373, 8715, 2100, 316, 27, 1

Offset: 0

Philippe Deléham, Jan 28 2007

Subtriangle of the triangle given by [0,3,-1/3,1/3,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Unsigned version of A123965.
From Philippe Deléham, Feb 19 2012: (Start)
Riordan array (1/(1-3*x+x^2), x/(1-3*x+x^2)).
Equals A078812*A007318 as infinite lower triangular matrices.
Triangle of coefficients of Chebyshev's S(n,x+3) polynomials (exponents of x in increasing order). (End)
For 1 <= k <= n, T(n,k) equals the number of (n-1)-length  words over {0,1,2,3} containing k-1 letters equal 3 and avoiding 01. - Milan Janjic, Dec 20 2016

			Triangle begins:
   1;
   3,  1;
   8,  6,  1;
  21, 25,  9,  1;
  55, 90, 51, 12,  1;
  ...
Triangle [0,3,-1/3,1/3,0,0,0,...] DELTA [1,0,0,0,0,0,...] begins:
  1;
  0,  1;
  0,  3,  1;
  0,  8,  6,  1;
  0, 21, 25,  9,  1;
  0, 55, 90, 51, 12,  1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Diagonal sums: A000244(powers of 3).
Row sums: A001353 (n+1).
Diagonals: A001906(n+1), A001871.
Cf. A000012, A008585, A062728.
Cf. Triangle of coefficients of Chebyshev's S(n,x+k) polynomials: A207824, A207823, A125662, A078812, A101950, A049310, A104562, A053122, A207815, A159764, A123967 for k = 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5 respectively.

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
A125662:= func< n,k | Abs( Coefficient(R!( Evaluate(ChebyshevU(n+1), (3-x)/2) ), k) ) >;
[A125662(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Aug 20 2023

With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 3 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)
Table[Abs[CoefficientList[ChebyshevU[n,(x-3)/2], x]], {n,0,12}]//Flatten (* G. C. Greubel, Aug 20 2023 *)

def A125662(n,k): return abs( ( chebyshev_U(n, (3-x)/2) ).series(x, n+2).list()[k] )
flatten([[A125662(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 20 2023

T(n,k) = T(n-1,k-1) + 3*T(n-1,k) - T(n-2,k); T(0,0)=1; T(n,k)=0 if k < 0 or k > n.
Sum_{k=0..n} T(n, k) = A001353(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000244(n+1).
G.f.: 1/(1-3*x+x^2-y*x). - Philippe Deléham, Feb 19 2012
From G. C. Greubel, Aug 20 2023: (Start)
T(n, k) = abs( [x^k]( ChebyshevU(n, (3-x)/2) ) ).
Sum_{k=0..n} (-1)^k*T(n, k) = A000027(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000225(n). (End)

a(45) corrected and a(51) added by Philippe Deléham, Feb 19 2012

A152063 Triangle read by rows. Coefficients of the Fibonacci product polynomials F(n) = Product_{k=1..(n - 1)/2} (1 + 4cos^2(kPi/n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 5, 1, 6, 8, 1, 8, 19, 13, 1, 9, 25, 21, 1, 11, 42, 65, 34, 1, 12, 51, 90, 55, 1, 14, 74, 183, 210, 89, 1, 15, 86, 234, 300, 144, 1, 17, 115, 394, 717, 654, 233, 6, 18, 130, 480, 951, 954, 377, 1, 20, 165, 725, 1825, 2622, 1985, 610, 1, 21, 183, 855

Offset: 1

Gary W. Adamson and Roger L. Bagula, Nov 22 2008

The triangle A125076 is formed by reading upward sloping diagonals. - Gary W. Adamson, Nov 26 2008

Bisection of the triangle: odd-indexed rows are reversals of the rows of A126124, even-indexed rows are the reversals of the rows of A123965. - Gary W. Adamson, Aug 15 2010

			First few rows of the triangle are:
1;
1;
1, 2;
1, 3;
1, 5, 5;
1, 6, 8;
1, 8, 19, 13;
1, 9, 25, 21;
1, 11, 42, 65, 34;
1, 12, 51, 90, 55;
1, 14, 74, 183, 210, 89;
1, 15, 86, 234, 300, 144;
1, 17, 115, 394, 717, 654, 233;
1, 18, 130, 480, 951, 954, 377;
1, 20, 165, 725, 1825, 2622, 1985, 610;
1, 21, 183, 855, 2305, 3573, 2939, 987;
...
By row, alternate signs (+,-,+,-,...) with descending exponents. Rows with n terms have exponents (n-1), (n-2), (n-3),...;
Example: There are two rows with 4 terms corresponding to the polynomials
x^3 - 8x^2 + 19x - 13 (roots associated with the heptagon); and
x^3 - 9x^2 + 25x - 21 (roots associated with the 9-gon (nonagon)).

James P. Bradshaw, Philipp Lampe, and Dusan Ziga, Snake graphs and their characteristic polynomials, arXiv:1910.11823 [math.CO], 2019. See 4.7 p. 16.
N. D. Cahill and D. A. Narayan, Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants, Fibonacci Quarterly, 42(3):216-221, 2004.
M. X. He, D. Simon and P. E. Ricci, Dynamics of the zeros of Fibonacci polynomials, Fibonacci Quarterly, 35(2):160-168, 1997.
V. E. Hoggatt and C. T. Long, Divisibility Properties of Generalized Fibonacci Polynomials, Fibonacci Quarterly, 12:113-120, 1974.

Cf. A000045, A002530 (row sums), A125076, A126124, A123965, A011973.

P := proc(n) option remember; if n < 5 then return
ifelse(n < 3, 1, ifelse(n = 3, 1 + 2*q, 1 + 3*q)) fi;
(1 + 3*q)*P(n - 2) - q^2*P(n - 4) end:
T := n -> local k; seq(coeff(P(n), q, k), k = 0..(n-1)/2):
for n from 1 to 12 do T(n) od;  # (after F. Chapoton)  Peter Luschny, May 27 2024
# Alternative:
P := n -> local k; add(binomial(n-k,k)*(1+x)^(floor(n/2)-k)*x^k, k=0..floor(n/2)):
T := n -> local k; seq(coeff(P(n), x, k), k = 0..n/2):
for n from 0 to 12 do T(n) od; # (after F. Chapoton) Peter Luschny, May 28 2024

Recurrence (as monic polynomials) P(n+4) = (1 + 3*q)*P(n+2) - q^2*P(n). - F. Chapoton, May 27 2024

As monic polynomials, these are the numerators of the polynomials from A011973 evaluated at 1/(1+q). - F. Chapoton, May 28 2024

cp's OEIS Frontend

A126124 Triangle, matrix inverse of A124733, companion to A123965.

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A124025 Duplicate of A123965.

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A091965 Triangle read by rows: T(n,k) = number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and three types of steps H=(1,0) (left factors of 3-Motzkin steps).

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A125662 A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).

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A152063 Triangle read by rows. Coefficients of the Fibonacci product polynomials F(n) = Product_{k=1..(n - 1)/2} (1 + 4*cos^2(k*Pi/n)).

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A152063 Triangle read by rows. Coefficients of the Fibonacci product polynomials F(n) = Product_{k=1..(n - 1)/2} (1 + 4cos^2(kPi/n)).