A066170 -id:A066170 - cp's OEIS frontend

A069009 Let M denote the 6 X 6 matrix with rows / 1,1,1,1,1,1 / 1,1,1,1,1,0 / 1,1,1,1,0,0 / 1,1,1,0,0,0 / 1,1,0,0,0,0 / 1,0,0,0,0,0 / and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1); then a(n) = t(n).

1, 3, 15, 59, 250, 1030, 4283, 17752, 73658, 305513, 1267344, 5257031, 21806850, 90457205, 375227042, 1556484658, 6456477531, 26782210229, 111095686086, 460837670465, 1911607611040, 7929568022610, 32892759309540

Offset: 0

Benoit Cloitre, Apr 02 2002

This sequence is related to the tridecagon or triskaidecagon (13-gon).

The lengths of the diagonals of the regular tridecagon are r[k] = sin(k*Pi/13)/sin(Pi/13), 1 <= k <= 6, where r[1] = 1 is the length of the edge.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for linear recurrences with constant coefficients, signature (3, 6, -4, -5, 1, 1).

Cf. A066170.
Cf. A006359, A069007, A069008, A069009, A070778, A006359 (offset), for x(n), y(n), z(n), t(n), u(n), v(n).
Cf. A120747 (m = 5: hendecagon or 11-gon)

nmax:=22: with(LinearAlgebra): M:=Matrix([[1,1,1,1,1,1], [1,1,1,1,1,0], [1,1,1,1,0,0], [1,1,1,0,0,0], [1,1,0,0,0,0], [1,0,0,0,0,0]]): v:= Vector[row]([1,1,1,1,1,1]): for n from 0 to nmax do b:=evalm(v&*M^n); a(n):=b[4] od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Aug 03 2011
nmax:=24: m:=6: for k from 1 to m-1 do T(1,k):=0 od: T(1,m):=1: for n from 2 to nmax do for k from 1 to m do T(n,k):= add(T(n-1,k1), k1=m-k+1..m) od: od: for n from 1 to nmax/3 do seq(T(n,k), k=1..m) od; for n from 2 to nmax do a(n-2):=T(n,3) od: seq(a(n), n=0..nmax-2); # Johannes W. Meijer, Aug 03 2011

b = {1, -3, -6, 4, 5, -1, -1}; p[x_] := Sum[x^(n - 1)*b[[8 - n]], {n, 1, 7}] q[x_] := ExpandAll[x^6*p[1/x]] Table[ SeriesCoefficient[ Series[x/q[x], {x, 0, 30}], n], {n, 0, 30}] (* Roger L. Bagula and Gary W. Adamson, Sep 19 2006 *)
CoefficientList[Series[1/(1 - 3 x - 6 x^2 + 4 x^3 + 5 x^4 - x^5 - x^6), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 19 2015 *)

Vec(1/(1-3*x-6*x^2+4*x^3+5*x^4-x^5-x^6)+O(x^33)) \\ Joerg Arndt, Sep 19 2015

G.f.: 1/(1-3*x-6*x^2+4*x^3+5*x^4-x^5-x^6). - Roger L. Bagula and Gary W. Adamson, Sep 19 2006
a(n-2) = T(n,3) with T(n,k) = sum(T(n-1,k1), k1=7-k..6), T(1,1) = T(1,2) = T(1,3) = T(1,4) = T(1,5) = 0 and T(1,6) = 1, n>=1 and 1 <= k <= 6. [Steinbach]
sum(T(n,k)*r[k], k=1..6) = r[6]^n, n>=1, with r[k] = sin(k*Pi/13)/sin(Pi/13). [Steinbach]

Edited by Henry Bottomley, May 06 2002

Information added by Johannes W. Meijer, Aug 03 2011

A030111 Triangular array in which k-th entry in n-th row is C([ (n+k)/2 ],k) (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 3, 4, 1, 1, 3, 6, 4, 5, 1, 1, 4, 6, 10, 5, 6, 1, 1, 4, 10, 10, 15, 6, 7, 1, 1, 5, 10, 20, 15, 21, 7, 8, 1, 1, 5, 15, 20, 35, 21, 28, 8, 9, 1, 1, 6, 15, 35, 35, 56, 28, 36, 9, 10, 1, 1, 6, 21, 35, 70, 56, 84, 36, 45, 10, 11, 1, 1, 7, 21, 56, 70, 126, 84, 120, 45, 55, 11, 12, 1

Offset: 1

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Same as A046854, but missing the initial column of ones.

Riordan array (1/((1-x)(1-x^2)),x/(1-x^2)). Diagonal sums are A052551. - Paul Barry, Sep 30 2006

			1;
1 1;
2 1 1;
2 3 1 1;
3 3 4 1 1;
3 6 4 5 1 1;
...

Indranil Ghosh, Rows 0..125, flattened

Cf. A066170.

Flatten[Table[Binomial[Floor[(n+k)/2],k],{n,20},{k,n}]] (* Harvey P. Dale, Jun 03 2014 *)

{T(n, k) = binomial((n+k)\2, k)}; /* Michael Somos, Jul 23 1999 */

printp(matrix(8,8,n,k,binomial((n+k)\2,k)))

for(n=1,7, for(k=1,n,print1(binomial((n+k)\2,k)); if(k==n,print1("; ")); print1(" ")))

G.f.: 1 / (1 - x - xy - x^2 + x^2y + x^3). - Ralf Stephan, Feb 13 2005

Sum(k=1, n, T(n, k)) = F(n+2)-1 where F(n) is the n-th Fibonacci number. - Benoit Cloitre, Oct 07 2002

Description corrected by Michael Somos, Jul 23 1999

Corrected and extended by Harvey P. Dale, Jun 03 2014

A120747 Sequence relating to the 11-gon (or hendecagon).

Original entry on oeis.org

0, 1, 4, 14, 50, 175, 616, 2163, 7601, 26703, 93819, 329615, 1158052, 4068623, 14294449, 50221212, 176444054, 619907431, 2177943781, 7651850657, 26883530748, 94450905714, 331837870408, 1165858298498, 4096053203771, 14390815650209, 50559786403254

Offset: 1

Gary W. Adamson, Jul 01 2006

The hendecagon is an 11-sided polygon. The preferred word in the OEIS is 11-gon.
The lengths of the diagonals of the regular 11-gon are r[k] = sin(k*Pi/11)/sin(Pi/11), 1 <= k <= 5, where r[1] = 1 is the length of the edge.
The value of limit(a(n)/a(n-1),n=infinity) equals the longest diagonal r[5].
The a(n) equal the matrix elements M^n[1,2], where M = Matrix([[1,1,1,1,1], [1,1,1,1,0], [1,1,1,0,0], [1,1,0,0,0], [1,0,0,0,0]]). The characteristic polynomial of M is (x^5 - 3x^4 - 3x^3 + 4x^2 + x - 1) with roots x1 = -r[4]/r[3], x2 = -r[2]/r[4], x3 = r[1]/r[2], x4 = r[3]/r[5] and x5 = r[5]/r[1].
Note that M^4*[1,0,0,0,0] = [55, 50, 41, 29, 15] which are all terms of the 5-wave sequence A038201. This is also the case for the terms of M^n*[1,0,0,0,0], n>=1.

			From _Johannes W. Meijer_, Aug 03 2011: (Start)
The lengths of the regular hendecagon edge and diagonals are:
  r[1] = 1.000000000, r[2] = 1.918985948, r[3] = 2.682507066,
  r[4] = 3.228707416, r[5] = 3.513337092.
The first few rows of the T(n,k) array are, n>=1, 1 <= k <=5:
    0,   0,   0,   0,   1, ...
    1,   1,   1,   1,   1, ...
    1,   2,   3,   4,   5, ...
    5,   9,  12,  14,  15, ...
   15,  29,  41,  50,  55, ...
   55, 105, 146, 175, 190, ...
  190, 365, 511, 616, 671, ... (End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jay Kappraff, Slavik Jablan, Gary W. Adamson and Radmila Sazdanovich, Golden Fields, Generalized Fibonacci Sequences and Chaotic Matrices, Forma, Vol. 19 No. 4, pp. 367-387, 2004.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, MR 1439165
Eric Weisstein's World of Mathematics, Hendecagon.
Index entries for linear recurrences with constant coefficients, signature (3,3,-4,-1,1).

Cf. A006358, A033304, A038201, A052963, A065941, A066170.
From Johannes W. Meijer, Aug 03 2011: (Start)
Cf. A006358 (T(n+2,1) and T(n+1,5)), A069006 (T(n+1,2)), A038342 (T(n+1,3)), this sequence (T(n,4)) (m=5: hendecagon or 11-gon).
Cf. A000045 (m=2; pentagon or 5-gon); A006356, A006054 and A038196 (m=3: heptagon or 7-gon); A006357, A076264, A091024 and A038197 (m=4: enneagon or 9-gon); A006359, A069007, A069008, A069009, A070778 (m=6; tridecagon or 13-gon); A025030 (m=7: pentadecagon or 15-gon); A030112 (m=8: heptadecagon or 17-gon). (End)

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x^2*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5) )); // G. C. Greubel, Nov 13 2022

nmax:=27: m:=5: for k from 1 to m-1 do T(1,k):=0 od: T(1,m):=1: for n from 2 to nmax do for k from 1 to m do T(n,k):= add(T(n-1,k1), k1=m-k+1..m) od: od: for n from 1 to nmax/3 do seq(T(n,k), k=1..m) od; for n from 1 to nmax do a(n):=T(n,4) od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Aug 03 2011

LinearRecurrence[{3, 3, -4, -1, 1}, {0, 1, 4, 14, 50}, 41] (* G. C. Greubel, Nov 13 2022 *)

def A120747_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5) ).list()
A120747_list(40) # G. C. Greubel, Nov 13 2022

a(n) = 3*a(n-1) + 3*a(n-2) - 4*a(n-3) - a(n-4) + a(n-5).
G.f.: x^2*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
From Johannes W. Meijer, Aug 03 2011: (Start)
a(n) = T(n,4) with T(n,k) = Sum_{k1 = 6-k..6} T(n-1, k1), T(1,1) = T(1,2) = T(1,3) = T(1,4) = 0 and T(1,5) = 1, n>=1 and 1 <= k <= 5. [Steinbach]
Sum_{k=1..5} T(n,k)*r[k] = r[5]^n, n>=1. [Steinbach]
r[k] = sin(k*Pi/11)/sin(Pi/11), 1 <= k <= 5. [Kappraff]
Sum_{k=1..5} T(n,k) = A006358(n-1).
Limit_{n -> 00} T(n,k)/T(n-1,k) = r[5], 1 <= k <= 5.
sequence(sequence( T(n,k), k=2..5), n>=1) = A038201(n-4).
G.f.: (x^2*(x - x1)*(x - x2))/((x - x3)*(x - x4)*(x - x5)*(x - x6)*(x - x7)) with x1 = phi, x2 = (1-phi), x3 = r[1] - r[3], x4 = r[3] - r[5], x5 = r[5] - r[4], x6 = r[4] - r[2], x7 = r[2], where phi = (1 + sqrt(5))/2 is the golden ratio A001622. (End)

Edited and information added by Johannes W. Meijer, Aug 03 2011

A122765 Triangle read by rows: Let p(k, x) = x*p(k-1, x) - p(k-2, x). Then T(k,x) = dp(k,x)/dx.

Original entry on oeis.org

1, -1, 2, -2, -2, 3, 2, -6, -3, 4, 3, 6, -12, -4, 5, -3, 12, 12, -20, -5, 6, -4, -12, 30, 20, -30, -6, 7, 4, -20, -30, 60, 30, -42, -7, 8, 5, 20, -60, -60, 105, 42, -56, -8, 9, -5, 30, 60, -140, -105, 168, 56, -72, -9, 10

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 22 2006

Based on the coefficients of derivatives of the polynomials in A130777.

			Triangle begins as:
   1;
  -1,   2;
  -2,  -2,   3;
   2,  -6,  -3,   4;
   3,   6, -12,  -4,   5;
  -3,  12,  12, -20,  -5,   6;
  -4, -12,  30,  20, -30,  -6,   7;
   4, -20, -30,  60,  30, -42,  -7,   8;
   5,  20, -60, -60, 105,  42, -56,  -8,  9;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Cf. A000217, A026741, A046854, A066170, A076118, A122766, A130777, A165202.

A122765:= func< n,k | k*(-1)^Binomial(n-k+1, 2)*Binomial(Floor((n+k)/2), k) >;
[A122765(n,k): k in [1..n], n in [1..14]]; // G. C. Greubel, Dec 30 2022

(* First program *)
p[0,x]=1; p[1,x]=x-1; p[k_,x_]:= p[k, x]= x*p[k-1,x] -p[k-2,x]; a = Table[Expand[p[n, x]], {n, 0, 10}]; Table[CoefficientList[D[a[[n]], x], x], {n, 2, 10}]//Flatten
(* Second program *)
T[n_, k_]:= k*(-1)^Binomial[n-k+1,2]*Binomial[Floor[(n+k)/2], k];
Table[T[n, k], {n,14}, {k,n}]//Flatten (* G. C. Greubel, Dec 30 2022 *)

tpol(n) = if (n<=0, 1, if (n==1, x-1, x*tpol(n-1) -tpol(n-2)));
lista(nn) = {for(n=0, nn, pol = deriv(tpol(n)); for (k=0, poldegree(pol), print1(polcoeff(pol, k), ", ");););} \\ Michel Marcus, Feb 07 2014

def A122765(n, k): return k*(-1)^binomial(n-k+1, 2)*binomial(((n+k)//2), k)
flatten( [[A122765(n,k) for k in range(1,n+1)] for n in range(1,15)] ) # G. C. Greubel, Dec 30 2022

From G. C. Greubel, Dec 30 2022: (Start)
T(n, k) = coefficient [x^k]( p(n, x) ), where p(n,x) = (2/(x^2-4))*((n+1)*chebyshev_T(n+1,x/2) -n*chebyshev_T(n,x/2) - (x/2)*(chebyshev_U(n,x/2) - chebyshev_U(n-1,x/2))).
T(n, k) = k*(-1)^binomial(n-k+1, 2)*binomial(floor((n+k)/2), k).
T(n, n) = n.
T(n, n-1) = -(n-1).
T(n, n-2) = -2*A000217(n-2).
T(n, n-3) = 2*A000217(n-3).
T(n, 1) = (-1)^binomial(n, 2)*floor((n+1)/2).
T(n, 2) = 2*(-1)^binomial(n-1, 2)*binomial(floor((n+2)/2), 2).
Sum_{k=1..n} T(n, k) = A076118(n).
Sum_{k=1..n} (-1)^k*T(n, k) = (-1)^(n-1)*A165202(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = [n=1] - [n=2].
Sum_{k=1..floor((n+1)/2)} (-1)^k*T(n-k+1, k) = (-1)^binomial(n+1, 2)*b(n), where b(n) = 4^floor(n/4)*A026741(n/2) if n is even and b(n) = 4^floor((n-1)/4)*A026741((n-1)/4) if n is odd. (End)

Name corrected and more terms from Michel Marcus, Feb 07 2014

A052963 a(0)=1, a(1)=2, a(2)=5, a(n) = 3*a(n+2) - a(n+3).

Original entry on oeis.org

1, 2, 5, 14, 40, 115, 331, 953, 2744, 7901, 22750, 65506, 188617, 543101, 1563797, 4502774, 12965221, 37331866, 107492824, 309513251, 891207887, 2566130837, 7388879260, 21275429893, 61260158842, 176391597266, 507899361905

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Equals the INVERT transform of the Pell sequence prefaced with a "1": (1, 1, 2, 5, 12, 29, ...). - Gary W. Adamson, May 01 2009

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1034
Elena Barcucci, Antonio Bernini, and Renzo Pinzani, Sequences from Fibonacci to Catalan: A combinatorial interpretation via Dyck paths, RAIRO-Theor. Inf. Appl. (2024) Vol. 58, Art. No. 8. See p. 14.
Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 14.
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16; arXiv preprint, arXiv:1302.2274 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Cf. A066170, A076264.

spec := [S,{S=Sequence(Union(Prod(Sequence(Union(Prod(Z,Z),Z)),Z),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

LinearRecurrence[{3,0,-1},{1,2,5},30] (* Harvey P. Dale, Dec 26 2015 *)

G.f.: -(-1+x+x^2)/(1-3*x+x^3).
a(n) = Sum((1/9)*(1+2*_alpha+_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-3*_Z+_Z^3)). [in Maple notation]
a(n)/a(n-1) tends to 2.8793852... = 1/(2*cos(4*Pi/9)), a root of x^3 -3x^2 + 1 (the characteristic polynomial of the 3 X 3 matrix). The latter polynomial is a factor (with (x + 1)) of the 4th degree polynomial of A066170: x^4 - 2x^3 - 3x^2 + x + 1. Given the 3 X 3 matrix [0 1 0 / 0 0 1 / -1 0 3], (M^n)*[1 1 1] = [a(n-2), a(n-1), a(n)]. - Gary W. Adamson, Feb 29 2004
a(n) = A076264(n)-A076264(n-1)-A076264(n-2). - R. J. Mathar, Feb 27 2019

More terms from James Sellers, Jun 05 2000

A122610 Triangle read by rows: T(n,k) is coefficient of x^k in Sum_{m=0..n} x^m(1-x)^(n-m)(-1)^[(m+1)/2]*binomial(m-[(m+1)/2],[m/2]).

Original entry on oeis.org

1, 1, -2, 1, -3, 1, 1, -4, 3, 1, 1, -5, 6, 1, -2, 1, -6, 10, -1, -6, 1, 1, -7, 15, -6, -11, 6, 1, 1, -8, 21, -15, -15, 18, 1, -2, 1, -9, 28, -29, -15, 39, -6, -9, 1, 1, -10, 36, -49, -7, 69, -30, -21, 9, 1, 1, -11, 45, -76, 14, 105, -84, -30, 36, 1, -2, 1, -12, 55, -111, 54, 140, -182, -15, 96, -14, -12, 1, 1, -13, 66, -155

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 20 2006

			1;
1, -2;
1, -3,  1;
1, -4,  3,   1;
1, -5,  6,   1,  -2;
1, -6, 10,  -1,  -6,  1;
1, -7, 15,  -6, -11,  6, 1;
1, -8, 21, -15, -15, 18, 1, -2;

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Cf. A066170.

T[n_, k_] := (-1)^Floor[(k + 1)/2]*Binomial[n - Floor[(k + 1)/2], Floor[k/2]]; a = Table[CoefficientList[Sum[T[n, k]*p^k*(1 - p)^(n -k), {k, 0, n}], p], {n, 0, 10}]; Flatten[a]

{T(n,k)=local(A); if(k<0||k>n, 0, A=sum(k=0, n, x^k*(1-x)^(n-k)*(-1)^((k+1)\2)*binomial(n-((k+1)\2),k\2)); polcoeff(A,k))}

@CachedFunction
def T(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 2*T(n-1,k) if n==1 else T(n-1,k)
    return T(n-1,k-1) - T(n-2,k) - h
A122610 = lambda n,k: T(n,n-k)
for n in (0..9): [A122610(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Edited by N. J. A. Sloane, Sep 24 2006

Offset set to 0 by Michel Marcus, Feb 07 2014

A122595 Expansion of x/(1 - 3*x + x^2 + x^3 - x^4).

Original entry on oeis.org

1, 3, 8, 20, 50, 125, 313, 784, 1964, 4920, 12325, 30875, 77344, 193752, 485362, 1215865, 3045825, 7630000, 19113672, 47881056, 119945321, 300471235, 752701000, 1885567500, 4723475586, 11832629493, 29641546393, 74254101600

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 19 2006

			x + 3*x^2 + 8*x^3 + 20*x^4 + 50*x^5 + 125*x^6 + 313*x^7 + 784*x^8 + 1964*x^9 + ...

Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,1).

Cf. A066170.

I:=[1, 3, 8, 20]; [n le 4 select I[n] else 3*Self(n-1)-Self(n-2)-Self(n-3)+Self(n-4): n in [1..40]] // Vincenzo Librandi, Aug 07 2012

nn = 30; CoefficientList[Series[x/(1 - 3*x + x^2 + x^3 - x^4), {x, 0, nn}], x]
LinearRecurrence[{3,-1,-1,1},{1,3,8,20},40] (* Harvey P. Dale, Nov 09 2020 *)

a(n)=polcoeff(x/(1-3*x+x^2+x^3-x^4)+O(x^(n+1)),n) \\ Charles R Greathouse IV, Aug 06 2012

G.f.: x/(1 - 3*x + x^2 + x^3 - x^4).
a(n) ~ 0.50556... * 2.505068...^n. - Charles R Greathouse IV, Aug 06 2012
a(n) = 3*a(n-1) - a(n-2) - a(n-3) + a(n-4) for n>4. - Wesley Ivan Hurt, Sep 18 2015

Edited by N. J. A. Sloane, Sep 21 2006; definition corrected Aug 06 2012

A122766 Triangle read by rows: let p(n, x) = x*p(n-1, x) - p(n-2, x), then T(n, x) = d^2/dx^2 (p(n, x)).

Original entry on oeis.org

2, -2, 6, -6, -6, 12, 6, -24, -12, 20, 12, 24, -60, -20, 30, -12, 60, 60, -120, -30, 42, -20, -60, 180, 120, -210, -42, 56, 20, -120, -180, 420, 210, -336, -56, 72, 30, 120, -420, -420, 840, 336, -504, -72, 90, -30, 210, 420, -1120, -840, 1512, 504, -720, -90, 110

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 22 2006

			Triangle begins as:
    2;
   -2,    6;
   -6,    6,   12;
    6,  -24,  -12,   20;
   12,   24,  -60,  -20,   30;
   12,   60,   60, -120,  -30,   42;
  -20,  -60,  180,  120, -210,  -42,  56;
   20, -120, -180,  420,  210, -336, -56,  72;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Cf. A000217, A046854, A066170, A122765, A130777.

A122766:= func< n,k | 2*(-1)^Binomial(n-k+1, 2)*Binomial(k+1,2)*Binomial(Floor((n+k+2)/2), k+1) >;
[A122766(n,k): k in [1..n], n in [1..14]]; // G. C. Greubel, Dec 31 2022

(* First program *)
p[0, x]=1; p[1, x]=x-1; p[k_, x_]:= p[k, x]= x*p[k-1, x] -p[k-2, x]; b = Table[Expand[p[n,x]], {n,0,15}]; Table[CoefficientList[D[b[[n]], {x,2}], x], {n,2,14}]//Flatten
(* Second program *)
T[n_, k_]:= 2*(-1)^Binomial[n-k+1,2]*Binomial[k+1,2]*Binomial[Floor[(n +k+2)/2], k+1]; Table[T[n,k], {n,14}, {k,n}]//Flatten (* G. C. Greubel, Dec 31 2022 *)

tpol(n) = if (n <= 0, 1, if (n == 1, x -1, x*tpol(n-1) - tpol(n-2)));
lista(nn) = {for(n=0, nn, pol = deriv(deriv(tpol(n))); for (k=0, poldegree(pol), print1(polcoeff(pol, k), ", ");););} \\ Michel Marcus, Feb 07 2014

def A122766(n, k): return 2*(-1)^binomial(n-k+1,2)*binomial(k+1,2)*binomial(((n+k+2)//2), k+1)
flatten([[A122766(n, k) for k in range(1, n+1)] for n in range(1, 15)]) # G. C. Greubel, Dec 31 2022

From G. C. Greubel, Dec 31 2022: (Start)
T(n, k) = 2*(-1)^binomial(n-k+1, 2)*binomial(k+1,2)*binomial(floor((n+k +2)/2), k+1).
T(n, 1) = 2*(-1)^binomial(n,2)*binomial(floor((n+3)/2), 2)
T(n, n) = 2*A000217(n).
Sum_{k=1..n} T(n, k) = 2*A104555(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = 2*([n=1] - [n=2]). (End)

Edited by N. J. A. Sloane, Oct 01 2006

Name corrected and more terms from Michel Marcus, Feb 07 2014

A124038 Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

Original entry on oeis.org

1, -2, 1, -1, -2, 1, 2, -2, -2, 1, 1, 4, -3, -2, 1, -2, 3, 6, -4, -2, 1, -1, -6, 6, 8, -5, -2, 1, 2, -4, -12, 10, 10, -6, -2, 1, 1, 8, -10, -20, 15, 12, -7, -2, 1, -2, 5, 20, -20, -30, 21, 14, -8, -2, 1, -1, -10, 15, 40, -35, -42, 28, 16, -9, -2, 1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Nov 03 2006

			Triangular sequence begins as:
   1;
  -2,   1;
  -1,  -2,   1;
   2,  -2,  -2,   1;
   1,   4,  -3,  -2,   1;
  -2,   3,   6,  -4,  -2,   1;
  -1,  -6,   6,   8,  -5,  -2,  1;
   2,  -4, -12,  10,  10,  -6, -2,  1;
   1,   8, -10, -20,  15,  12, -7, -2,  1;
  -2,   5,  20, -20, -30,  21, 14, -8, -2,  1;
  -1, -10,  15,  40, -35, -42, 28, 16, -9, -2, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A005809, A045721, A066170.
Row reversal of: A374439.
Columns are related to: A000034 (k=0), A029578 (k=1), A131259 (k=2).
Diagonals are related to: A113679 (k=n-1), A022958 (k=n-2), A005843 (k=n-3), A000217 (k=n-4), -A002378 (k=n-5).
Sums include: (-1)^floor((n+1)/2)*A016116 (signed diagonal), A057079 (row), A119910 (signed row), (-1)^n*A130706 (diagonal).

function T(n,k) // T = A124038
  if k lt 0 or k gt n then return 0;
  elif k ge n-2 then return k-n + (-1)^(n+k);
  else return T(n-1,k-1) - T(n-2,k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 22 2025

(* First program *)
t[n_, m_, d_]:= If[n==m && n>1 && m>1, x, If[n==m-1 || n==m+1, -1, If[n==m== 1, x-2, 0]]];
M[d_]:= Table[t[n,m,d], {n,d}, {m,d}];
Join[{{1}}, Table[CoefficientList[Table[Det[M[d]], {d,10}][[d]], x], {d,10}]]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k] = If[k<0 || k>n, 0, If[k>n-2, k-n+(-1)^(n-k), T[n-1, k- 1] -T[n-2,k]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 22 2025 *)

@CachedFunction
def A124038(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 2*A124038(n-1,k) if n==1 else 0
    return A124038(n-1,k-1) - A124038(n-2,k) - h
for n in (0..9): [A124038(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

from sage.combinat.q_analogues import q_stirling_number2
def A124038(n,k): return (1 + ((n-k)%2))*q_stirling_number2(n+1, n-k+1, -1)
print(flatten([[A124038(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 22 2025

From G. C. Greubel, Jan 22 2025: (Start)
T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.
T(n, k) = (-1)^floor((n-k+1)/2)*(1 + (n-k mod 2))*qStirling2(n+1, n-k+1,-1).
T(2*n, n) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n)*A005809(n/2) - 2*(1-(-1)^n)* A045721((n-1)/2) ). (End)

Edited by G. C. Greubel, Jan 22 2025

A122160 Identity matrices minus Steinbach matrices as characteristic polynomials to give a triangular array I[n]-An[i,j]-> P[k,x] P[k,n]->T[n,m).

Original entry on oeis.org

1, 0, -1, -1, -1, 1, -1, 2, 1, -1, -2, 7, -3, -2, 1, -1, 7, -13, 5, 2, -1, -1, 12, -34, 30, -6, -3, 1, 0, 5, -30, 60, -45, 9, 3, -1, 1, 1, -41, 130, -155, 78, -10, -4, 1, 1, -6, -3, 87, -220, 229, -106, 14, 4, -1, 2, -19, 45, 54, -378, 609, -455, 160, -15, -5, 1, 1, -15, 73, -123, -89, 609, -889, 615, -205, 20, 5, -1, 1, -24, 164, -460

Offset: 1

Gary W. Adamson and Roger L. Bagula, Oct 17 2006

Remember? 1/(1-x)=Sum[x^n,{n,0,Infinitity}] So to try with the Steinbach field: (I-A[i,j])^(-1)=Sun[A[i,j]^n,{n,0,Infinity}] It doesn't appear it should be finite? But I-A[i,j] is finite--> zero? {{1,0,0}, {{1,1,1}, {{0,-1,-1}, {0,1,0}, {1,1,0}, {-1,0,0}, {0,0,1}} - 1,0,0}}= { -1,0,1}} Matrices: {{0, -1}, {-1, 1}}, {{0, -1, -1}, {-1, 0, 0}, {-1, 0, 1}}, {{0, -1, -1, -1}, {-1, 0, -1, 0}, {-1, -1, 1, 0}, {-1, 0, 0, 1}}, {{0, -1, -1, -1, -1}, {-1, 0, -1, -1, 0}, {-1, -1, 0, 0, 0}, {-1, -1, 0, 1, 0}, {-1, 0, 0, 0, 1}}

			{1},
{0, -1},
{-1, -1, 1},
{-1, 2, 1, -1},
{-2, 7, -3, -2, 1},
{-1, 7, -13, 5, 2, -1},
{-1, 12, -34, 30, -6, -3, 1},
{0, 5, -30, 60, -45, 9,3, -1},
{1, 1, -41, 130, -155, 78, -10, -4, 1}

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Cf. A065941, A122767, A123218, A066170, A122610.

An[d_] := Table[If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[IdentityMatrix[d] - An[d], x], x], {d, 1, 20}]]; Flatten[%]

I[n]-An[i,j]-> P[k,x] P[k,n]->T[n,m)

cp's OEIS Frontend

A069009 Let M denote the 6 X 6 matrix with rows / 1,1,1,1,1,1 / 1,1,1,1,1,0 / 1,1,1,1,0,0 / 1,1,1,0,0,0 / 1,1,0,0,0,0 / 1,0,0,0,0,0 / and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1); then a(n) = t(n).

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A030111 Triangular array in which k-th entry in n-th row is C([ (n+k)/2 ],k) (1<=k<=n).

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A120747 Sequence relating to the 11-gon (or hendecagon).

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A122765 Triangle read by rows: Let p(k, x) = x*p(k-1, x) - p(k-2, x). Then T(k,x) = dp(k,x)/dx.

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A052963 a(0)=1, a(1)=2, a(2)=5, a(n) = 3*a(n+2) - a(n+3).

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A122610 Triangle read by rows: T(n,k) is coefficient of x^k in Sum_{m=0..n} x^m*(1-x)^(n-m)*(-1)^[(m+1)/2]*binomial(m-[(m+1)/2],[m/2]).

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A122610 Triangle read by rows: T(n,k) is coefficient of x^k in Sum_{m=0..n} x^m(1-x)^(n-m)(-1)^[(m+1)/2]*binomial(m-[(m+1)/2],[m/2]).