A213667 -id:A213667 - cp's OEIS frontend

A265278 Expansion of (x^4+x^3-x^2+x)/(x^3+x^2-3*x+1).

0, 1, 2, 6, 16, 40, 98, 238, 576, 1392, 3362, 8118, 19600, 47320, 114242, 275806, 665856, 1607520, 3880898, 9369318, 22619536, 54608392, 131836322, 318281038, 768398400, 1855077840, 4478554082, 10812186006, 26102926096, 63018038200, 152139002498, 367296043198

Offset: 0

Alois P. Heinz, Apr 06 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Marika Diepenbroek, Monica Maus, and Alex Stoll, Pattern Avoidance in Reverse Double Lists, Preprint 2015. See Table 3.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 41.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Agrees with A213667 except for initial terms.

Cf. A002605, A265106, A265107, A270810.

Table[2 Fibonacci[n-1, 2] + LucasL[n-1, 2]/2 + KroneckerDelta[n-1] - 1, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)
LinearRecurrence[{3,-1,-1},{0,1,2,6,16},40] (* Harvey P. Dale, Mar 18 2018 *)

concat(0, Vec(x*(1-x+x^2+x^3)/((1-x)*(1-2*x-x^2)) + O(x^50))) \\ Colin Barker, Apr 12 2016

From Colin Barker, Apr 12 2016: (Start)
a(n) = (-2 + (1-sqrt(2))^n + (1+sqrt(2))^n)/2 for n>1.
a(n) = 3*a(n-1)-a(n-2)-a(n-3) for n>4.
(End)
E.g.f.: x + (cosh(sqrt(2)*x) - 1)*exp(x). - Ilya Gutkovskiy, Sep 16 2016

A065113 Sum of the squares of the a(n)-th and the (a(n)+1)st triangular numbers (A000217) is a perfect square.

Original entry on oeis.org

6, 40, 238, 1392, 8118, 47320, 275806, 1607520, 9369318, 54608392, 318281038, 1855077840, 10812186006, 63018038200, 367296043198, 2140758220992, 12477253282758, 72722761475560, 423859315570606, 2470433131948080, 14398739476117878, 83922003724759192

Offset: 1

Robert G. Wilson v, Nov 12 2001

The sequence of square roots of the sum of the squares of the n-th and the (n+1)st triangular numbers is A046176.

			T6 = 21 and T7 = 28, 21^2 + 28^2 = 441 + 784 = 1225 = 35^2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT] (2008)
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A001652, A002315, A003499 (first differences), A065651.

CoefficientList[ Series[2*(x - 3)/(-1 + 7x - 7x^2 + x^3), {x, 0, 24} ], x]
LinearRecurrence[{7,-7,1},{6,40,238},41] (* Harvey P. Dale, Dec 27 2011 *)

a(n)=-1+subst(poltchebi(abs(n+1))-poltchebi(abs(n)),x,3)/2

Vec(2*x*(3-x)/((1-6*x+x^2)*(1-x)) + O(x^40)) \\ Colin Barker, Mar 05 2016

a(n) = 2*A001652(n) = -1 + A002315(n).
a(n) - a(n-1) = A003499(n).
From Michael Somos, Apr 07 2003: (Start)
G.f.: 2*x*(3-x)/((1-6*x+x^2)*(1-x)).
a(n) = 6*a(n-1) - a(n-2) + 4.
a(-1-n) = -a(n) - 2. (End)
a(1)=6, a(2)=40, a(3)=238, a(n) = 7*a(n-1)-7*a(n-2)+a(n-3). - Harvey P. Dale, Dec 27 2011
a(n)^2 + (a(n)+2)^2 = A075870(n+1)^2 = A165518(n+1). - Joerg Arndt, Feb 15 2012
a(n) = (-2-(3-2*sqrt(2))^n*(-1+sqrt(2))+(1+sqrt(2))*(3+2*sqrt(2))^n)/2. - Colin Barker, Mar 05 2016
From Klaus Purath, Sep 05 2021: (Start)
(a(n+1) - a(n) - a(n-1) + a(n-2))/8 = A005319(n), for n >= 3.
((a(n) - a(n-1))^2)/2 - 2 = A005319(n)^2 = 2*A132592(n), for n>= 2.
a(n) = A265278(2*n+1).
a(n) = A293004(2*n+1).
a(n) = A213667(2*n).
a(n) = Sum_{k=1..n} A003499(k). (End)

A231690 Cardinalities of the sub-operad FF_6 of the operad MFF.

Original entry on oeis.org

1, 6, 56, 640, 8158, 111258, 1588544, 23446248, 354855218, 5477342222, 85893429256, 1364577254040, 21916000458014, 355251287893170, 5804407209709312, 95493879511032240, 1580592247322440642, 26301843121772151254, 439764358275666481496, 7384252698468635017936, 124469446338979722639294

Offset: 1

N. J. A. Sloane, Nov 14 2013

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..301
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

Cf. A213667, A231691.

InverseSeries[x(1 - 3x - x^2 + x^3)/(1 + 3x + x^2 - x^3) + O[x]^22] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Oct 24 2018, from PARI *)

N=22; x='x+O('x^N);
Vec(serreverse(Ser(x*(1-3*x-x^2+x^3)/(1+3*x+x^2-x^3))))  \\ Gheorghe Coserea, Jan 13 2017

A(x) = serreverse(A213667(-x))(-x). - Gheorghe Coserea, Jan 13 2017

Offset changed and more terms from Gheorghe Coserea, Jan 13 2017

A213666 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) obtained by taking n copies of the path P_3 and identifying one of their endpoints (a star with n branches of length 2).

Original entry on oeis.org

1, 3, 1, 0, 3, 8, 5, 1, 0, 0, 7, 20, 18, 7, 1, 0, 0, 0, 15, 48, 56, 32, 9, 1, 0, 0, 0, 0, 31, 112, 160, 120, 50, 11, 1, 0, 0, 0, 0, 0, 63, 256, 432, 400, 220, 72, 13, 1, 0, 0, 0, 0, 0, 0, 127, 576, 1120, 1232, 840, 364, 98, 15, 1

Offset: 1

Emeric Deutsch, Jul 01 2012

Rows also give the coefficients of the domination polynomial of the n-helm graph (divided by x, i.e., with initial 0 dropped from rows). - Eric W. Weisstein, May 28 2017
Row n contains 2n + 1 entries (first n-1 of which are 0).
Sum of entries in row n = 2*3^{n-1} - 1 = A048473(n).
Sum of entries in column k = A213667(k).

			Row 2 is 0,3,8,5,1 because G(2) is the path P_5 abcde; no domination subset of size 1, three of size 2 (ad, bd, be), all subsets of size 3 with the exception of abc and cde are dominating (binomial(5,3)-2=8), all binomial(5,4)=5 subsets of size 4 are dominating, and abcde is dominating.
Triangle starts:
  1, 3, 1;
  0, 3, 8,  5,  1;
  0, 0, 7, 20, 18,  7,  1;
  0, 0, 0, 15, 48, 56, 32, 9, 1;

S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251 [math.CO], 2009.
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
T. Kotek, J. Preen, F. Simon, P. Tittmann, and M. Trinks, Recurrence relations and splitting formulas for the domination polynomial, arXiv:1206.5926 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Domination Polynomial.
Eric Weisstein's World of Mathematics, Helm Graph.

Cf. A048473, A213667.

T := proc (n, k) if k = n then 2^n-1 else 2^(2*n-k)*(2*binomial(n, k-n-1) + binomial(n, k-n)) end if end proc: for n to 10 do seq(T(n, k), k = 1 .. 2*n+1) end d; # yields sequence in triangular form

T[n_, n_] := 2^n - 1;
T[n_, k_] := 2^(2*n - k)*(2*Binomial[n, k - n - 1] + Binomial[n, k - n]);
Table[T[n, k], {n, 1, 10}, {k, 1, 2*n + 1}] // Flatten (* Jean-François Alcover, Dec 02 2017 *)

T(n,k) = 2^(2*n-k)*(2*binomial(n,k-n-1)+binomial(n,k-n)) if k > n; T(n,n)=2^n - 1.
The generating polynomial of row n is g[n] = g[n,x] = (1+x)(x*(2+x))^n - x^n (= domination polynomial of the graph G(n)).
Bivariate g.f.: G(x,z) = x*z*(1+x)*(2+x)/(1-2*x*z-x^2*z)-x*z/(1-xz).

A281261 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 2, 2, 1, 5, 2, 5, 9, 2, 1, 15, 14, 2, 7, 35, 20, 2, 1, 28, 70, 27, 2, 9, 84, 126, 35, 2, 1, 45, 210, 210, 44, 2, 11, 165, 462, 330, 54, 2, 1, 66, 495, 924, 495, 65, 2, 13, 286, 1287, 1716, 715, 77, 2, 1, 91, 1001, 3003, 3003, 1001, 90, 2, 15, 455, 3003, 6435, 5005, 1365, 104, 2, 1, 120, 1820, 8008, 12870, 8008, 1820, 119, 2

Offset: 1

Gheorghe Coserea, Jan 18 2017

Row n>1 contains floor((n+3)/2) terms.

			A(x;t) = x + (2*t+2)*x^2 + (t^2+5*t+2)*x^3 + (5*t^2+9*t+2)*x^4 + ...
Triangle starts:
n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]      [8]
[1]  1;
[2]  2,       2;
[3]  1,       5,       2;
[4]  5,       9,       2;
[5]  1,       15,      14,      2;
[6]  7,       35,      20,      2;
[7]  1,       28,      70,      27,      2;
[8]  9,       84,      126,     35,      2;
[9]  1,       45,      210,     210,     44,      2;
[10] 11,      165,     462,     330,     54,      2;
[11] 1,       66,      495,     924,     495,     65,      2;
[12] 13,      286,     1287,    1716,    715,     77,      2;
[13] 1,       91,      1001,    3003,    3003,    1001,    90,      2;
[14] 15,      455,     3003,    6435,    5005,    1365,    104,     2;
[15] ...

Gheorghe Coserea, Rows n = 1..202, flattened
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

Reverse[CoefficientList[#, t]]& /@ CoefficientList[x*((1-t)*x^3 + (t^2 - 2*t - 1)*x^2 + (2*t - 1)*x + 1)/((t-1)*x^3 + (3-t)*x^2 - 3*x + 1) + O[x]^16, x] // Rest // Flatten (* Jean-François Alcover, Feb 18 2019 *)

N=16; x='x+O('x^N); concat(apply(p->Vec(p),  Vec(Ser(x*((1-t)*x^3 + (t^2-2*t-1)*x^2 + (2*t-1)*x + 1)/((t-1)*x^3 + (3-t)*x^2 - 3*x + 1)))))

N = 14; concat(1, concat(vector(N, n, Vec(substpol(((1+t)^(n+2) + (1-t)^(n+2))/2 - t^2 + 1, t^2, t)))))

A(x;t) = Sum{n>=1} P_n(t)*x^n = x*((1-t)*x^3 + (t^2-2*t-1)*x^2 + (2*t-1)*x + 1)/((t-1)*x^3 + (3-t)*x^2 - 3*x + 1).
A278457(x;t) = serreverse(A(-x;t))(-x).
A151821(n) = P_n(1), A213667(n) = P_n(2).
P_n(t^2) = ((1+t)^(n+1) + (1-t)^(n+1))/2 - t^2 + 1, for n>1.

cp's OEIS Frontend

A265278 Expansion of (x^4+x^3-x^2+x)/(x^3+x^2-3*x+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A065113 Sum of the squares of the a(n)-th and the (a(n)+1)st triangular numbers (A000217) is a perfect square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A231690 Cardinalities of the sub-operad FF_6 of the operad MFF.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A213666 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) obtained by taking n copies of the path P_3 and identifying one of their endpoints (a star with n branches of length 2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A281261 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

PARI

PARI

Formula