A155112 -id:A155112 - cp's OEIS frontend

A155020 a(n) = 2a(n-1) + 2a(n-2) for n > 2, a(0)=1, a(1)=1, a(2)=3.

1, 1, 3, 8, 22, 60, 164, 448, 1224, 3344, 9136, 24960, 68192, 186304, 508992, 1390592, 3799168, 10379520, 28357376, 77473792, 211662336, 578272256, 1579869184, 4316282880, 11792304128, 32217174016, 88018956288, 240472260608, 656982433792, 1794909388800, 4903783645184, 13397386067968

Offset: 0

Philippe Deléham, Jan 19 2009

Equals 1 followed by A028859. - Klaus Brockhaus, Jul 18 2009
a(n) is the number of ways to arrange 1- and 2-cent postage stamps (totaling n cents) in a row so that the first stamp is correctly placed and any subsequent stamp may (or not) be placed upside down.
Number of compositions of n into parts k >= 1 where there are F(k+1) = A000045(k+1) sorts of part k. - Joerg Arndt, Sep 30 2012
a(n) is the top-left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 1, 0] or of the 3 X 3 matrix [1, 1, 1; 1, 0, 1; 1, 1, 1].
From Tom Copeland, Nov 08 2014: (Start)
(Setting a(0)=0.)
This array is one of a family of Catalan arrays related by compositions of the special fractional linear (Möbius) transformations P(x,t) = x/(1-t*x); its inverse Pinv(x,t) = P(x,-t); and an o.g.f. of the Catalan numbers A000108, C(x) = (1-sqrt(1-4x))/2; and its inverse Cinv(x) = x*(1-x). (Cf. A091867.)
O.g.f.: G(x) = -P(P(Cinv(-x),1),1) = -P(Cinv(-x),2) = x(1+x)/(1-2x(1+x)) = (x+x^2)/(1-2(x+x^2)) = x + 3*x^2 + 8*x^3 + ... = A155020(x) with a(0)=0.
Ginv(x) = -C(P(P(-x,-1),-1)) = -C(P(-x,-2)) = (-1+sqrt(1+4*x/(1+2*x)))/2 = x*A064613(-x).
G(x) = x*(1+x) + 2*(x*(1+x))^2 + 2^2*(x*(1+x))^3 - ..., and so this array contains the row sums of A030528 * Diag(1, 2^1, 2^2, 2^3, ...). (End)
INVERT transform of Fibonacci(n+1). - Alois P. Heinz, Feb 11 2021

			a(2) = 3 because we have {1,1}, {1,_1} and {2}.
a(3) = 8 because we can order the stamps in eight ways: {1,1,1}  {1,1,_1}  {1,_1,1}  {1,_1,_1}  {2,1}   {2,_1}  {1,2}   {1,_2}, where _1 and _2 are upside down stamps.
a(4) = 22 = 2*3 + 2*8 because we can append 2 or _2 to the a(2) examples and 1 or _1 to the a(3) examples. - _Jon Perry_, Nov 10 2014

Indranil Ghosh, Table of n, a(n) for n = 0..2287
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 14.
I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239v1 [math.CO] 17 Sep 2015.
Juan B. Gil and Jessica A. Tomasko, Fibonacci colored compositions and applications, arXiv:2108.06462 [math.CO], 2021.
Jeffrey Shallit, Proof of Irvine's conjecture via mechanized guessing, arXiv preprint arXiv:2310.14252 [math.CO], October 22 2023.
Index entries for linear recurrences with constant coefficients, signature (2,2).

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: this sequence (m=2), A155116 (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), A155157 (m=10).
Cf. A028859 (essentially the same sequence). - Klaus Brockhaus, Jul 18 2009
Cf. A000045, A000108, A030528, A064613, A091867, A154929.
Row sums of A155112.

I:=[1,1,3,8]; [n le 4 select I[n] else 2*Self(n-1)+2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 10 2014

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*combinat[fibonacci](1+i), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021

CoefficientList[Series[(1 -x -x^2)/(1 -2x -2x^2), {x,0,20}], x]
With[{m=2}, LinearRecurrence[{m, m}, {1, m-1, m^2-1}, 30]] (* G. C. Greubel, Mar 25 2021 *)

makelist(sum(binomial(n-k,k)*2^(n-k-1),k,0,floor(n/2)),n,1,12); /* Emanuele Munarini, Feb 04 2014 */

Vec( (1-x-x^2)/(1-2*x-2*x^2) + O(x^66) )  /* Joerg Arndt, Sep 30 2012 */

[1]+[(-1)*(sqrt(2)*i)^(n-2)*chebyshev_U(n, -sqrt(2)*i/2) for n in (1..30)] # G. C. Greubel, Mar 25 2021

G.f.: (1 - x - x^2)/(1 - 2*x - 2*x^2).
G.f.: 1/( 1 - Sum_{k>=1} (x+x^2)^k ) - 1/( 1 - Sum_{k>=1} F(k+1)*x^k ) where F(k) = A000045(k). - Joerg Arndt, Sep 30 2012
a(n+1) = Sum_{k=0..n} A154929(n,k) = A028859(n).
a(n) = Sum_{k=0..floor(n/2)} ( binomial(n-k,k)*2^(n-k-1) ) for n > 0. - Emanuele Munarini, Feb 04 2014
a(n) = (1/2)*[n=0] - (sqrt(2)*i)^(n-2)*ChebyshevU(n, -sqrt(2)*i/2). - G. C. Greubel, Mar 25 2021
E.g.f.: (3 + exp(x)*(3*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)))/6. - Stefano Spezia, Mar 02 2024

A300453 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x + 1)^n + x^2 - 1.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 2, 2, 0, 3, 4, 1, 0, 4, 7, 4, 1, 0, 5, 11, 10, 5, 1, 0, 6, 16, 20, 15, 6, 1, 0, 7, 22, 35, 35, 21, 7, 1, 0, 8, 29, 56, 70, 56, 28, 8, 1, 0, 9, 37, 84, 126, 126, 84, 36, 9, 1, 0, 10, 46, 120, 210, 252, 210, 120, 45, 10, 1, 0, 11, 56, 165

Offset: 0

Franck Maminirina Ramaharo, Mar 06 2018

This is essentially the usual Pascal triangle A007318, horizontally shifted and with the first three columns altered.
Let P(n;x) = (x + 1)^n + x^2 - 1. Then P(n;x) = P(n-1;x) + x*(x + 1)^(n - 1), with P(0;x) = x^2.
Let a (2,n)-torus knot be projected on the plane. The resulting projection is a planar diagram with n double points. Then, T(n,k) gives the number of state diagrams having k components that are obtained by deleting each double point, then pasting the edges in one of the two ways as shown below.
         \  /     \_/                   \  /     \   /
   (1)    \/  ==>                 (2)      \/  ==>  | |
          /\       _                     /\       | |
         /  \     /   \                   /  \     /   \
See example for the case n = 2.

			The triangle T(n,k) begins
n\k  0   1   2    3     4     5     6     7     8     9    10   11  12  13 14
0:   0   0   1
1:   0   1   1
2:   0   2   2
3:   0   3   4    1
4:   0   4   7    4     1
5:   0   5  11   10     5     1
6:   0   6  16   20    15     6     1
7:   0   7  22   35    35    21     7     1
8:   0   8  29   56    70    56    28     8     1
9:   0   9  37   84   126   126    84    36     9     1
10:  0  10  46  120   210   252   210   120    45    10     1
11:  0  11  56  165   330   462   462   330   165    55    11    1
12:  0  12  67  220   495   792   924   792   495   220    66   12   1
13:  0  13  79  286   715  1287  1716  1716  1287   715   286   78  13   1
14:  0  14  92  364  1001  2002  3003  3432  3003  2002  1001  364  91  14  1
...
The states of the (2,2)-torus knot (Hopf Link) are the last four diagrams:
                                    ____  ____
                                   /    \/    \
                                  /     /\     \
                                 |     |  |     |
                                 |     |  |     |
                                  \     \/     /
                                   \____/\____/
                           ___    ____         __________
                         /    \  /    \       /    __    \
                        /     /  \     \     /    /  \    \
                       |      |  |      |   |     |  |     |
                       |      |  |      |   |     |  |     |
                        \      \/      /     \     \/     /
                         \_____/\_____/       \____/\____/
      ____    ____        ____    ____        ____________        __________
     /    \  /    \      /    \  /    \      /     __     \      /    __    \
    /     /  \     \    /     /  \     \    /     /  \     \    /    /  \    \
   |     |    |     |  |     |    |     |  |     |    |     |  |    |    |    |
   |     |    |     |  |     |    |     |  |     |    |     |  |    |    |    |
    \     \  /     /    \     \__/     /    \     \  /     /    \    \__/    /
     \____/  \____/      \____________/      \____/  \____/      \__________/
There are 2 diagrams that consist of two components, and 2 diagrams that consist of one component.

Colin Adams, The Knot Book, W. H. Freeman and Company, 1994.
Louis H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.

Michael De Vlieger, Table of n, a(n) for n = 0..11327 (rows 0 <= n <= 150, flattened).
Agnijo Banerjee, Knot theory [Foil knot family].
Allison Henrich, Rebecca Hoberg, Slavik Jablan, Lee Johnson, Elizabeth Minten and Ljiljana Radovic, The Theory of Pseudoknots, arXiv preprint arXiv:1210.6934 [math.GT], 2012.
Abdullah Kopuzlu, Abdulgani Şahin and Tamer Ugur, On polynomials of K(2,n) torus knots, Applied Mathematical Sciences, Vol. 3 (2009), 2899-2910.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Franck Ramaharo, Note on sequences A123192, A137396 and A300453, arXiv:1911.04528 [math.CO], 2019.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.
Wikipedia, Torus knot.
Xinfei Li, Xin Liu and Yong-Chang Huang, Tackling tangledness of cosmic strings by knot polynomial topological invariants, arxiv preprint arXiv:1602.08804 [hep-th], 2016.

Row sums: A000079 (powers of 2).
Triangles related to the regular projection of some knots: A299989 (connected summed trefoils); A300184 (chain links); A300454 (twist knot).
When n = 3 (trefoil), the corresponding 4-tuplet (0,3,4,1) appears in several triangles: A030528 (row 5), A256130 (row 3), A128908 (row 3), A186084 (row 6), A284938 (row 7), A101603 (row 3), A155112 (row 3), A257566 (row 3).
Cf. A001477, A007318, A007318, A152947.

f[n_] := CoefficientList[ Expand[(x + 1)^n + x^2 - 1], x]; Array[f, 12, 0] // Flatten (* or *)
CoefficientList[ CoefficientList[ Series[(x^2 + y*x/(1 - y*(x + 1)))/(1 - y), {y, 0, 11}, {x, 0, 11}], y], x] // Flatten (* Robert G. Wilson v, Mar 08 2018 *)

P(n, x) := (x + 1)^n + x^2 - 1$
T : []$
for i:0 thru 20 do
  T : append(T, makelist(ratcoef(P(i, x), x, n), n, 0, max(2, i)))$
T;

row(n) = Vecrev((x + 1)^n + x^2 - 1);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Mar 12 2018

T(n,1) = A001477(n).
T(n,2) = A152947(n).
T(n,k) = A007318(n,k-1), k >= 1.
T(n,0) = 0, T(0,1) = 1, T(0,2) = 1 and T(n,k) = T(n-1,k) + A007318(n-1,k-1).
G.f.: (x^2 + y*x/(1 - y*(x + 1)))/(1 - y).

A155000 a(n) = 8a(n-1) + 56a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=15.

Original entry on oeis.org

1, 1, 15, 176, 2248, 27840, 348608, 4347904, 54305280, 677924864, 8464494592, 105679749120, 1319449690112, 16473663471616, 205678490419200, 2567953077764096, 32061620085587968, 400298333039493120

Offset: 0

Philippe Deléham, Jan 18 2009

nonn

The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - R. J. Mathar, Jan 20 2009

G. C. Greubel, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (8,56).

Cf. A154996, A154997, A154999, A155001, A155112.

I:=[1,15]; [1] cat [n le 2 select I[n] else 8*(Self(n-1) +7*Self(n-2)): n in [1..30]]; // G. C. Greubel, Apr 20 2021

m:=30; S:=series( (1-7*x-49*x^2)/(1-8*x-56*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 20 2021

Join[{1},LinearRecurrence[{8,56},{1,15},20]] (* Harvey P. Dale, Dec 11 2012 *)

def A155000_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-7*x-49*x^2)/(1-8*x-56*x^2) ).list()
A155000_list(30) # G. C. Greubel, Apr 20 2021

a(n) = Sum_{k=0..n} A155112(n,k)*7^(n-k). - Philippe Deléham, Jan 27 2009

G.f.: 1 + x*(1+7*x)/(1-8*x-56*x^2). - Harvey P. Dale, Dec 11 2012

More terms from Philippe Deléham, Jan 27 2009

A262910 a(n) = Sum_{k=0..n} binomial(k+n-1,k)binomial(k+n,2k).

Original entry on oeis.org

1, 2, 10, 59, 366, 2337, 15205, 100235, 667222, 4474733, 30188335, 204646532, 1392850785, 9511878729, 65144238981, 447263887479, 3077459618886, 21215286546705, 146500755609415, 1013180180867125, 7016536189029551, 48650933146617728, 337709155342663620

Offset: 0

Vladimir Kruchinin, Oct 04 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1176

Cf. A007863, A155112.

a := n -> hypergeom([-n, n, n+1], [1/2, 1], -1/4):
seq(round(evalf(a(n), 32)), n=0..21); # Peter Luschny, Oct 08 2015

Table[Sum[Binomial[k+n-1,k]*Binomial[k+n,2*k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Oct 04 2015 *)

B(x):=sum(sum(binomial(i+n-1,i)*binomial(i+n,2*i+1),i,0,n-1)/n*x^n,n,1,30);
taylor(x*diff(B(x),x)/B(x),x,0,20);

a(n) = sum(k=0, n, binomial(k+n-1,k)*binomial(k+n,2*k));
vector(50, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

G.f.: A(x) = x*B'(x)/B(x), where B(x)/x is g.f. of A007863.
Recurrence: 5*(n-1)*n*(35*n^2 - 143*n + 138)*a(n) = 2*(n-1)*(630*n^3 - 2889*n^2 + 3746*n - 1200)*a(n-1) - 2*(70*n^4 - 426*n^3 + 811*n^2 - 589*n + 150)*a(n-2) + 2*(n-3)*(2*n - 3)*(35*n^2 - 73*n + 30)*a(n-3). - Vaclav Kotesovec, Oct 04 2015
a(n) = hypergeom([-n, n, n+1], [1/2, 1], -1/4). - Peter Luschny, Oct 08 2015
a(n) = A155112(2n,n). - Alois P. Heinz, Sep 29 2022

A291385 a(n) = (1/4)*A073388(n+1).

Original entry on oeis.org

1, 4, 14, 47, 152, 480, 1488, 4548, 13744, 41152, 122272, 360944, 1059584, 3095552, 9005568, 26101824, 75404544, 217191424, 623928832, 1788071680, 5113137152, 14592352256, 41569120256, 118219097088, 335685021696, 951817715712, 2695241605120, 7622609858560

Offset: 0

Clark Kimberling, Sep 04 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,-8,-4)

Cf. A019590, A291382, A073388, A155112.

z = 60; s = x + x^2; p = (1 - 2 s)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A019590 *)
u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A073388 *)
u / 2  (* A291385 *)
LinearRecurrence[{4,0,-8,-4},{1,4,14,47},30] (* Harvey P. Dale, Aug 24 2022 *)

G.f.: -(((1 + x) (-1 + x + x^2))/(-1 + 2 x + 2 x^2)^2).
a(n) = 4*a(n-1) - 8*a(n-3) + 4*a(n-4) for n >= 5.
a(n) = Sum_{k=0..n+1} k * A155112(n+1,k). - Alois P. Heinz, Sep 29 2022

A292399 p-INVERT of (1,2,3,5,8,...) (distinct Fibonacci numbers), where p(S) = (1 - S)^2.

Original entry on oeis.org

2, 7, 22, 69, 212, 644, 1936, 5772, 17088, 50288, 147232, 429136, 1245888, 3604544, 10396160, 29900992, 85784064, 245548800, 701402624, 1999734016, 5691409408, 16172221440, 45885403136, 130011401216, 367902195712, 1039836672000, 2935713865728, 8279592292352

Offset: 0

Clark Kimberling, Sep 30 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,-8,-4)

Cf. A000045, A155112, A292400.

z = 60; s = x (x + 1)/(1 - x - x^2); p = (1 - s)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000045 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292399 *)

G.f.: -(((1 + x) (-2 + 3 x + 3 x^2))/(-1 + 2 x + 2 x^2)^2).
a(n) = 4*a(n-1) - 8*a(n-3) - 4*a(n-4) for n >= 5.
a(n) = Sum_{k=0..n+1} (k+1) * A155112(n+1,k). - Alois P. Heinz, Sep 29 2022

A370173 Riordan array (1-x-x^2, x*(1+x)).

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 0, -2, 1, 1, 0, -1, -2, 2, 1, 0, 0, -3, -1, 3, 1, 0, 0, -1, -5, 1, 4, 1, 0, 0, 0, -4, -6, 4, 5, 1, 0, 0, 0, -1, -9, -5, 8, 6, 1, 0, 0, 0, 0, -5, -15, -1, 13, 7, 1, 0, 0, 0, 0, -1, -14, -20, 7, 19, 8, 1, 0, 0, 0, 0, 0, -6, -29, -21, 20, 26, 9, 1

Offset: 0

Philippe Deléham, Feb 27 2024

Triangle T(n,k) read by rows : matrix product of A155112*A130595.

Triangle T(n,k), read by rows, given by [-1, 2, -1/2, -1/2, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

			Triangle T(n,k) begins:
  1;
 -1,  1;
 -1,  0,  1;
  0, -2,  1,  1;
  0, -1, -2,  2, 1;
  0,  0, -3, -1, 3, 1;
...

Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.

Cf. A084938, A130595, A155112.

from functools import cache
@cache
def T(n, k):
    if k > n: return 0
    if n == 0: return 1
    if k == 0: return -1 if n == 1 or n == 2 else 0
    return T(n-1, k-1) + T(n-2, k-1)
for n in range(9):
    print([T(n, k) for k in range(n+1)])  # Peter Luschny, Feb 28 2024

T(n,k) = T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = -1, T(n,0) = 0 for n>2, T(n,k) = 0 if k>n.
T(n,k) = Sum_{j = k..n} A155112(n,j)*A130595(j,k).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A155020(n), A155116(n), A155117(n), A155119(n), A155127(n), A155130(n), A155132(n), A155144(n), A155157(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.

cp's OEIS Frontend

A155020 a(n) = 2*a(n-1) + 2*a(n-2) for n > 2, a(0)=1, a(1)=1, a(2)=3.

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A300453 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x + 1)^n + x^2 - 1.

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A155000 a(n) = 8*a(n-1) + 56*a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=15.

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A262910 a(n) = Sum_{k=0..n} binomial(k+n-1,k)*binomial(k+n,2*k).

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A291385 a(n) = (1/4)*A073388(n+1).

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A292399 p-INVERT of (1,2,3,5,8,...) (distinct Fibonacci numbers), where p(S) = (1 - S)^2.

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A370173 Riordan array (1-x-x^2, x*(1+x)).

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A155020 a(n) = 2a(n-1) + 2a(n-2) for n > 2, a(0)=1, a(1)=1, a(2)=3.

A155000 a(n) = 8a(n-1) + 56a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=15.

A262910 a(n) = Sum_{k=0..n} binomial(k+n-1,k)binomial(k+n,2k).