A094811 -id:A094811 - cp's OEIS frontend

A005022 Number of walks of length 2n+6 in the path graph P_7 from one end to the other.

6, 26, 100, 364, 1288, 4488, 15504, 53296, 182688, 625184, 2137408, 7303360, 24946816, 85196928, 290926848, 993379072, 3391793664, 11580678656, 39539651584, 134998297600, 460915984384, 1573671536640, 5372862566400, 18344123969536, 62630804299776

Offset: 1

N. J. A. Sloane, Simon Plouffe

			Example: a(1)=6 because in the path ABCDEFG we have ABABCDEFG, ABCBCDEFG, ABCDCDEFG, ABCDEDEFG, ABCDEFEFG and ABCDEFGFG. - _Emeric Deutsch_, Apr 02 2004

W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. J. Everett, P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45.
C. J. Everett, P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45. [Annotated scanned copy]
P. Flajolet, J.-C. Raoult, and J. Vuillemin, The number of registers required for evaluating arithmetic expressions, Theoret. Comput. Sci. 9 (1979), no. 1, 99-125.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Xavier Gérard Viennot, A Strahler bijection between Dyck paths and planar trees. Formal power series and algebraic combinatorics (Barcelona, 1999). Discrete Math. 246 (2002), no. 1-3, 317--329. MR1887493 (2003b:05013).
Index entries for linear recurrences with constant coefficients, signature (6,-10,4).

See A094811 for another version.

I:=[6, 26, 100]; [n le 3 select I[n] else 6*Self(n-1)-10*Self(n-2)+4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 08 2013

a:=k->sum(binomial(6+2*k,8*j+k-2),j=ceil((2-k)/8)..floor((8+k)/8))-sum(binomial(6+2*k,8*j+k-1),j=ceil((1-k)/8)..floor((7+k)/8)): seq(a(k),k=1..28);
A005022:=-1/((2*z-1)*(2*z**2-4*z+1)) -1; # [Conjectured (correctly) by Simon Plouffe in his 1992 dissertation. Gives sequence with an additional leading term of 1.]

CoefficientList[Series[-(2 (2 z^2 - 5 z + 3))/(4 z^3 - 10 z^2 + 6 z - 1), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 27 2011 *)
CoefficientList[Series[(1 / x) (1 / (1 - 6 x + 10 x^2 - 4 x^3) - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)

Vec(2*(1-x)*(3-2*x) / ((1-2*x)*(1-4*x+2*x^2)) + O(x^50)) \\ Colin Barker, Apr 27 2016

G.f.: 1/(1-6x+10x^2-4x^3)-1.
a(n) = 6*a(n-1)-10*a(n-2)+4*a(n-3). - Emeric Deutsch, Apr 02 2004
a(k) = sum(binomial(6+2*k, 8*j+k-2)-binomial(6+2*k, 8*j+k-1), j=-infinity..infinity) (a finite sum).
The g.f. x^3/(1-6*x+10*x^2-4*x^3) occurs on page 320 of Viennot, 2002.
a(n) = -2^(1+n)+(3/2-sqrt(2))*(2-sqrt(2))^n+(3/2+sqrt(2))*(2+sqrt(2))^n - Colin Barker, Apr 27 2016
E.g.f.: (-2 + 3*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x))*exp(2*x) - 1. - Ilya Gutkovskiy, Apr 27 2016

Edited by Emeric Deutsch, Apr 28 2004

A217257 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 7, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,3) = T(0,4) = T(0,5) = T(0,6) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 5, 0, 0, 0, 1, 5, 9, 5, 0, 0, 0, 0, 6, 14, 14, 0, 0, 0, 0, 0, 6, 20, 28, 14, 0, 0, 0, 0, 0, 0, 26, 48, 42, 0, 0, 0, 0, 0, 0, 0, 26, 74, 90, 42, 0, 0, 0, 0, 0, 0, 0, 0, 100, 164, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 264, 296, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 364, 560, 428, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 17 2013

A hexagon arithmetic of E. Lucas.

			Square array begins:
1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=0
0, 1, 2, 3, 4, 5, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=1
0, 0, 2, 5, 9, 14, 20, 26, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=2
0, 0, 0, 5, 14, 28, 48, 74, 100, 100, 0, 0, 0, 0, 0, 0, 0, ... row n=3
0, 0, 0, 0, 14, 42, 90, 162, 264, 364, 364, 0, 0, 0, 0, 0, ... row n=4
0, 0, 0, 0, 0, 42, 132, 296, 560, 924, 1288, 1288, 0, 0, 0, ... row n=5
...

E. Lucas, Théorie des nombres, A. Blanchard, Paris, 1958, p.89

E. Lucas, Théorie des nombres, Tome 1, Jacques Gabay, Paris, p. 89

Cf. similar sequences: A216230, A216228, A216226, A216238, A216054.

T(n,n) = A024175(n).
T(n,n+1) = A024175(n+1).
T(n,n+2) = A094803(n+1).
T(n,n+3) = A007070(n).
T(n,n+4) = A094806(n+2).
T(n,n+5) = T(n,n+6) = A094811(n+2).
Sum_{k, 0<=k<=n} T(n-k,k) = A030436(n).

a(69) = 0 deleted by Georg Fischer, Oct 16 2021

A259475 Array read by antidiagonals: row n gives coefficients of Taylor series expansion of 1/F_{n+1}(t), where F_i(t) is a Fibonacci polynomial defined by F_0=1, F_1=1, F_{i+1} = F_i-t*F_{i-1}.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 8, 1, 0, 1, 5, 13, 21, 16, 1, 0, 1, 6, 19, 40, 55, 32, 1, 0, 1, 7, 26, 66, 121, 144, 64, 1, 0, 1, 8, 34, 100, 221, 364, 377, 128, 1, 0, 1, 9, 43, 143, 364, 728, 1093, 987, 256, 1, 0, 1, 10, 53, 196, 560, 1288, 2380, 3280, 2584, 512, 1, 0

Offset: 0

N. J. A. Sloane, Jul 03 2015

Table 3.1 in Hopkins thesis is the same below the main diagonal. - F. Chapoton, Sep 04 2025

			The first few antidiagonals are:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  1,  0;
  1, 3,  4,  1,   0;
  1, 4,  8,  8,   1,   0;
  1, 5, 13, 21,  16,   1,  0;
  1, 6, 19, 40,  55,  32,  1, 0;
  1, 7, 26, 66, 121, 144, 64, 1, 0;
  ...
Square array starts:
  [0] 1, 0,  0,   0,    0,    0,     0,     0,      0,       0,       0, ...
  [1] 1, 1,  1,   1,    1,    1,     1,     1,      1,       1,       1, ...
  [2] 1, 2,  4,   8,   16,   32,    64,   128,    256,     512,    1024, ...
  [3] 1, 3,  8,  21,   55,  144,   377,   987,   2584,    6765,   17711, ...
  [4] 1, 4, 13,  40,  121,  364,  1093,  3280,   9841,   29524,   88573, ...
  [5] 1, 5, 19,  66,  221,  728,  2380,  7753,  25213,   81927,  266110, ...
  [6] 1, 6, 26, 100,  364, 1288,  4488, 15504,  53296,  182688,  625184, ...
  [7] 1, 7, 34, 143,  560, 2108,  7752, 28101, 100947,  360526, 1282735, ...
  [8] 1, 8, 43, 196,  820, 3264, 12597, 47652, 177859,  657800, 2417416, ...
  [9] 1, 9, 53, 260, 1156, 4845, 19551, 76912, 297275, 1134705, 4292145, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Noureddine Chair, Explicit Computations for the Intersection Numbers on Grassmannians, and on the Space of Holomorphic Maps from CP¹ into G_r(Cⁿ), arXiv:hep-th/9808170, 1998. (Cf. Table 4).
E. Hopkins, Free Complexes over the Exterior Algebra with Small Homology, Ph.D. thesis, University of Nebraska, 2021. (Cf. Table 3.1.)
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.

The initial rows of the array are A000007, A000012, A000079, A001906, A003432, A005021, A094811, A094256.
A(n,n) gives A274969.
Cf. A309896.
A188843 is a variant without the first two rows and the first column, and the antidiagonals read in opposite direction.

F:= proc(n) option remember;
      `if`(n<2, 1, expand(F(n-1)-t*F(n-2)))
    end:
A:= (n, k)-> coeff(series(1/F(n+1), t, k+1), t, k):
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jul 04 2015

F[n_] := F[n] = If[n<2, 1, Expand[F[n-1] - t*F[n-2]]]; A[n_, k_] := SeriesCoefficient[1/F[n+1], { t, 0, k}]; Table[A[d-k, k], {d, 0, 12}, {k, 0, d}] // Flatten (* Jean-François Alcover, Feb 17 2016, after Alois P. Heinz *)

# The lower triangular array computed by F. Chapoton's formula:
from math import comb as binomial
def T(n: int, k: int) -> int:
    if k < 0:  return 0
    if k == 0: return 1
    if k == 1: return n
    return (binomial(n + 2*k, k - 2) * (n**2 + 3*n - 2*k + 2)) // (k * (k - 1))
for n in range(9): print([T(n, k) for k in range(n+1)])  # Peter Luschny, Sep 06 2025

@cached_function
def F(n, k):
    if k <  0: return 0
    if k == 0: return 1
    return sum((-1)^j*binomial(n-1-j,j+1)*F(n,k-2-2*j) for j in (0..(n-2)/2))
def A(n, k): return F(n+1, 2*k)
print([A(n-k, k) for n in (0..11) for k in (0..n)]) # Peter Luschny, Aug 21 2019

Let F(n, k) = Sum_{j=0..(n-2)/2} (-1)^j*binomial(n-1-j, j+1)*F(n, k-2-2*j) for k > 0; F(n, 0) = 1 and F(n, k) = 0 if k < 0. Then A(n, k) = F(n+1, 2*k). See [Shibukawa] and A309896. - Peter Luschny, Aug 21 2019

For n >= k >= 2, A(n, k) = binomial(n+2*k,k-2)*(n^2+3*n-2*k+2)/(k*(k-1)). - F. Chapoton, Sep 05 2025

More terms from Alois P. Heinz, Jul 04 2015

cp's OEIS Frontend

A005022 Number of walks of length 2n+6 in the path graph P_7 from one end to the other.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A217257 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 7, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,3) = T(0,4) = T(0,5) = T(0,6) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A259475 Array read by antidiagonals: row n gives coefficients of Taylor series expansion of 1/F_{n+1}(t), where F_i(t) is a Fibonacci polynomial defined by F_0=1, F_1=1, F_{i+1} = F_i-t*F_{i-1}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

SageMath

Formula

Extensions