A055587 -id:A055587 - cp's OEIS frontend

A049600 Array T read by diagonals; T(i,j) is the number of paths from (0,0) to (i,j) consisting of nonvertical segments (x(k),y(k))-to-(x(k+1),y(k+1)) such that 0 = x(1) < x(2) < ... < x(n-1) < x(n)=i, 0 = y(1) <= y(2) <= ... <= y(n-1) <= y(n)=j, for i >= 0, j >= 0.

0, 0, 1, 0, 1, 2, 0, 1, 3, 4, 0, 1, 4, 8, 8, 0, 1, 5, 13, 20, 16, 0, 1, 6, 19, 38, 48, 32, 0, 1, 7, 26, 63, 104, 112, 64, 0, 1, 8, 34, 96, 192, 272, 256, 128, 0, 1, 9, 43, 138, 321, 552, 688, 576, 256, 0, 1, 10, 53, 190, 501, 1002, 1520, 1696, 1280, 512

Offset: 0

Clark Kimberling

Essentially array A059576 divided by sequence A011782.
[Hetyei] calls a variant of this array (omitting the initial row of zeros) the asymmetric Delannoy numbers and shows how they arise in certain lattice path enumeration problems and a face enumeration problem associated to Jacobi polynomials. - Peter Bala, Oct 29 2008
Essentially triangle in A208341. - Philippe Deléham, Mar 23 2012
T(n+k,n) is the dot product of a vector from the n-th row of Pascal's triangle A007318 with a vector created by the first n+1 values evaluated from the cycle index of symmetry group S(k). Example: T(4+3,4) = T(7,4) = {1,4,6,4,1}.{1,4,10,20,35} = 192. - Richard Turk, Sep 21 2017
The formula T(n,k) = Sum_{r=0..n-1} C(k+r,r)*C(n-1,r) (Paul D. Hanna, Oct 06 2006) counts the paths of the title by number, r, of interior vertices in the path. - David Callan, Nov 25 2021

			Diagonals (each starting on row 1): {0}; {0,1}; {0,1,2}; ...
Array begins:
    0     0     0     0     0     0     0     0     0     0     0 ...
    1     1     1     1     1     1     1     1     1     1     1 ...
    2     3     4     5     6     7     8     9    10    11    12 ...
    4     8    13    19    26    34    43    53    64    76    89 ...
    8    20    38    63    96   138   190   253   328   416   518 ...
   16    48   104   192   321   501   743  1059  1462  1966  2586 ...
   32   112   272   552  1002  1683  2668  4043  5908  8378 11584 ...
   64   256   688  1520  2972  5336  8989 14407 22180 33028 47818 ...
Triangle begins:
  0;
  0, 1;
  0, 1, 2;
  0, 1, 3,  4;
  0, 1, 4,  8,  8;
  0, 1, 5, 13, 20,  16;
  0, 1, 6, 19, 38,  48,  32;
  0, 1, 7, 26, 63, 104, 112, 64;
  ...
(1, 0, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) where DELTA is the operator defined in A084938 begins:
  1;
  1, 0;
  1, 2,  0;
  1, 3,  4,  0;
  1, 4,  8,  8,   0;
  1, 5, 13, 20,  16,   0;
  1, 6, 19, 38,  48,  32,  0;
  1, 7, 26, 63, 104, 112, 64, 0;

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
David Callan, Some bijections for lattice paths, arXiv:2112.05241 [math.CO], 2021.
David Callan, A bijection for Delannoy paths, arXiv:2202.04649 [math.CO], 2022.
R. Cori and G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
R. Cori and G. Hetyei, How to count genus one partitions, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
Robert Cori and Gabor Hetyei, Genus one partitions, in 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AT, pp. 333-344, .
Sergio Falcon, On the complex k-Fibonacci numbers, Cogent Mathematics, (2016), 3: 1201944. See Table 1.
G. Hetyei, Central Delannoy numbers, Legendre polynomials and a balanced join operation preserving the Cohen-Macauley property, Annals of Combinatorics, 10 (2006), 443-462.
G. Hetyei, Central Delannoy numbers and balanced Cohen-Macaulay complexes, Ann. Comb. 10 (2006), 443-462.
G. Hetyei, Links we almost missed between Delannoy numbers and Legendre polynomials
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
M. Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Clark Kimberling, Enumeration of paths, compositions of integers and Fibonacci numbers, Fib. Quarterly 39 (5) (2001) 430-435, Figure 1.
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 3C.
Benjamin Lyons and McCabe Olsen, Self-Reachable Chip Configurations on Trees, arXiv:2409.00763 [math.CO], 2024. See p. 18.
Thomas Selig, Combinatorial aspects of sandpile models on wheel and fan graphs, arXiv:2202.06487 [math.CO], 2022.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Diagonal sums are even-indexed Fibonacci numbers. Alternating (+-) diagonal sums are signed Fibonacci numbers.
T(n, n-1) = A001850(n) (Delannoy numbers). T(n, n)=A047781. Cf. A035028, A055587.
Cf. A208341. A055587.

a049600 n k = a049600_tabl !! n !! k
a049600_row n = a049600_tabl !! n
a049600_tabl = [0] : map (0 :) a208341_tabl
-- Reinhard Zumkeller, Apr 15 2014

A049600 := proc(n,k)
    add(binomial(k+j,j)*binomial(n-1,j),j=0..n-1) ;
end proc: # R. J. Mathar, Oct 26 2015

t[n_, k_] := Hypergeometric2F1[ n-k+1, 1-k, 1, -1] // Floor; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 09 2013 *)
t[n_, k_] := Sum[LaguerreL[n-k, i, 0]* LaguerreL[k-i, i, 0], {i,0,k}] //Floor; Table[t[n,k], {n, 0, 16}, {k, -1, n}] (* Richard Turk, Sep 08 2017 *)
T[n_, k_] := If[k == 0, 0, JacobiP[k - 1, 0, 1 - 2*k + n, 3]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Nov 25 2021 *)

{A(i, j) = polcoeff( (x / (1 - 2*x)) * ((1 - x) / (1 - 2*x))^j + x * O(x^i), i)}; /* Michael Somos, Oct 01 2003 */

T(n,k)=sum(j=0,n-1,binomial(k+j,j)*binomial(n-1,j)) \\ Paul D. Hanna, Oct 06 2006

T(n,k) = Sum_{j=0..n-1} C(k+j,j)*C(n-1,j). - Paul D. Hanna, Oct 06 2006
T(i,j) = 2*T(i-1,j) + T(i,j-1) - T(i-1,j-1) with T(0,0)=1 and T(i,j)=0 if one of i,j<0. - Theodore Kolokolnikov, Jul 05 2010
O.g.f.: t*x/(1 - (2*t+1)*x + t*x^2) = t*x + (t + 2*t^2)*x^2 + (t + 3*t^2 + 4*t^3)*x^3 + .... Taking the row reverse of this triangle (with an additional column of 1's) gives A055587. - Peter Bala, Sep 10 2012
T(i,0) = 2^(i-1) and for j>0, T(i,j) = T(i,j-1) + Sum_{k=0..i-1} T(k,j). - Glen Whitney, Aug 17 2021
T(n, k) = JacobiP(k - 1, 0, 1 - 2*k + n, 3) for k >= 1. - Peter Luschny, Nov 25 2021

A055588 a(n) = 3*a(n-1) - a(n-2) - 1 with a(0) = 1 and a(1) = 2.

Original entry on oeis.org

1, 2, 4, 9, 22, 56, 145, 378, 988, 2585, 6766, 17712, 46369, 121394, 317812, 832041, 2178310, 5702888, 14930353, 39088170, 102334156, 267914297, 701408734, 1836311904, 4807526977, 12586269026, 32951280100, 86267571273

Offset: 0

Wolfdieter Lang, May 30 2000; Barry E. Williams, Jun 04 2000

Number of directed column-convex polyominoes with area n+2 and having two cells in the bottom row. - Emeric Deutsch, Jun 14 2001
a(n) is the length of the list generated by the substitution: 3->3, 4->(3,4,6), 6->(3,4,6,6): {3, 4}, {3, 3, 4, 6}, {3, 3, 3, 4, 6, 3, 4, 6, 6}, {3, 3, 3, 3, 4, 6, 3, 4, 6, 6, 3, 3, 4, 6, 3, 4, 6, 6, 3, 4, 6, 6}, etc. - Wouter Meeussen, Nov 23 2003
Equals row sums of triangle A144955. - Gary W. Adamson, Sep 27 2008
Equals the INVERT transform of A034943 and the INVERTi transform of A094790. - Gary W. Adamson, Apr 01 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 20.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
M. M. Mogbonju, I. A. Ogunleke, and O. A. Ojo, Graphical Representation Of Conjugacy Classes In The Order-Preserving Full Transformation Semigroup, International Journal of Scientific Research and Engineering Studies (IJSRES), Volume 1(5) (2014), ISSN: 2349-8862.
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.02067 [math.CO], 2015 (1st line of Table 1 is 3*a(n-2)).
László Németh, Pascal pyramid in the space H^2 x R, arXiv:1701.06022 [math.CO], 2017 (1st line of Table 1 is a(n-2)).
Yan X Zhang, Four Variations on Graded Posets, arXiv:1508.00318 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A001906, A008346, A019827, A034943, A055587, A094790, A144955.
Partial sums of A001519.
Apart from the first term, same as A052925.

List([0..40], n-> Fibonacci(2*n)+1 ); # G. C. Greubel, Jun 06 2019

[Fibonacci(2*n)+1: n in [0..40]]; // Vincenzo Librandi, Sep 30 2017

g:=z/(1-3*z+z^2): gser:=series(g, z=0, 43): seq(abs(coeff(gser, z, n)+1), n=0..27); # Zerinvary Lajos, Mar 22 2009

Table[Fibonacci[2n] +1, {n, 0, 40}] (* or *) LinearRecurrence[{4, -4, 1}, {1, 2, 4}, 40] (* Vincenzo Librandi, Sep 30 2017 *)

vector(40, n, n--; fibonacci(2*n)+1) \\ G. C. Greubel, Jun 06 2019

[lucas_number1(n,3,1)+1 for n in range(40)] # Zerinvary Lajos, Jul 06 2008

a(n) = (((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n)/sqrt(5) + 1.
a(n) = Sum_{m=0..n} A055587(n, m) = 1 + A001906(n).
G.f.: (1 - 2*x)/((1 - 3*x + x^2)*(1-x)).
From Paul Barry, Oct 07 2004: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3);
a(n) = Sum_{k=0..floor(n/3)} binomial(n-k, 2*k)2^(n-3*k). (End)
From Paul Barry, Oct 26 2004: (Start)
a(n) = A001906(n) + 1.
a(n) = Sum_{k=0..n} Fibonacci(2*k+2)*(2*0^(n-k) - 1).
a(n) = A008346(2*n). (End)
a(n) = Sum_{k=0..2*n+1} ((-1)^(k+1))*Fibonacci(k). - Michel Lagneau, Feb 03 2014
E.g.f.: cosh(x) + sinh(x) + 2*exp(3*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, May 14 2024
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/10) (A019827). - Amiram Eldar, Nov 28 2024

A106195 Riordan array (1/(1-2x), x(1-x)/(1-2*x)).

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 20, 13, 5, 1, 32, 48, 38, 19, 6, 1, 64, 112, 104, 63, 26, 7, 1, 128, 256, 272, 192, 96, 34, 8, 1, 256, 576, 688, 552, 321, 138, 43, 9, 1, 512, 1280, 1696, 1520, 1002, 501, 190, 53, 10, 1, 1024, 2816, 4096, 4048, 2972, 1683, 743, 253, 64, 11

Offset: 0

Gary W. Adamson, Apr 24 2005; Paul Barry, May 21 2006

Extract antidiagonals from the product P * A, where P = the infinite lower triangular Pascal's triangle matrix; and A = the Pascal's triangle array:
  1, 1,  1,  1, ...
  1, 2,  3,  4, ...
  1, 3,  6, 10, ...
  1, 4, 10, 20, ...
  ...
Row sums are Fibonacci(2n+2). Diagonal sums are A006054(n+2). Row sums of inverse are A105523. Product of Pascal triangle A007318 and A046854.
A106195 with an appended column of ones = A055587. Alternatively, k-th column (k=0, 1, 2) is the binomial transform of bin(n, k).
T(n,k) is the number of ideals in the fence Z(2n) having k elements of rank 1. - Emanuele Munarini, Mar 22 2011
Subtriangle of the triangle given by (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 22 2012

			Triangle begins
   1;
   2,   1;
   4,   3,   1;
   8,   8,   4,  1;
  16,  20,  13,  5,  1;
  32,  48,  38, 19,  6, 1;
  64, 112, 104, 63, 26, 7, 1;
(0, 2, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, ...) begins :
  1;
  0,  1;
  0,  2,   1;
  0,  4,   3,   1;
  0,  8,   8,   4,  1;
  0, 16,  20,  13,  5,  1;
  0, 32,  48,  38, 19,  6, 1;
  0, 64, 112, 104, 63, 26, 7, 1. - _Philippe Deléham_, Mar 22 2012

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Column 0 = 1, 2, 4...; (binomial transform of 1, 1, 1...); column 1 = 1, 3, 8, 20...(binomial transform of 1, 2, 3...); column 2: 1, 4, 13, 38...= binomial transform of bin(n, 2): 1, 3, 6...

Cf. A001792, A001906, A002620, A007318, A029653, A049612, A055587, A078812, A208341.

a106195 n k = a106195_tabl !! n !! k
a106195_row n = a106195_tabl !! n
a106195_tabl = [1] : [2, 1] : f [1] [2, 1] where
   f us vs = ws : f vs ws where
     ws = zipWith (-) (zipWith (+) ([0] ++ vs) (map (* 2) vs ++ [0]))
                      ([0] ++ us ++ [0])
-- Reinhard Zumkeller, Dec 16 2013

[ (&+[Binomial(n-k, n-j)*Binomial(j, k): j in [0..n]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 15 2020

T := (n, k) -> hypergeom([-n+k, k+1],[1],-1):
seq(lprint(seq(simplify(T(n, k)), k=0..n)), n=0..7); # Peter Luschny, May 20 2015

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := u[n - 1, x] + (x + 1) v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207605 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A106195 *)
(* Clark Kimberling, Feb 19 2012 *)
Table[Hypergeometric2F1[-n+k, k+1, 1, -1], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 15 2020 *)

create_list(sum(binomial(i,k)*binomial(n-k,n-i),i,0,n),n,0,8,k,0,n); /* Emanuele Munarini, Mar 22 2011 */

from sympy import Poly, symbols
x = symbols('x')
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + (x + 1)*v(n - 1, x)
def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017

from mpmath import hyp2f1, nprint
def T(n, k): return hyp2f1(k - n, k + 1, 1, -1)
for n in range(13): nprint([int(T(n, k)) for k in range(n + 1)]) # Indranil Ghosh, May 28 2017, after formula from Peter Luschny

[[sum(binomial(n-k,n-j)*binomial(j,k) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 15 2020

T(n,k) = Sum_{j=0..n} C(n-k,n-j)*C(j,k).
From Emanuele Munarini, Mar 22 2011: (Start)
T(n,k) = Sum_{i=0..n-k} C(k,i)*C(n-k,i)*2^(n-k-i).
T(n,k) = Sum_{i=0..n-k} C(k,i)*C(n-i,k)*(-1)^i*2^(n-k-i).
Recurrence: T(n+2,k+1) = 2*T(n+1,k+1)+T(n+1,k)-T(n,k). (End)
From Clark Kimberling, Feb 19 2012: (Start)
Define u(n,x) = u(n-1,x)+v(n-1,x), v(n,x) = u(n-1,x)+(x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1. Then v matches A106195 and u matches A207605. (End)
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1). - Philippe Deléham, Mar 22 2012
T(n+k,k) is the coefficient of x^n y^k in 1/(1-2x-y+xy). - Ira M. Gessel, Oct 30 2012
T(n, k) = A208341(n+1,n-k+1), k = 0..n. - Reinhard Zumkeller, Dec 16 2013
T(n, k) = hypergeometric_2F1(-n+k, k+1, 1 , -1). - Peter Luschny, May 20 2015
G.f. 1/(1-2*x+x^2*y-x*y). - R. J. Mathar, Aug 11 2015
Sum_{k=0..n} T(n, k) = Fibonacci(2*n+2) = A088305(n+1). - G. C. Greubel, Mar 15 2020

Edited by N. J. A. Sloane, Apr 09 2007, merging two sequences submitted independently by Gary W. Adamson and Paul Barry

A055852 Convolution of A055589 with A011782.

Original entry on oeis.org

0, 1, 7, 34, 138, 501, 1683, 5336, 16172, 47264, 134048, 370688, 1003136, 2664192, 6960384, 17922048, 45552640, 114442240, 284508160, 700579840, 1710161920, 4141416448, 9955639296, 23770693632, 56400543744, 133041225728

Offset: 0

Wolfdieter Lang May 30 2000

Seventh column of triangle A055587.

T(n,5) of array T as in A049600.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).

Cf. A011782, A049600, A055587, A055589.

a:=[1,7,34,138,501,1683];; for n in [7..30] do a[n]:=12*a[n-1] -60*a[n-2] +160*a[n-3] -240*a[n-4] +192*a[n-5] -64*a[n-6]; od; Concatenation([0], a); # G. C. Greubel, Jan 16 2020

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x)^5/(1-2*x)^6 )); // G. C. Greubel, Jan 16 2020

seq(coeff(series(x*(1-x)^5/(1-2*x)^6, x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 16 2020

CoefficientList[Series[x*(1-x)^5/(1-2*x)^6, {x,0,30}], x] (* G. C. Greubel, Jan 16 2020 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-x)^5/(1-2*x)^6)) \\ G. C. Greubel, Jan 16 2020

def A055852_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)^5/(1-2*x)^6 ).list()
A055852_list(30) # G. C. Greubel, Jan 16 2020

a(n) = T(n, 5) = A055587(n+5, 6).

G.f.: x*(1-x)^5/(1-2*x)^6.

A055589 Convolution of A049612 with A011782.

Original entry on oeis.org

0, 1, 6, 26, 96, 321, 1002, 2972, 8472, 23392, 62912, 165504, 427264, 1085184, 2717184, 6718464, 16427008, 39763968, 95387648, 226951168, 535953408, 1257046016, 2929852416, 6789267456, 15648423936, 35888562176, 81927340032

Offset: 0

Wolfdieter Lang May 30 2000

Sixth column of triangle A055587. T(n,4) of array T as in A049600.

Index entries for linear recurrences with constant coefficients, signature (10, -40, 80, -80, 32).

Cf. A049600, A049612, A055587, A011782.

a(n)= T(n, 4)= A055587(n+4, 5).

G.f.: x*((1-x)^4)/(1-2*x)^5.

A055853 Convolution of A055852 with A011782.

Original entry on oeis.org

0, 1, 8, 43, 190, 743, 2668, 8989, 28814, 88720, 264224, 765088, 2162624, 5986304, 16268800, 43499264, 114629120, 298147840, 766361600, 1948794880, 4907171840, 12245598208, 30305419264, 74425892864, 181481635840, 439603953664

Offset: 0

Wolfdieter Lang May 30 2000

Eighth column of triangle A055587.

T(n,6) of array T as in A049600.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

Cf. A011782, A049600, A055587, A055852.

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x)^6/(1-2*x)^7 )); // G. C. Greubel, Jan 16 2020

seq(coeff(series(x*(1-x)^6/(1-2*x)^7, x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 16 2020

CoefficientList[Series[x*(1-x)^6/(1-2*x)^7, {x,0,30}], x] (* G. C. Greubel, Jan 16 2020 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-x)^6/(1-2*x)^7)) \\ G. C. Greubel, Jan 16 2020

def A055853_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)^6/(1-2*x)^7 ).list()
A055853_list(30) # G. C. Greubel, Jan 16 2020

a(n) = T(n, 6)= A055587(n+6, 7).

G.f.: x*(1-x)^6/(1-2*x)^7.

A055854 Convolution of A055853 with A011782.

Original entry on oeis.org

0, 1, 9, 53, 253, 1059, 4043, 14407, 48639, 157184, 489872, 1480608, 4358752, 12541184, 35364864, 97960192, 267050240, 717619200, 1903452160, 4989337600, 12937052160, 33212530688, 84484882432, 213090238464, 533236219904

Offset: 0

Wolfdieter Lang May 30 2000

Ninth column of triangle A055587.

T(n,7) of array T as in A049600.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Cf. A011782, A049600, A055587, A055853.

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x)^7/(1-2*x)^8 )); // G. C. Greubel, Jan 16 2020

seq(coeff(series(x*(1-x)^7/(1-2*x)^8, x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 16 2020

CoefficientList[Series[x*(1-x)^7/(1-2*x)^8, {x,0,30}], x] (* G. C. Greubel, Jan 16 2020 *)
LinearRecurrence[{16,-112,448,-1120,1792,-1792,1024,-256},{0,1,9,53,253,1059,4043,14407,48639,157184},40] (* Harvey P. Dale, Nov 04 2023 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-x)^7/(1-2*x)^8)) \\ G. C. Greubel, Jan 16 2020

def A055854_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)^7/(1-2*x)^8 ).list()
A055854_list(30) # G. C. Greubel, Jan 16 2020

a(n)= T(n, 7)= A055587(n+7, 8).

G.f.: x*(1-x)^7/(1-2*x)^8.

A055855 Convolution of A055854 with A011782.

Original entry on oeis.org

0, 1, 10, 64, 328, 1462, 5908, 22180, 78592, 265729, 864146, 2719028, 8316200, 24814832, 72453344, 207502016, 584094080, 1618757120, 4423347200, 11932579840, 31812874240, 83901227008, 219074805760, 566754967552

Offset: 0

Wolfdieter Lang May 30 2000

Tenth column of triangle A055587.

T(n,8) of array T as in A049600.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-144,672,-2016,4032,-5376,4608,-2304,512).

Cf. A011782, A049600, A055584, A055857.

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x)^8/(1-2*x)^9 )); // G. C. Greubel, Jan 16 2020

seq(coeff(series(x*(1-x)^8/(1-2*x)^9, x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 16 2020

CoefficientList[Series[x*(1-x)^8/(1-2*x)^9, {x,0,30}], x] (* G. C. Greubel, Jan 16 2020 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-x)^8/(1-2*x)^9)) \\ G. C. Greubel, Jan 16 2020

def A055855_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)^8/(1-2*x)^9 ).list()
A055855_list(30) # G. C. Greubel, Jan 16 2020

a(n) = T(n, 8) = A055587(n+8, 9).

G.f.: x*(1-x)^8/(1-2*x)^9.

A156906 Transform of Fibonacci(n+1) with Hankel transform (-1)^binomial(n+1,2) * Fibonacci(n+1).

Original entry on oeis.org

1, 0, 1, -1, 0, 0, 2, -2, -3, 3, 11, -11, -31, 31, 101, -101, -328, 328, 1102, -1102, -3760, 3760, 13036, -13036, -45750, 45750, 162262, -162262, -580638, 580638, 2093802, -2093802, -7601043, 7601043, 27756627, -27756627, -101888163

Offset: 0

Paul Barry, Feb 17 2009

Hankel transform is (-1)^binomial(n+1,2) * Fibonacci(n+1).

Image of Fibonacci(n+1) by the Riordan array (1/(1 + x*c(-x^2)), x*c(-x^2)/(1 + x*c(-x^2))) = (1/(1-x), x*(1-x)/(1-2*x))^{-1} = A055587^{-1}.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000045, A000108, A055587.

cx := (1-sqrt(1-4*x))/2/x ;
A156906 := proc(n)
        1+x^2*subs(x=-x^2,cx)/(1+x) ;
        coeftayl(%,x=0,n) ;
end proc:
seq(A156906(n), n=0..40);  # R. J. Mathar, Jul 28 2016

a[n_]:= (1/2)*(2*Boole[n==0] -(-1)^n + Sum[(-1)^(n+j+1)*Binomial[2*j, j]/(2*j-1), {j, 0, Floor[n/2]}]); Table[a[n], {n, 0, 60}] (* G. C. Greubel, Jun 14 2021 *)

def A156906(n): return (1/2)*( 2*bool(n==0) - (-1)^n + sum( (-1)^(n+j+1)*binomial( 2*j, j)/(2*j-1) for j in (0..n//2)) )
[A156906(n) for n in (0..40)] # G. C. Greubel, Jun 14 2021

G.f.: (1 + 2*x + sqrt(1+4*x^2))/(2*(1+x)) = 1 + x^2*c(-x^2)/(1+x) where c(x) is the g.f. of A000108;
G.f.: 1/(1 - x^2/(1 + x + 2*x^2/(1 - x/2 + 3*x^2/4/(1 + x/6 + 10*x^2/9/(1 - x/15 + 24/25*x^2/(1 + ...)))))) (continued fraction);
In the continued fraction expansion of the g.f. the general term is 1 - x*if(n=0, 0, (-1)^n/(Fibonacci(n)*Fibonacci(n+1))) + x^2*(-0^n + Fibonacci(n)*Fibonacci(n+2) )/Fibonacci(n+1)^2.
a(n) = Sum_{k=0..n} (-1)^(n-k)*b(k), where b(n) = binomial(1,n) -A000108((n-2)/2) * (-1)^(n/2) * (1+(-1)^n)/2 and b(0) = 1.
n*a(n) = -n*a(n-1) -4*(n-3)*a(n-2) -4*(n-3)*a(n-3). - R. J. Mathar, Nov 14 2011
a(n) = (1/2)*( 2*[n=0] - (-1)^n + Sum_{j=0..floor(n/2)} (-1)^(n+j+1)*binomial(2*j, j)/(2*j-1) ). - G. C. Greubel, Jun 14 2021

cp's OEIS Frontend

A049600 Array T read by diagonals; T(i,j) is the number of paths from (0,0) to (i,j) consisting of nonvertical segments (x(k),y(k))-to-(x(k+1),y(k+1)) such that 0 = x(1) < x(2) < ... < x(n-1) < x(n)=i, 0 = y(1) <= y(2) <= ... <= y(n-1) <= y(n)=j, for i >= 0, j >= 0.

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A055588 a(n) = 3*a(n-1) - a(n-2) - 1 with a(0) = 1 and a(1) = 2.

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A106195 Riordan array (1/(1-2*x), x*(1-x)/(1-2*x)).

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A055852 Convolution of A055589 with A011782.

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A055589 Convolution of A049612 with A011782.

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A055853 Convolution of A055852 with A011782.

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A106195 Riordan array (1/(1-2x), x(1-x)/(1-2*x)).