A162551 -id:A162551 - cp's OEIS frontend

A352626 a(n) = (n+1)(3n-2)C(2n,n-1)/(4*n-2).

1, 8, 42, 200, 910, 4032, 17556, 75504, 321750, 1361360, 5727436, 23984688, 100053772, 416024000, 1725013800, 7135405920, 29452939110, 121347523440, 499132441500, 2050025300400, 8408638258020, 34448503964160, 140974630569240, 576340150932000, 2354075068866300

Offset: 1

Natalya Ter-Saakov, Mar 24 2022

This is half the number of edges of the origami flip graph of the all-equal-angle single-vertex crease pattern of degree 2n.

Michael De Vlieger, Table of n, a(n) for n = 1..1658
Thomas C. Hull, Manuel Morales, Sarah Nash, and Natalya Ter-Saakov, Maximal origami flip graphs of flat-foldable vertices: properties and algorithms, arXiv:2203.14173 [math.CO], 2022.

Number of vertices of the origami flip graph of the all-equal-angle single-vertex crease pattern of degree 2n is A162551.

a:=n->(n+1)*(3*n-2)*binomial(2*n,n-1)/(4*n-2):
seq(a(n),n=1..33);

a[n_] := (n + 1)*(3*n - 2)*Binomial[2*n, n - 1]/(4*n - 2); Array[a, 25] (* Amiram Eldar, Mar 25 2022 *)

a(n) = (n+1)*(3*n-2)*binomial(2*n,n-1)/(4*n-2); \\ Michel Marcus, Mar 25 2022

G.f.: x*(2*x+1)/(1-4*x)^(3/2).
D-finite with recurrence (-n+1)*a(n) +2*(n+2)*a(n-1) +4*(2*n-5)*a(n-2)=0. - R. J. Mathar, Mar 29 2022
a(n) = A002457(n-1)+2*A002457(n-2). - R. J. Mathar, Mar 29 2022

A380120 Triangle read by rows: T(n, k) is the number of walks of length n on the Z X Z grid with unit steps in all four directions (NSWE) starting at (0, 0). k is the absolute value of the x-coordinate of the endpoint of the walk.

Original entry on oeis.org

1, 2, 2, 6, 8, 2, 20, 30, 12, 2, 70, 112, 56, 16, 2, 252, 420, 240, 90, 20, 2, 924, 1584, 990, 440, 132, 24, 2, 3432, 6006, 4004, 2002, 728, 182, 28, 2, 12870, 22880, 16016, 8736, 3640, 1120, 240, 32, 2, 48620, 87516, 63648, 37128, 17136, 6120, 1632, 306, 36, 2

Offset: 0

Peter Luschny, Jan 17 2025

			Triangle starts:
  [0] [    1]
  [1] [    2,     2]
  [2] [    6,     8,     2]
  [3] [   20,    30,    12,     2]
  [4] [   70,   112,    56,    16,     2]
  [5] [  252,   420,   240,    90,    20,    2]
  [6] [  924,  1584,   990,   440,   132,   24,    2]
  [7] [ 3432,  6006,  4004,  2002,   728,  182,   28,   2]
  [8] [12870, 22880, 16016,  8736,  3640, 1120,  240,  32,  2]
  [9] [48620, 87516, 63648, 37128, 17136, 6120, 1632, 306, 36, 2]
.
For n = 0 there is only the empty walk w = '' with start and end point (x=0, y=0).
For n = 3 the walks depending on the x-coordinate of the endpoint are:
W(x= 3) = {WWW},
W(x= 2) = {NWW,SWW,WNW,WSW,WWN,WWS},
W(x= 1) = {NNW,NSW,NWN,NWS,SNW,SSW,SWN,SWS,WNN,WNS,WSN,WSS,WWE,WEW,EWW},
W(x= 0) = {NNN,NNS,NSN,NSS,NWE,NEW,SNN,SNS,SSN,SSS,SWE,SEW,WNE,WSE,WEN,WES,ENW,ESW,EWN,EWS},
W(x=-1) = {NNE,NSE,NEN,NES,SNE,SSE,SEN,SES,WEE,ENN,ENS,ESN,ESS,EWE,EEW},
W(x=-2) = {NEE,SEE,ENE,ESE,EEN,EES},
W(x=-3) = {EEE}.
T(3, 0) = card(W(x=0)) = 20, T(3, 1) = card(W(x=1)) + card(W(x=-1)) = 30,
T(3, 2) = card(W(x=2)) + card(W(x=-2)) = 12, T(3, 3) = card(W(x=3)) + card(W(x=-3)) = 2.

Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Philippe Flajolet, Institut Henri Poincaré, March 28, 2013.
Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009; Discrete Mathematics & Theoretical Computer Science, DMTCS Proceedings vol. AK, (FPSAC 2009).
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

Related triangles: A052174 (N X N), A378067 (Z X N), A379822 (Z X Z, variant), A380119.

Cf. A000984 (column 0), A162551 (column 1), A006659 (column 2), A000302 (row sums), A068551 (row sum without column 0), A040000 (row minimum).

T := (n, k) -> ifelse(k = 0, binomial(2*n, n - k), 2*binomial(2*n, n - k)):
seq(print(seq(T(n, k), k = 0..n)), n = 0..9);

from dataclasses import dataclass
@dataclass
class Walk:
    s: str = ""
    x: int = 0
    y: int = 0
def Trow(n: int) -> list[int]:
    W = [Walk()]
    row = [0] * (n + 1)
    for w in W:
        if len(w.s) == n:
            row[abs(w.x)] += 1
        else:
            for s in "NSWE":
                x = y = 0
                match s:
                    case "W": x =  1
                    case "E": x = -1
                    case "N": y =  1
                    case "S": y = -1
                    case _  : pass
                W.append(Walk(w.s + s, w.x + x, w.y + y))
    return row
for n in range(10): print(Trow(n))

T(n, k) = binomial(2*n, n - k) if k = 0, otherwise 2*binomial(2*n, n - k).
Assuming the columns starting at n = 0, i.e. prepended by k zeros:
T(n, k) = [x^n] (2^(2*k+1)*x^k / (sqrt(1-4*x)*(1+sqrt(1-4*x))^(2*k))) for k >= 1.
T(n, k) = n! * [x^n] (2*BesselI(k, 2*x)*exp(2*x)) for k >= 1.

A217234 Triangle of expansion coefficients of the sum of an n X n array with equal top row and left column (extended by the rule of Pascal's triangle) in terms of the top row elements.

Original entry on oeis.org

1, 1, 4, 1, 12, 6, 1, 40, 20, 8, 1, 140, 70, 30, 10, 1, 504, 252, 112, 42, 12, 1, 1848, 924, 420, 168, 56, 14, 1, 6864, 3432, 1584, 660, 240, 72, 16, 1, 25740, 12870, 6006, 2574, 990, 330, 90, 18, 1, 97240, 48620, 22880, 10010, 4004, 1430, 440, 110, 20

Offset: 1

J. M. Bergot, Sep 28 2012

Define a finite n X n square array with indeterminate elements A(1, c), c=1..n in the top row, the same elements A(r,1 ) = A(1,r) in the first column, r=1..n, and the remaining elements defined by the Pascal triangle rule: A(r,c) = A(r,c-1)+A(r-1,c).
The triangle T(n,m) gives the coefficients in the formula Sum_{r=1..n} Sum_{c=1..n} A(r,c) = Sum_{m=1..n} T(n,m) * A(1,m).
It says how many times the first, second, third, etc. element of the first row (or the first column) contributes to the sum of the n X n array.

			1;
1,4;
1,12,6;
1,40,20,8;
1,140,70,30,10;
1,504,252,112,42,12;
1,1848,924,420,168,56,14;
1,6864,3432,1584,660,240,72,16;
1,25740,12870,6006,2574,990,330,90,18;
1,97240,48620,22880,10010,4004,1430,440,110,20;

Cf. A100320 (2nd column), A000984 (third column), A162551 (third column), A024483 (4th column), A006659 (5th column), A002058 (6th column), A030662 (row sums).

A217234_row := proc(n)
    local A,r,c,s ;
    A := array({},1..n,1..n) ;
    for r from 2 to n do
        A[r,1] := A[1,r] ;
    end do:
    for r from 2 to n do
        for c from 2 to n do
            A[r,c] := A[r,c-1]+A[r-1,c] ;
        end do:
    end do:
    s := add(add( A[r,c],c=1..n) ,r=1..n) ;
    for c from 1 to n do
        printf("%d,", coeff(s,A[1,c]) ) ;
    end do:
    return ;
end proc:
for n from 1 to 10 do
    A217234_row(n) ;
    printf(";\n") ;
end do; # R. J. Mathar, Oct 13 2012

A217234row [n_] := Module[{A, x, r, c, s }, A = Array[x, {n, n}]; Do[A[[r, 1]] = A[[1, r]], {r, 2, n}]; Do[A[[r, c]] = A[[r, c - 1]] + A[[r - 1, c]], {r, 2, n}, {c, 2, n}]; s = Sum[A[[r, c]], {r, 1, n}, {c, 1, n}]; If[n == 1, {1}, List @@ s /. x[, ] -> 1]];
Table[A217234row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 04 2023, after R. J. Mathar *)

Edited by R. J. Mathar, Oct 13 2012

Typo in data corrected by Jean-François Alcover, Nov 04 2023

A239453 Convolution of the generalized Catalan numbers A057977 with themselves.

Original entry on oeis.org

1, 2, 3, 8, 11, 30, 43, 112, 172, 420, 694, 1584, 2809, 6006, 11379, 22880, 46088, 87516, 186562, 335920, 754646, 1293292, 3050238, 4992288, 12319816, 19315400, 49725004, 74884320, 200571541, 290845350, 808559299, 1131445440, 3257808976, 4407922860, 13119940234

Offset: 0

Peter Luschny, Mar 19 2014

nonn

Cf. A162551.

def A239453_list(n):
    r = sqrt(1-4*x^2)
    g = lambda x: (1-r)*(r+x)/(2*x^2*r)
    s = taylor(g(x), x, 0, n+1)
    f = [s.coefficient(x, j) for j in (0..n+1)]
    return [add(f[k]*f[j-k] for k in (0..j)) for j in (0..n)]
A239453_list(36)

D-finite with recurrence -(n+4)*(137*n-1035)*a(n) +(383*n^2-1622*n-5421)*a(n-1) +4*(274*n^2-1385*n-3534)*a(n-2) +4*(-766*n^2+4393*n+555)*a(n-3) +16*(-137*n^2+898*n-606)*a(n-4) +16*(n-3)*(383*n-1622)*a(n-5)=0. - R. J. Mathar, Feb 03 2025

A304202 a(n) = (2n-3)4^(n-1) - 2binomial(2n, n-1).

Original entry on oeis.org

18, 208, 1372, 7632, 39050, 190112, 895524, 4120528, 18629652, 83088096, 366560568, 1602837280, 6956911962, 30007067456, 128736063316, 549740689872, 2338025684540, 9907917740128, 41853370268424, 176294674155104, 740683257681988, 3104678088923328, 12986226585328232

Offset: 3

Vincenzo Librandi, May 08 2018

Sihuang Hu and Gabriele Nebe, Strongly perfect lattices sandwiched between Barnes-Wall lattices, arXiv:1805.01196 [math.NT], 2018. See p. 21.

Cf. A144704, A162551.

[(2*n-3)*4^(n-1)-2*Binomial(2*n, n-1): n in [3..20]];

Table[(2 n - 3) 4^(n - 1) - 2 Binomial[2 n, n - 1], {n, 3, 40}]

a(n) = (2*n-3)*4^(n-1) - 2*binomial(2*n, n-1) \\ Charles R Greathouse IV, Oct 23 2023

E.g.f.: (3 + 12*x + 8*x^2 - 3*exp(4*x) + 8*exp(4*x)*x - 8*exp(2*x)*I_1(2*x) )/4, where I_1(.) is the modified Bessel function of the first kind. - Bruno Berselli, May 08 2018

(n+1)*(2*n^2-7*n+7)*a(n) - 2*n*(4*n-5)*(2*n-3)*a(n-1) + 8*(2*n-3)*(2*n^2-3*n+2)*a(n-2) = 0. - R. J. Mathar, May 08 2018

A337500 a(n) is the number of ballot sequences of length n tied or won by at most 3 votes.

Original entry on oeis.org

1, 2, 4, 8, 14, 30, 50, 112, 182, 420, 672, 1584, 2508, 6006, 9438, 22880, 35750, 87516, 136136, 335920, 520676, 1293292, 1998724, 4992288, 7696444, 19315400, 29716000, 74884320, 115000920, 290845350, 445962870

Offset: 0

Nachum Dershowitz, Aug 30 2020

Also the number of n-step walks on a path graph ending within 3 steps of the origin.

Also the number of monotonic paths of length n ending within 3 steps of the diagonal.

Cf. A337499, A024483.

Bisections give A162551 (odd part, starting from second element), A051924 (even part).

a(n) = A337499(n) + (n mod 2)*A024483(floor((n+3)/2)).

Conjecture: D-finite with recurrence -(n+3)*(n-4)*a(n) +2*(n^2-2*n-11)*a(n-1) +4*(n-1)^2*a(n-2) -8*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 27 2020

cp's OEIS Frontend

A352626 a(n) = (n+1)*(3*n-2)*C(2*n,n-1)/(4*n-2).

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A380120 Triangle read by rows: T(n, k) is the number of walks of length n on the Z X Z grid with unit steps in all four directions (NSWE) starting at (0, 0). k is the absolute value of the x-coordinate of the endpoint of the walk.

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A217234 Triangle of expansion coefficients of the sum of an n X n array with equal top row and left column (extended by the rule of Pascal's triangle) in terms of the top row elements.

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Maple

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A239453 Convolution of the generalized Catalan numbers A057977 with themselves.

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Sage

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A304202 a(n) = (2*n-3)*4^(n-1) - 2*binomial(2*n, n-1).

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Magma

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A337500 a(n) is the number of ballot sequences of length n tied or won by at most 3 votes.

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A352626 a(n) = (n+1)(3n-2)C(2n,n-1)/(4*n-2).

A304202 a(n) = (2n-3)4^(n-1) - 2binomial(2n, n-1).