A151403 -id:A151403 - cp's OEIS frontend

A156275 a(n) = 10^n*Catalan(n).

1, 10, 200, 5000, 140000, 4200000, 132000000, 4290000000, 143000000000, 4862000000000, 167960000000000, 5878600000000000, 208012000000000000, 7429000000000000000, 267444000000000000000, 9694845000000000000000, 353576700000000000000000

Offset: 0

Philippe Deléham, Feb 07 2009

In general, for m >= 1, Sum_{k>=0} 1/(m^k * Catalan(k)) = 2*m*(8*m + 1) / (4*m - 1)^2 + 24 * m^2 * arcsin(1/(2*sqrt(m))) / (4*m - 1)^(5/2). - Vaclav Kotesovec, Nov 23 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vincent Pilaud, Pebble trees, arXiv:2205.06686 [math.CO], 2022.

Cf. A000108, A005159, A085880, A151374, A151403, A156058, A156128, A156266, A156270, A156273.

Column k=10 of A290605.

[10^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011

Table[10^n CatalanNumber[n],{n,0,20}] (* Harvey P. Dale, Mar 12 2013 *)

a(n) = 10^n*A000108(n).
From Gary W. Adamson, Jul 18 2011: (Start)
a(n) is the upper left term in M^n, M = an infinite square production matrix:
  10, 10,  0,  0,  0, ...
  10, 10, 10,  0,  0, ...
  10, 10, 10, 10,  0, ...
  10, 10, 10, 10, 10, ...
  ... (End)
E.g.f.: KummerM(1/2, 2, 40*x). - Peter Luschny, Aug 26 2012
G.f.: c(10*x) with c(x) the o.g.f. of A000108 (Catalan). - Philippe Deléham, Nov 15 2013
a(n) = Sum_{k=0..n} A085880(n,k)*9^k. - Philippe Deléham, Nov 15 2013
G.f.: 1/(1 - 10*x/(1 - 10*x/(1 - 10*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 08 2017
Sum_{n>=0} 1/a(n) = 180/169 + 800*arctan(1/sqrt(39)) / (507*sqrt(39)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 1580/1681 - 2400*arctanh(1/sqrt(41)) / (1681*sqrt(41)). - Amiram Eldar, Jan 25 2022
D-finite with recurrence (n+1)*a(n) +20*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022

a(15) corrected by Vincenzo Librandi, Jul 19 2011

A354735 a(0) = a(1) = 1; a(n) = 4 * Sum_{k=0..n-2} a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, 4, 8, 36, 96, 416, 1312, 5504, 19200, 79168, 293888, 1203712, 4648448, 19027968, 75411456, 309487616, 1248411648, 5144133632, 21011775488, 86971449344, 358540509184, 1490753372160, 6189315784704, 25843660619776, 107902536122368, 452308820819968, 1897178275512320

Offset: 0

Ilya Gutkovskiy, Jun 04 2022

nonn

Cf. A007477, A052704, A151403, A354733, A354734, A354736.

a[0] = a[1] = 1; a[n_] := a[n] = 4 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 27}]
nmax = 27; CoefficientList[Series[(1 - Sqrt[1 - 16 x^2 (1 + x)])/(8 x^2), {x, 0, nmax}], x]

G.f. A(x) satisfies: A(x) = 1 + x + 4 * (x * A(x))^2.
G.f.: (1 - sqrt(1 - 16 * x^2 * (1 + x))) / (8 * x^2).
a(n) ~ sqrt((2+3*r)*(1+r)) / (sqrt(Pi) * n^(3/2) * r^n), where r = 2*cos(arccos(-5/32)/3)/3 - 1/3. - Vaclav Kotesovec, Jun 04 2022

A240558 a(n) = 2^nn!/((floor(n/2)+1)floor(n/2)!^2).

Original entry on oeis.org

1, 2, 4, 24, 32, 320, 320, 4480, 3584, 64512, 43008, 946176, 540672, 14057472, 7028736, 210862080, 93716480, 3186360320, 1274544128, 48432676864, 17611882496, 739699064832, 246566354944, 11342052327424, 3489862254592, 174493112729600, 49855175065600

Offset: 0

Peter Luschny, Apr 14 2014

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

Cf. A000108, A000984, A056040, A057977, A069723, A099045, A151374, A151403.

A240558 := n -> 2^n*n!/((iquo(n,2)+1)*iquo(n,2)!^2):
seq(A240558(n), n=0..30);

Table[SeriesCoefficient[((I*(2*x*(8*x+1)-1))/Sqrt[16*x^2-1]-2*x+1) /(8*x^2), {x,0,n}], {n,0,22}]

x='x+O('x^50); Vec(round((I*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2)) \\ G. C. Greubel, Apr 05 2017

def A240558():
    x, n = 1, 1
    while True:
        yield x
        m = 2*n if is_odd(n) else 8/(n+2)
        x *= m
        n += 1
a = A240558(); [next(a) for i in range(36)]

O.g.f.: ((i*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2), where i=sqrt(-1).
For a recurrence see the Sage program.
a(n) = 2^n*A057977(n)
a(2*k) = A151403(k) = 2^k*A151374(k) = 4^k*A000108(k).
a(2*k+1) = A099045(k+1) = 2^k*A069723(k+2) = 4^k*A000984(k+1).
From Peter Luschny, Jan 31 2015: (Start)
a(n) = Sum_{k=0..n} A056040(n)*C(n,k)/(floor(n/2)+1).
a(n) = Sum_{k=0..n} n!*C(n,k)/((floor(n/2)+1)*(floor(n/2)!)^2).
a(n) = 2^n*n!*[x^n]((x+1)*hypergeom([],[2],x^2)).
a(n) ~ 2^(n+N)/((n+1)^*sqrt(Pi*(2*N+1))); here  = 1 if n is even, 0 otherwise and N = n++1. (End)
Conjecture: -(n+2)*(n^2-5)*a(n) +8*(-2*n-1)*a(n-1) +16*(n-1)*(n^2+2*n-4)*a(n-2)=0. - R. J. Mathar, Jun 14 2016

A375393 a(0) = 1; a(n) = Sum_{k=0..n-1} (4k+3) a(k) * a(n-k-1).

Original entry on oeis.org

1, 3, 30, 483, 10314, 268686, 8167068, 281975715, 10863651474, 461227101210, 21377716429860, 1073816307452430, 58106804389870500, 3370330005649001532, 208635817503306332088, 13731856676157543219747, 957698874584753026878306, 70562301536089812703526370, 5477354759932929856218644820

Offset: 0

Ilya Gutkovskiy, Sep 09 2024

nonn

Cf. A000699, A088716, A151403, A215648, A376086, A376087.

a[0] = 1; a[n_] := a[n] = Sum[(4 k + 3) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; A[] = 0; Do[A[x] = 1 + 3 x A[x]^2 + 4 x^2 A'[x] A[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + 3 * x * A(x)^2 + 4 * x^2 * A'(x) * A(x).

A376087 a(0) = 1; a(n) = Sum_{k=0..n-1} (4k+1) a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 6, 65, 994, 19386, 456940, 12594465, 396969930, 14078044862, 554782989908, 24053551260186, 1138039204281236, 58353983394380500, 3223791843357228120, 190914111715994215905, 12065701995815379444954, 810602692757305194731094, 57688894099612173692496580

Offset: 0

Ilya Gutkovskiy, Sep 09 2024

nonn

Cf. A000699, A088716, A151403, A215648, A375393, A376086.

a[0] = 1; a[n_] := a[n] = Sum[(4 k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 4 x^2 A'[x] A[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + x * A(x)^2 + 4 * x^2 * A'(x) * A(x).

A178657 Irregular triangle: the coefficient [x^k] of the polynomial (1-x)^(2n-1) Sum_{s>=0} A001263(n+2s,2s+1)*x^s in row n >= 1 and column k >= 0.

Original entry on oeis.org

1, 1, 3, 1, 15, 15, 1, 1, 43, 161, 105, 10, 1, 96, 855, 1680, 855, 96, 1, 1, 185, 3191, 13387, 17655, 7623, 945, 21, 1, 323, 9570, 72254, 188188, 188188, 72254, 9570, 323, 1, 1, 525, 24675, 302359, 1345605, 2499861, 2036125, 715725, 99414, 4410, 36, 1, 808

Offset: 1

Roger L. Bagula, Jun 01 2010

Row sums are 1, 4, 32, 320, 3584, 43008, 540672, 7028736, 93716480, 1274544128, ... (see A151403, A052704).

The sequence is the Narayana number analog of A034839.

			1;
1, 3;
1, 15, 15, 1;
1, 43, 161, 105, 10;
1, 96, 855, 1680, 855, 96, 1;
1, 185, 3191, 13387, 17655, 7623, 945, 21;
1, 323, 9570, 72254, 188188, 188188, 72254, 9570, 323, 1;
1, 525, 24675, 302359, 1345605, 2499861, 2036125, 715725, 99414, 4410, 36;
1, 808, 56896, 1055320, 7329975, 22338816, 32152848, 22338816, 7329975, 1055320, 56896, 808, 1;

Cf. A001263, A034839.

A001263 := proc(n,k) if n <=0 or k <=0 then 0 ; elif k > n then 0 ; else binomial(n-1,k-1)*binomial(n,k-1)/k ; end if; end proc:
A178657 := proc(n,k) (1-x)^(2*n-1)*add(A001263(n+2*l,2*l+1)*x^l,l=0..20) ; expand(%) ; coeftayl(%,x=0,k) ; end proc: # R. J. Mathar, Aug 30 2011

p[x_, n_] = (1 - x)^(2*n - 1)*Sum[(Binomial[2*k + n, 2*k] Binomial[ 2*k + n, 1 + 2*k]/(2*k + n))*x^k, {k, 0, Infinity}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 2, 10}];
Flatten[%]

A212694 Number of 2-colored Dyck n-paths with up steps (U, u), down steps (D, d), and avoiding UU and DD.

Original entry on oeis.org

1, 4, 25, 197, 1745, 16580, 165115, 1700809, 17971466, 193710087, 2121585340, 23543198588, 264138223362, 2991130956918, 34143543312267, 392458689992396, 4538574332686469, 52768896995910303, 616471818881678085, 7232838546289017796, 85188401983572928395

Offset: 0

Alois P. Heinz, May 23 2012

nonn

Upper case letters denote one color and lower case letters the other.

			a(1) = 4: ud, Ud, uD, UD.
a(2) = 25: uudd, Uudd, uUdd, uuDd, UuDd, uUDd, udud, Udud, uDud, UDud, udUd, UdUd, uDUd, UDUd, uudD, UudD, uUdD, uduD, UduD, uDuD, UDuD, udUD, UdUD, uDUD, UDUD.

Alois P. Heinz, Table of n, a(n) for n = 0..910
Wikipedia, Counting lattice paths

Cf. A000108, A007863, A151403.

b:= proc(x, y, t) option remember; `if`(x=0, 1,
      `if`(y<1  , 0, b(x-1, y-1, 0)+`if`(t=1, 0, b(x-1, y-1, 1)))+
      `if`(x b(2*n, 0$2):
seq(a(n), n=0..30);

b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, If[y < 1, 0, b[x - 1, y - 1, 0] + If[t == 1, 0, b[x - 1, y - 1, 1]]] + If[x < y + 2, 0, b[x - 1, y + 1, 0] + If[t == 2, 0, b[x - 1, y + 1, 2]]]];
a[n_] := b[2n, 0, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Recurrence: 5*n*(n+1)*(n+2)*(11856*n^6 - 462048*n^5 + 6376819*n^4 - 42412433*n^3 + 147659510*n^2 - 260432089*n + 184927095)*a(n) = n*(n+1)*(1683552*n^7 - 66428880*n^6 + 937628962*n^5 - 6466755025*n^4 + 23922419618*n^3 - 47274340850*n^2 + 44490285903*n - 13108829940)*a(n-1) - n*(14310192*n^8 - 585624672*n^7 + 8802419365*n^6 - 66744441981*n^5 + 284660409448*n^4 - 703230360993*n^3 + 976943147665*n^2 - 682900257024*n + 175305383460)*a(n-2) + 2*(17238624*n^9 - 746332752*n^8 + 12295273466*n^7 - 105941663163*n^6 + 537013083761*n^5 - 1677467513328*n^4 + 3243096592679*n^3 - 3743377415442*n^2 + 2332897216785*n - 592736810400)*a(n-3) - (37974768*n^9 - 1728832752*n^8 + 30714335273*n^7 - 291055104422*n^6 + 1652510618897*n^5 - 5890634883203*n^4 + 13267453974662*n^3 - 18291224811263*n^2 + 14073816074160*n - 4638445144200)*a(n-4) + (23308896*n^9 - 1108835760*n^8 + 20950004098*n^7 - 213362099351*n^6 + 1311302140489*n^5 - 5085585201440*n^4 + 12508042515937*n^3 - 18889708965719*n^2 + 15984167161110*n - 5834960787600)*a(n-5) - (8927568*n^9 - 442319616*n^8 + 8817227331*n^7 - 95321285525*n^6 + 623567997102*n^5 - 2575734547859*n^4 + 6742249614729*n^3 - 10822975984520*n^2 + 9730758446550*n - 3782772932400)*a(n-6) + 4*(2*n - 13)*(248976*n^8 - 11244288*n^7 + 197289135*n^6 - 1812121625*n^5 + 9661474889*n^4 - 30841990357*n^3 + 57959845045*n^2 - 59304520365*n + 25796647800)*a(n-7) - 4*(n-7)*(2*n - 15)*(2*n - 13)*(11856*n^6 - 390912*n^5 + 4244419*n^4 - 21288517*n^3 + 54240485*n^2 - 69082196*n + 35668710)*a(n-8). - Vaclav Kotesovec, Jul 16 2014

Limit n->infinity a(n)^(1/n) = (13+3*sqrt(17))/2 = 12.68465843842649... . - Vaclav Kotesovec, Jul 16 2014

A380119 Triangle read by rows: T(n, k) is the number of walks of length 2*n on the N X N grid with unit steps in all four directions (NSWE) starting at (0, 0). k is the common value of the x- and the y-coordinate of the endpoint of the walk.

Original entry on oeis.org

1, 2, 2, 10, 16, 6, 70, 140, 90, 20, 588, 1344, 1134, 448, 70, 5544, 13860, 13860, 7392, 2100, 252, 56628, 151008, 169884, 109824, 42900, 9504, 924, 613470, 1717716, 2108106, 1561560, 750750, 231660, 42042, 3432, 6952660, 20225920, 26546520, 21781760, 12155000, 4667520, 1191190, 183040, 12870

Offset: 0

Peter Luschny, Jan 19 2025

			The triangle starts:
  [0] [      1]
  [1] [      2,        2]
  [2] [     10,       16,        6]
  [3] [     70,      140,       90,      20]
  [4] [    588,     1344,     1134,      448,       70]
  [5] [   5544,    13860,    13860,     7392,     2100,     252]
  [6] [  56628,   151008,   169884,   109824,    42900,    9504,     924]
  [7] [ 613470,  1717716,  2108106,  1561560,   750750,  231660,   42042,   3432]
  [8] [6952660, 20225920, 26546520, 21781760, 12155000, 4667520, 1191190, 183040, 12870]
.
For n = 2 the walks depending on the x-coordinate of the endpoint are:
W(x=0) = {NNSS,NSNS,NSWE,NWSE,NWES,WNSE,WNES,WWEE,WENS,WEWE},
W(x=1) = {NNSW,NNWS,NSNW,NSWN,NWNS,NWSN,NWWE,NWEW,WNNS,WNSN,WNWE,WNEW,WWNE,WWEN,WENW,WEWN},
W(x=2) = {NNWW,NWNW,NWWN,WNNW,WNWN,WWNN}.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008.
Richard K. Guy, Christian Krattenthaler and Bruce E. Sagan, Lattice paths, reflections, & dimension-changing bijections, Ars Combin. 34 (1992), 3-15.

Related triangles: A380120.

Cf. A005568 (column 0), A000984 (main diagonal), A253487 (sub diagonal), A151403 (row sums).

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) == 2*n:
            if w.x == w.y: row[w.y] += 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
                if (w.y + y >= 0) and (w.x + x >= 0):
                    W.append(Walk(w.s + s, w.x + x, w.y + y))
    return row
for n in range(6): print(Trow(n))

A243312 The total T(d,n) of expressions with n instances of a digit d between 6 and 9 (and using the four arithmetic operators) which have a defined value when evaluated.

Original entry on oeis.org

1, 4, 31, 305, 3345, 39505, 487935, 6245118

Offset: 1

David Lyness, Jun 03 2014

Bounded above by A151403 (which is equal to the *total* number of expressions, not just the ones that evaluate to a valid numeric value).
An expression has an undefined value when it contains a "division by zero" operation.
For 0 <= d <= 5 and d' > 5, it is possible that T(d,n) < T(d',n). See link below.
T(6,10) = 1097128463 < T(7,10) = 1097153419 < T(8,10) = 1097155971 < T(9,10) = 1097157195. - Hiroaki Yamanouchi, Oct 01 2014

			For n = 2 and digit d != 0, there are four possible expressions:
(d + d)
(d - d)
(d * d)
(d / d)
None of these include a "division by zero" operation, and so all four of the above expressions can be considered valid. Therefore, T(d, 2) = 4 for d > 0.

David Lyness, Experimenting with bracketing and binary operators

# coding=utf-8
 
import itertools
 
 
def all_expressions(generic_expression, operation_combinations):
    """
    Merges a source expression and combinations of binary operators to generate a list of all possible expressions.
    @param ((str,)) operation_combinations: all combinations of binary operators to consider
    @param str generic_expression: source expression with placeholder binary operators
    @rtype: (str,)
    """
    expression_combinations = []
    for combination in operation_combinations:
        expression_combinations.append(generic_expression.format(*combination))
    return expression_combinations
 
 
def all_bracketings(expr):
    """
    Generates all possible permutations of parentheses for an expression.
    @param str expr: the non-bracketed source expression
    @rtype: str
    """
    if len(expr) == 1:
        yield expr
    else:
        for i in range(1, len(expr), 2):
            for left_expr in all_bracketings(expr[:i]):
                for right_expr in all_bracketings(expr[i + 1:]):
                    yield "({}{}{})".format(left_expr, expr[i], right_expr)
 
 
def num_valid_expressions(num_digits):
    """Perform all calculations with the given operations and in the range of digits specified.
    @param int num_digits: the number of digits in the expression
    @rtype: int
    """
    operations = ["+", "-", "*", "/"]
    digit = 9
    operation_iterable = itertools.product(*[operations] * (num_digits - 1))
    operation_combinations = []
    valid_expression_count = 0
    template = " ".join(str(digit) * num_digits)
    for operation in operation_iterable:
        operation_combinations.append(operation)
    for bracketed_expression in all_bracketings(template):
        for expression in all_expressions(bracketed_expression.replace(" ", "{}"), operation_combinations):
            try:
                eval(expression)
                valid_expression_count += 1
            except ZeroDivisionError:
                pass
    return valid_expression_count
 
 
if _name_ == "_main_":
    min_num_digits = 1
    max_num_digits = 6
    operation_set = ["+", "-", "*", "/"]
    for num in range(min_num_digits, max_num_digits + 1):
        print(num_valid_expressions(num))

A338415 Triangle read by rows: T(n,m) = C(n+m+1,n)C(2n-m,n)C(3n+1,n)C(4n+2,2m+1)/(2(n+1)C(2n,n)C(2n+2m+2,2n)).

Original entry on oeis.org

1, 2, 2, 7, 18, 7, 30, 130, 130, 30, 143, 884, 1530, 884, 143, 728, 5880, 14896, 14896, 5880, 728, 3876, 38760, 131100, 193200, 131100, 38760, 3876, 21318, 254562, 1085238, 2153250, 2153250, 1085238, 254562, 21318, 120175, 1669800, 8627300, 21755800, 29370330, 21755800, 8627300, 1669800, 120175

Offset: 0

Vladimir Kruchinin, Oct 25 2020

			1,
2, 2,
7, 18, 7,
30, 130, 130, 30,
143, 884, 1530, 884, 143

Cf. A000108, A001263, A006013, A008459, A151403.

A(x,y) := ((x*(27*y^4-108*y^3+162*y^2+(-108)*y+27)-2*y^3+66*y^2+66*y-2)/(54*y^6-324*y^5+810*y^4+(-1080)*y^3+810*y^2+(-324)*y+54)+sqrt(27*x^2*y^4+((-108)*x^2-4*x)*y^3+(162*x^2+132*x)*y^2+((-108)*x^2+132*x-16)*y+27*x^2+(-4)*x)/(2*3^(3/2)*(y-1)^4))^(1/3)+(y^2+14*y+1)/((9*y^4-36*y^3+54*y^2+(-36)*y+9)*((x*(27*y^4-108*y^3+162*y^2+(-108)*y+27)-2*y^3+66*y^2+66*y-2)/(54*y^6-324*y^5+810*y^4+(-1080)*y^3+810*y^2+(-324)*y+54)+sqrt(27*x^2*y^4+((-108)*x^2-4*x)*y^3+(162*x^2+132*x)*y^2+((-108)*x^2+132*x-16)*y+27*x^2+(-4)*x)/(2*3^(3/2)*(y-1)^4))^(1/3))+(2*y+2)/(3*(y^2-2*y+1));
taylor(A(x,y),x,0,7,y,0,7);

T(n,m):=(binomial(n+m+1,n)*binomial(2*n-m,n)*binomial(3*n+1,n)* binomial(4*n+2,2*m+1))/((2*n+2)*binomial(2*n,n)*binomial(2*n+2*m+2,2*n));

G.f. satisfies A(x,y)=x/(A(x,y)^2*y^2-2*A(x,y)^2*y-2*A(x,y)*y+A(x,y)^2-2*A(x,y)+1).

A(x,y) = ((x*(27*y^4-108*y^3+162*y^2+(-108)*y+27)-2*y^3+66*y^2+66*y-2)/(54*y^6-324*y^5+810*y^4+(-1080)*y^3+810*y^2+(-324)*y+54)+sqrt(27*x^2*y^4+((-108)*x^2-4*x)*y^3+(162*x^2+132*x)*y^2+((-108)*x^2+132*x-16)*y+27*x^2+(-4)*x)/(2*3^(3/2)*(y-1)^4))^(1/3)+(y^2+14*y+1)/((9*y^4-36*y^3+54*y^2+(-36)*y+9)*((x*(27*y^4-108*y^3+162*y^2+(-108)*y+27)-2*y^3+66*y^2+66*y-2)/(54*y^6-324*y^5+810*y^4+(-1080)*y^3+810*y^2+(-324)*y+54)+sqrt(27*x^2*y^4+((-108)*x^2-4*x)*y^3+(162*x^2+132*x)*y^2+((-108)*x^2+132*x-16)*y+27*x^2+(-4)*x)/(2*3^(3/2)*(y-1)^4))^(1/3))+(2*y+2)/(3*(y^2-2*y+1)).

cp's OEIS Frontend

A156275 a(n) = 10^n*Catalan(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A354735 a(0) = a(1) = 1; a(n) = 4 * Sum_{k=0..n-2} a(k) * a(n-k-2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A240558 a(n) = 2^n*n!/((floor(n/2)+1)*floor(n/2)!^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

A375393 a(0) = 1; a(n) = Sum_{k=0..n-1} (4*k+3) * a(k) * a(n-k-1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A376087 a(0) = 1; a(n) = Sum_{k=0..n-1} (4*k+1) * a(k) * a(n-k-1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A178657 Irregular triangle: the coefficient [x^k] of the polynomial (1-x)^(2*n-1) * Sum_{s>=0} A001263(n+2*s,2*s+1)*x^s in row n >= 1 and column k >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A212694 Number of 2-colored Dyck n-paths with up steps (U, u), down steps (D, d), and avoiding UU and DD.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A380119 Triangle read by rows: T(n, k) is the number of walks of length 2*n on the N X N grid with unit steps in all four directions (NSWE) starting at (0, 0). k is the common value of the x- and the y-coordinate of the endpoint of the walk.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

A240558 a(n) = 2^nn!/((floor(n/2)+1)floor(n/2)!^2).

A375393 a(0) = 1; a(n) = Sum_{k=0..n-1} (4k+3) a(k) * a(n-k-1).

A376087 a(0) = 1; a(n) = Sum_{k=0..n-1} (4k+1) a(k) * a(n-k-1).

A178657 Irregular triangle: the coefficient [x^k] of the polynomial (1-x)^(2n-1) Sum_{s>=0} A001263(n+2s,2s+1)*x^s in row n >= 1 and column k >= 0.

A338415 Triangle read by rows: T(n,m) = C(n+m+1,n)C(2n-m,n)C(3n+1,n)C(4n+2,2m+1)/(2(n+1)C(2n,n)C(2n+2m+2,2n)).