A059231 -id:A059231 - cp's OEIS frontend

A204057 Triangle derived from an array of f(x), Narayana polynomials.

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 14, 1, 1, 5, 19, 45, 42, 1, 1, 6, 29, 100, 197, 132, 1, 1, 7, 41, 185, 562, 903, 429, 1, 1, 8, 55, 306, 1257, 3304, 4279, 1430, 1, 1, 9, 71, 469, 2426, 8925, 20071, 20793, 4862, 1, 1, 10, 89, 680, 4237, 20076, 65445, 124996, 103049, 16796, 1

Offset: 1

Gary W. Adamson, Jan 09 2012

Row sums = (1, 2, 4, 10, 31, 113, 466, 2129, 10641, 138628, 335379, 2702364,...)
Another version of triangle in A008550. - Philippe Deléham, Jan 13 2012
Another version of A243631. - Philippe Deléham, Sep 26 2014

			First few rows of the array =
  1,....1,....1,.....1,.....1,...; = A000012
  1.....2,....5,....14,....42,...; = A000108
  1,....3,...11,....45,...197,...; = A001003
  1,....4,...19,...100,...562,...; = A007564
  1,....5,...29,...185,..1257,...; = A059231
  1,....6,...41,...306,..2426,...; = A078009
  ...
First few rows of the triangle =
  1;
  1, 1;
  1, 2,  1;
  1, 3,  5,   1;
  1, 4, 11,  14,    1;
  1, 5, 19,  45,   42,    1;
  1, 6, 29, 100,  197,  132,     1;
  1, 7, 41, 185,  562,  903,   429,     1;
  1, 8, 55, 306, 1257, 3304,  4279,  1430,    1;
  1, 9, 71, 469, 2426, 8952, 20071, 20793, 4862, 1;
  ...
Examples: column 4 of the array = A090197: (1, 14, 45, 100,...) = N(4,n) where N(4,x) is the 4th Narayana polynomial.
Term (5,3) = 29 is the upper left term of M^3, where M = the infinite square production matrix:
  1, 4, 0, 0, 0,...
  1, 1, 4, 0, 0,...
  1, 1, 1, 4, 0,...
  1, 1, 1, 1, 4,...
... generating row 5, A059231: (1, 5, 29, 185,...).

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000108, A001003, A007564, A028387, A059231, A078009, A090197, A090198, A090199, A090200.

Cf. A008550, A132745, A243631.

A204057:= func< n, k | n eq 0 select 1 else (&+[ Binomial(n, j)^2*k^j*(n-j)/(n*(j+1)): j in [0..n-1]]) >;
[A204057(k, n-k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 16 2021

Table[Hypergeometric2F1[1-k, -k, 2, n-k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Feb 16 2021 *)

def A204057(n, k): return 1 if n==0 else sum( binomial(n, j)^2*k^j*(n-j)/(n*(j+1)) for j in [0..n-1])
flatten([[A204057(k, n-k) for k in [1..n]] for n in [1..12]]) # G. C. Greubel, Feb 16 2021

The triangle is the set of antidiagonals of an array in which columns are f(x) of the Narayana polynomials; with column 1 = (1, 1, 1,...) column 2 = (1, 2, 3,..), column 3 = A028387, column 4 = A090197, then A090198, A090199,...
The array by rows is generated from production matrices of the form:
  1, (N-1)
  1, 1, (N-1)
  1, 1, 1, (N-1)
  1, 1, 1, 1, (N-1)
...(infinite square matrices with the rest zeros); such that if the matrix is M, n-th term in row N is the upper left term of M^n.
From G. C. Greubel, Feb 16 2021: (Start)
T(n, k) = Hypergeometric2F1([1-k, -k], [2], n-k).
Sum_{k=1..n} T(n, k) = A132745(n) - 1. (End)

Corrected by Philippe Deléham, Jan 13 2012

A371364 Expansion of (1/x) * Series_Reversion( x * (1-4x)^2 / (1-3x) ).

Original entry on oeis.org

1, 5, 49, 597, 8129, 118469, 1807665, 28512213, 461141761, 7606159365, 127454706609, 2163636679125, 37130370808257, 643099703566277, 11227141735655345, 197356077159062613, 3490230884900117505, 62054912214781757957, 1108568475427756051761

Offset: 0

Seiichi Manyama, Mar 19 2024

nonn

Index entries for reversions of series

Cf. A059231, A371365.

A371364 := proc(n)
    add(3^(n-k)*binomial(2*n+k+1,k)*binomial(2*n,n-k),k=0..n) ;
    %/(n+1) ;
end proc:
seq(A371364(n),n=0..60) ; # R. J. Mathar, Apr 22 2024

my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-4*x)^2/(1-3*x))/x)

a(n) = sum(k=0, n, 3^(n-k)*binomial(2*n+k+1, k)*binomial(2*n, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} 3^(n-k) * binomial(2*n+k+1,k) * binomial(2*n,n-k).

D-finite with recurrence (n+1)*(2*n+1)*a(n) +3*(-6*n^2-9*n+2)*a(n-1) -27*(7*n-9)*(2*n-3)*a(n-2) -243*(n-2)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Apr 22 2024

A086871 Row sums of A059450.

Original entry on oeis.org

1, 2, 10, 58, 370, 2514, 17850, 130890, 983650, 7536418, 58648810, 462306266, 3683602130, 29620138994, 240059315610, 1958940281322, 16081662931650, 132723191430210, 1100568370427850, 9164925012016506, 76612776253995570

Offset: 0

N. J. A. Sloane, Sep 16 2003

Hankel transform is A165928. - Paul Barry, Sep 30 2009

Number of skew Dyck paths of semilength n with the down steps coming in two colors. - David Scambler, Jun 21 2013

			G.f. = 1 + 2*x + 10*x^2 + 58*x^3 + 370*x^4 + 2514*x^5 + 17850*x^6 + 130890*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.
J. Machacek, Lattice walks ending on a coordinate hyperlane avoiding backtracking and repeats, arXiv:2105.02417 [math.CO], 2021. See Thm. 4.4 G(x,E^1).

Table[SeriesCoefficient[2/(1+Sqrt[(1-9*x)/(1-x)]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

{a(n) = if( n<0, 0, polcoeff( 2 / (1 + sqrt((1 - 9*x) / (1 - x) + x * O(x^n))), n))}; /* Michael Somos, Mar 06 2004 */

{a(n) = if( n<1, n==0, n++; 2 * polcoeff( serreverse( x * (1 - 4*x) / (1 - 3*x) + x * O(x^n)), n))}; /* Michael Somos, Mar 06 2004 */

a(n) = 2*A059231(n), if n>0.
G.f.: (1 - x - sqrt((1 - x) * (1 - 9*x))) / (4*x) = 2 / (1 + sqrt((1 - 9*x) / (1 - x))) =: y satisfies 0 = (1 - x) * (1 - y) + 2*x*y^2. - Michael Somos, Mar 06 2004
Moment representation: a(n) = (1/(4*Pi))*Integral_{x=1..9} x^n*sqrt(-x^2+10x-9)/x+(1/2)*0^n. - Paul Barry, Sep 30 2009
D-finite with recurrence Recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 3^(2*n+1)/(2*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
G.f.: 1/(2*x) - 1/2 + G(0)  where G(k) = 1 - 1/(x + x/(1 + 1/G(k+1) )) ; (continued fraction,3-step). - Sergei N. Gladkovskii, Nov 29 2012

More terms from Ray Chandler, Sep 17 2003

A224071 Number of Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.

Original entry on oeis.org

1, 2, 5, 15, 52, 201, 841, 3726, 17213, 82047, 400600, 1993377, 10071777, 51532938, 266462229, 1390174911, 7308741084, 38682855225, 205940368441, 1102091393574, 5925177392573, 31987877317887, 173337754977904

Offset: 0

José Luis Ramírez Ramírez, Mar 30 2013

nonn

Hankel transform is A006215. Invert transform of A155069. - Michael Somos, Apr 02 2013

			a(2) = 5 because we have HH, UDH, HUD, UDUD and UUDD.
G.f. = 1 + 2*x + 5*x^2 + 15*x^3 + 52*x^4 + 201*x^5 + 841*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Bloom and S. Elizalde, Pattern avoidance in matchings and partitions, arXiv:1211.3442 [math.CO], 2012; Theorem 6.1.
Paul Barry, A study of Integer Sequences, Riordan Arrays, Pascal-like Arrays and Hankel Transforms, Ph.D Thesis, University College, Cork, Republic of Ireland, 2009.
Arnauld Mesinga Mwafise, Computational and Combinatorial Enumeration of Poset Matrices, 2024. See p. 8.

Cf. A001003, A007564, A010892, A059231.

CoefficientList[Series[4/(3-5*x+Sqrt[x^2-6*x+1]), {x, 0, 20}], x] (* Vaclav Kotesovec, May 23 2013 *)
a[ n_] := SeriesCoefficient[ (3 - 5 x - Sqrt[ 1 - 6 x + x^2]) / (2 - 6 x + 6 x^2), {x, 0, n}]; (* Michael Somos, Mar 28 2014 *)

a(n):=sum((k+1)*((-1)^floor((k+2)/3)+(-1)^floor((k+1)/3))*sum(binomial(n+1,n-k-i)*binomial(n+i,n),i,0,n-k),k,0,n)/(2*(n+1)); /* Vladimir Kruchinin, Mar 08 2016*/

z='z+O('z^66); Vec(4/(3-5*z+sqrt(1-6*z+z^2))) /* Joerg Arndt, Mar 30 2013 */

G.f.: 4/(3-5*x+sqrt(1-6*x+x^2)).
Recurrence: n*a(n) = 9*(n-1)*a(n-1) - 2*(11*n-15)*a(n-2) + 3*(7*n-12)*a(n-3) - 3*(n-3)*a(n-4). - Vaclav Kotesovec, May 23 2013
a(n) ~ sqrt(884+627*sqrt(2)) * (3+2*sqrt(2))^n / (98*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, May 23 2013
0 = +a(n)*(+9*a(n+1) - 144*a(n+2) + 174*a(n+3) - 81*a(n+4) + 12*a(n+5)) + a(n+1)*(+18*a(n+1) + 399*a(n+2) - 597*a(n+3) + 318*a(n+4) - 57*a(n+5)) + a(n+2)*(-300*a(n+2) + 538*a(n+3) - 255*a(n+4) + 52*a(n+5)) + a(n+3)*(-126*a(n+3) + 73*a(n+4) - 18*a(n+5)) + a(n+4)*(+a(n+5)) if n>=0. - Michael Somos, Mar 28 2014
a(n) = Sum_{k=0..n}((k+1)*((-1)^floor((k+2)/3)+(-1)^floor((k+1)/3))*Sum_{i=0..n-k}(binomial(n+1,n-k-i)*binomial(n+i,n)))/(2*(n+1)). - Vladimir Kruchinin, Mar 08 2016

A366166 Decimal expansion of sqrt(Pi)/(3sqrt(3))(Gamma(1/3)/Gamma(5/6))^3.

Original entry on oeis.org

4, 5, 5, 9, 7, 9, 4, 4, 9, 9, 9, 5, 9, 8, 4, 5, 8, 1, 5, 4, 8, 1, 7, 3, 6, 4, 8, 4, 5, 5, 7, 2, 4, 8, 1, 1, 7, 6, 3, 6, 7, 4, 2, 3, 8, 0, 1, 6, 6, 1, 4, 0, 5, 6, 3, 5, 0, 5, 1, 8, 3, 8, 7, 6, 5, 4, 7, 2, 1, 1, 5, 9, 5, 9, 3, 5, 5, 7, 0, 4, 4, 9, 2, 3, 2, 4, 8, 7, 9, 6

Offset: 1

Hugo Pfoertner, Oct 13 2023

This constant c occurs in the probability that the "big player" in a game with 3 gamblers goes broke first, although he starts with an initial capital of N-2 units, whereas the other two gamblers start with one unit each. This probability is ~ c/N^3. See Diaconis link for details.

			4.5597944999598458154817364845572481176367423801661405635...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Persi Diaconis, Gambler's ruin with k gamblers (slide 3), talk in the Rutgers Experimental Mathematics Seminar, Fall 2023 Semester, Oct. 12, 2023.

Cf. A059231, A366566, A366567.

First[RealDigits[Gamma[1/3]^9/(2Pi)^4,10,100]] (* Paolo Xausa, Oct 14 2023 *)

sqrt(Pi)/(3*sqrt(3))*(gamma(1/3)/gamma(5/6))^3

Equals Gamma(1/3)^9 / (2*Pi)^4. - Peter Luschny, Oct 13 2023

A371365 Expansion of (1/x) * Series_Reversion( x * (1-4x)^3 / (1-3x) ).

Original entry on oeis.org

1, 9, 141, 2701, 57513, 1307553, 31083925, 763267077, 19208408721, 492817411705, 12842067417309, 338956669920189, 9042967461581753, 243464712274093713, 6606427290991922277, 180492205687604057013, 4960765361688213527073

Offset: 0

Seiichi Manyama, Mar 19 2024

nonn

Index entries for reversions of series

Cf. A059231, A371364.

Cf. A369023, A371363.

my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-4*x)^3/(1-3*x))/x)

a(n) = sum(k=0, n, 3^(n-k)*binomial(3*n+k+2, k)*binomial(3*n+1, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} 3^(n-k) * binomial(3*n+k+2,k) * binomial(3*n+1,n-k).

A257363 Number of 3-Motzkin paths with no level steps at height 1.

Original entry on oeis.org

1, 3, 10, 33, 110, 369, 1247, 4248, 14603, 50724, 178314, 635526, 2300829, 8477382, 31842897, 122103276, 478372886, 1915188093, 7831613468, 32674683984, 138871668314, 600140517762, 2631926843602, 11690520554421, 52498671870181, 237966449687118, 1087246253873875, 5001141997115010, 23137102115963262

Offset: 0

José Luis Ramírez Ramírez, Apr 20 2015

nonn

For n=2 we have 10 paths: H(1)H(1), H(1)H(2), H(2)H(1), H(2)H(2), H(1)H(3), H(3)H(1), H(3)H(3), H(2)H(3), H(3)H(2), UD.

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A117641, A217312, A253831, A059231.

rec:= (95+95*n)*a(n)+(-180-9*n)*a(n+1)+(-329-197*n)*a(n+2)+(369+144*n)*a(n+3)+(-117-36*n)*a(4+n)+(12+3*n)*a(n+5):
f:= gfun:-rectoproc({rec,a(0)=1,a(1)=3,a(2)=10,a(3)=33,a(4)=110},a(n),remember):
seq(f(n),n=0..100); # Robert Israel, Apr 28 2015

CoefficientList[Series[2*(3+x)/(6-17*x-9*x^2+x*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)

G.f.: 1/(1-3*x-x*F(x)), where F(x) is the g.f. of the sequence A117641.
G.f.: 2*(3+x)/(6-17*x-9*x^2+x*sqrt(1-6*x+5*x^2)).
a(n) ~ 5^(n+3/2)/(98*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
From Robert Israel, Apr 28 2015 (Start):
G.f.: (6-x*sqrt(1-6*x+5*x^2)-17*x-9*x^2)/(6-36*x+42*x^2+38*x^3).
3*(-n+1)*a(n) +9*(4*n-7)*a(n-1) +9*(-16*n+39)*a(n-2) +(197*n-656)*a(n-3) +9*(n+15)*a(n-4) +95*(-n+4)*a(n-5)=0. (End)

A086873 Triangle read by rows in which row n >= 1 gives coefficients in expansion of the polynomial Sum_{k=1..n} (1/n)binomial(n,k)binomial(n,k-1)x^(2k)(1+x)^(2n-2k) / x^2 in powers of x.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 9, 10, 5, 1, 6, 21, 44, 57, 42, 14, 1, 8, 38, 116, 240, 336, 308, 168, 42, 1, 10, 60, 240, 680, 1392, 2060, 2160, 1530, 660, 132, 1, 12, 87, 430, 1545, 4152, 8449, 13014, 14985, 12540, 7227, 2574, 429, 1, 14, 119, 700, 3045, 10122, 26173, 53048

Offset: 1

N. J. A. Sloane, Sep 16 2003

Row n has 2n-1 terms.

			For n=3 the polynomial is 1 + 4x + 9x^2 + 10x^3 + 5x^4.
  1;
  1,  2,  2;
  1,  4,  9,  10,    5;
  1,  6, 21,  44,   57,   42,   14;
  1,  8, 38, 116,  240,  336,  308,   168,    42;
  1, 10, 60, 240,  680, 1392, 2060,  2160,  1530,   660,  132;
  1, 12, 87, 430, 1545, 4152, 8449, 13014, 14985, 12540, 7227, 2574, 429;

C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.

A059231 gives row sums.

j := 0:f := n->sum(binomial(n,k)*binomial(n,k-1)/n*x^(2*k)*(1+x)^(2*n-2*k),k=1..n): for n from 1 to 15 do p := expand(f(n)/x^2):for l from 0 to 2*n-2 do j := j+1: a[j] := coeff(p,x,l):od:od:seq(a[l],l=1..j); # Sascha Kurz

for(n=1,8,p=sum(k=1,n,(1/n)*binomial(n,k)*binomial(n,k-1)*x^(2*k)*(1+x)^(2*n-2*k))/x^2; for(i=1,2*n-1,print1(polcoeff(p,i-1) ","); ); print; ); \\ Ray Chandler, Sep 17 2003

More terms from Vladeta Jovovic and Ray Chandler, Sep 17 2003

A257072 Number of 3-colored Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.

Original entry on oeis.org

1, 4, 17, 77, 374, 1959, 11085, 67500, 438485, 3004985, 21485222, 158744467, 1202966761, 9297312916, 72981656937, 580105886517, 4658713796790, 37736326098735, 307913254091925, 2528335636842300, 20875157745756429

Offset: 0

José Luis Ramírez Ramírez, Apr 20 2015

nonn

			a(1) = 4  because we have H1, H2, H2, UD.

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A224071, A250307, A059231

Table[8 DifferenceRoot[Function[{y, n}, {3645 (1 + n) (2 + n) y[n] + (-20736 - 21384 n - 5508 n^2) y[1 + n] + (6120 + 7857 n + 2061 n^2) y[2 + n] + (1330 - 165 n - 175 n^2) y[3 + n] + (-664 - 270 n - 26 n^2) y[4 + n] + 3 (4 + n) (5 + n) y[5 + n] == 0, y[0] == 1/8, y[1] == 1/2, y[2] == 17/8, y[3] == 77/8, y[4] == 187/4}]][k], {k, 0, 20}] (* Benedict W. J. Irwin, May 29 2016 *)
CoefficientList[Series[8/(7 -27*x +Sqrt[1 -10*x +9*x^2]), {x,0,50}], x] (* G. C. Greubel, May 29 2016 *)

a(n):=(sum(m*sum(((sum(binomial(j+m,k)*binomial(j-1,j-k)*4^(j-k),k,0,j))*3^(n-j-m)*binomial(n-j,m))/(j+m),j,0,n-m),m,1,n))+3^n; /* Vladimir Kruchinin, Mar 13 2016 */

G.f.: 8/(7-27*z+sqrt(1-10*z+9*z^2))=1/(1-3*z-z*F(z)), where F(z) is the g.f. of the sequence A059231.
a(n) = (3^n+Sum_{m=1..n}(m*Sum_{j=0..n-m}(((Sum_{k=0..j}(binomial(j+m,k)*binomial(j-1,j-k)*4^(j-k)))*3^(n-j-m)*binomial(n-j,m))/(j+m)))). - Vladimir Kruchinin, Mar 13 2016
a(n) ~ sqrt(2)*3^(2*n-1) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 13 2016
From Benedict W. J. Irwin, May 29 2016: (Start)
Let y(0)=1/8, y(1)=1/2, y(2)=17/8, y(3)=77/8, y(4)=187/4,
Let 3645*(n+1)*(n+2)*y(n)-(5508n^2+21384n+20736)*y(n+1)+(2061n^2+7857n+6120)*y(n+2)-(175n^2+165n-1330)*y(n+3)-(26n^2+270n+664)*y(n+4)+3*(n+4)*(n+5)*y(n+5) = 0,
a(n) = 8*y(n).
(End)
Conjecture: 3*n*a(n) +(-53*n+45)*a(n-1) +2*(151*n-213)*a(n-2) +9*(-73*n+144)*a(n-3) +405*(n-3)*a(n-4)=0. - R. J. Mathar, Sep 24 2016

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A204057 Triangle derived from an array of f(x), Narayana polynomials.

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A371364 Expansion of (1/x) * Series_Reversion( x * (1-4*x)^2 / (1-3*x) ).

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A086871 Row sums of A059450.

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A224071 Number of Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.

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Maxima

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A366166 Decimal expansion of sqrt(Pi)/(3*sqrt(3))*(Gamma(1/3)/Gamma(5/6))^3.

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A371365 Expansion of (1/x) * Series_Reversion( x * (1-4*x)^3 / (1-3*x) ).

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A257363 Number of 3-Motzkin paths with no level steps at height 1.

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A371364 Expansion of (1/x) * Series_Reversion( x * (1-4x)^2 / (1-3x) ).

A366166 Decimal expansion of sqrt(Pi)/(3sqrt(3))(Gamma(1/3)/Gamma(5/6))^3.

A371365 Expansion of (1/x) * Series_Reversion( x * (1-4x)^3 / (1-3x) ).

A086873 Triangle read by rows in which row n >= 1 gives coefficients in expansion of the polynomial Sum_{k=1..n} (1/n)binomial(n,k)binomial(n,k-1)x^(2k)(1+x)^(2n-2k) / x^2 in powers of x.