A001700 -id:A001700 - cp's OEIS frontend

A045720 3-fold convolution of A001700(n), n >= 0.

1, 9, 57, 312, 1578, 7599, 35401, 161052, 719790, 3173090, 13836426, 59803104, 256596276, 1094249019, 4642178601, 19605872724, 82483419846, 345839048094, 1445715336366, 6027524015664, 25070662980876, 104056307673654

Offset: 0

Wolfdieter Lang

Total number of 132 (or 213) patterns in the set of all 123-avoiding permutations of length (n+3). - Cheyne Homberger, Mar 16 2012

a(n) is the degree of the cyclic graphical Gaussian model for the (n+3) cycle. - Mateusz Michalek, Mar 04 2023

B. Sturmfels, and C. Uhler. Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry. Annals of the Institute of Statistical Mathematics 62.4 (2010): 603-638, Conjecture 2 proved in "Geometry of the Gaussian graphical model of the cycle"

Indranil Ghosh, Table of n, a(n) for n = 0..1500
José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
A. Ayyer, Towards a Human Proof of Gessel's Conjecture, JIS 12 (2009) 09.4.2
R. Dinu, M. Michalek, and M. Vodička. Geometry of the Gaussian graphical model of the cycle, arXiv preprint arXiv:2111.02937 [math.AG] (2021).
C. Homberger, Expected patterns in permutation classes, Electronic Journal of Combinatorics, 19(3) (2012), P43.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
D. R. Snow, Spreadsheets, Power Series, Generating Functions and Integers, The College Maths. J. 20 (1989) 149.

Cf. A000108, A001700, A008549.

Table[(n+5)*Binomial[2*(n+3),n+3]/4-3*2^(2n+3),{n,0,21}] (* Indranil Ghosh, Feb 18 2017 *)

x='x+O('x^30); Vec((((1-4*x)^(-1/2)-1)/(2*x))^3) \\ Altug Alkan, Sep 04 2018

import math
def C(n,r):
    f=math.factorial
    return f(n)/f(r)/f(n-r)
def A045720(n):
    return (n+5)*C(2*(n+3),n+3)/4-3*2**(2*n+3) # Indranil Ghosh, Feb 18 2017

a(n) = (n+5)*binomial(2*(n+3), n+3)/4 - 3*2^(2*n+3);
G.f.: (c(x)/sqrt(1-4*x))^3, where c(x) = g.f. for Catalan numbers A000108;
recursion: a(n)=(2*(2*n+7)/(n+3))*a(n-1)+(3/(n+3))*A008549(n+1), a(0)=1.

A045894 4-fold convolution of A001700(n), n >= 0.

Original entry on oeis.org

1, 12, 94, 608, 3525, 19044, 97954, 486000, 2345930, 11081880, 51447036, 235454848, 1064832173, 4767347796, 21160397050, 93223960784, 408037319262, 1775744775592, 7688699122724, 33140226601920, 142262721338146

Offset: 0

Wolfdieter Lang

Indranil Ghosh, Table of n, a(n) for n = 0..1500
José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Cf. A001700, A000108, A045720.

Table[(n + 11)*4^(n + 2) - (n + 5) Binomial[2 (n + 4), n + 4]/2, {n, 0, 20}] (* Michael De Vlieger, Feb 18 2017 *)

import math
def C(n,r):
    f=math.factorial
    return f(n)/f(r)/f(n-r)
def A045894(n):
    return (n+11)*4**(n+2)-(n+5)*C(2*(n+4),(n+4))/2 # Indranil Ghosh, Feb 18 2017

a(n) = (n+11)*4^(n+2) - (n+5)*binomial(2*(n+4), n+4)/2;
G.f.: c(x)^4/(1-4*x)^2, where c(x) = g.f. for Catalan numbers A000108;
recursion: a(n)= (2*(2*n+10)/(n+4))*a(n-1) + (4/(n+4))*A045720(n), a(0)=1.

A100033 Bisection of A001700.

Original entry on oeis.org

3, 35, 462, 6435, 92378, 1352078, 20058300, 300540195, 4537567650, 68923264410, 1052049481860, 16123801841550, 247959266474052, 3824345300380220, 59132290782430712, 916312070471295267, 14226520737620288370

Offset: 0

N. J. A. Sloane, Nov 20 2004

Cf. A001700, A002458, A187364, A187365.

a:=n->binomial(4*n+3,2*n+2): seq(a(n),n=0..19);

a(n) = binomial(4*n+3, 2*n+2). - Emeric Deutsch, Dec 09 2004
From Peter Bala, Mar 19 2023: (Start)
a(n) = (1/2)*Sum_{k = 0..2*n+2} binomial(2*n+2,k)^2.
a(n) = (1/2)*hypergeom([-2 - 2*n, -2 - 2*n], [1], 1).
a(n) = 2*(4*n + 1)*(4*n + 3)/((n + 1)*(2*n + 1)) * a(n-1). (End)
From Peter Bala, Mar 28 2023: (Start)
a(n) = (1/(2*n + 2))*Sum_{k = 0..2*n+2} k*binomial(2*n+2,k)^2.
a(n) = 2*(n + 1)*hypergeom([-1 - 2*n, -1 - 2*n], [2], 1). (End)

More terms from Emeric Deutsch, Dec 09 2004

A205945 Triangle read by rows related to A001700.

Original entry on oeis.org

1, 1, 2, 1, 7, 2, 1, 20, 12, 2, 1, 54, 53, 16, 2, 1, 143, 208, 88, 20, 2, 1, 376, 768, 415, 130, 24, 2, 1, 986, 2734, 1804, 700, 180, 28, 2

Offset: 1

Gary W. Adamson, Feb 01 2012

Row sums = A001700: (1, 3, 10, 35, 126, 462, ...).

			First few rows of the triangle =
1;
1, 2;
1, 7, 2;
1, 20, 12, 2;
1, 54, 53, 16, 2;
1, 143, 208, 88, 20, 2;
1, 376, 768, 415, 130, 24, 2;
1, 986, 2734, 1804, 700, 180, 28, 2;
...
Row 3 = (1, 7, 2) = row 5 of triangle A191314; and finite differences of column 5 of triangle A205573: (1, 8, 10, ...).

Cf. A001700, A191314, A205573.

Bisection of triangle A191314 extracting odd numbered rows. Accessing odd numbered columns of A205573, take finite differences of terms from the top -> down.

A246432 Convolution inverse of A001700.

Original entry on oeis.org

1, -3, -1, -2, -5, -14, -42, -132, -429, -1430, -4862, -16796, -58786, -208012, -742900, -2674440, -9694845, -35357670, -129644790, -477638700, -1767263190, -6564120420, -24466267020, -91482563640, -343059613650, -1289904147324, -4861946401452

Offset: 0

Michael Somos, Nov 14 2014

sign

			G.f. = 1 - 3*x - x^2 - 2*x^3 - 5*x^4 - 14*x^5 - 42*x^6 - 132*x^7 - 429*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1660

Cf. A000108, A001700, A115140, A115141.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 +Sqrt(1-4*x))/2 -2*x)); // G. C. Greubel, Aug 04 2018

CoefficientList[Series[(1 +Sqrt[1-4*x])/2 -2*x, {x, 0, 50}], x] (* G. C. Greubel, Aug 04 2018 *)

{a(n) = if( n<2, (n==0) - 3*(n==1), - binomial(2*n - 2, n-1) / n)};

{a(n) = if( n<0, 0, polcoeff( (1 + sqrt(1 - 4*x + x * O(x^n))) / 2 - 2*x, n))};

G.f.: (1 + sqrt(1 - 4*x)) / 2 - 2*x.
G.f.: -2*x + 1 - x / (1 - x / (1 - x / ...)) (continued fraction).
a(n) = A115140(n) = A115141(n) for all n in Z unless n=1.
a(n) = -A000108(n-1) for all n>1.

A035330 5-fold convolution of A001700(n), n >= 0.

Original entry on oeis.org

1, 15, 140, 1045, 6835, 40963, 230720, 1240740, 6437890, 32468470, 160010280, 773624615, 3680728375, 17274086235, 80119845080, 367821324040, 1673528845710, 7554110698850, 33858536700040, 150802994850570

Offset: 0

Wolfdieter Lang

Fifth column of triangular array A035324.

Michael De Vlieger, Table of n, a(n) for n = 0..1650
José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Cf. A000108, A045894, A035324.

Array[(#^2 + 27 # + 122) Binomial[2 (# + 5), # + 5]/24 - 5 (# + 8)*2^(2 # + 5) &, 20, 0] (* Michael De Vlieger, Sep 04 2018 *)

a(n) = (n^2+27*n+122)*binomial(2*(n+5), n+5)/24 - 5*(n+8)*2^(2*n+5) = A035324(n+5, 5);

G.f.: c(x)^5/(1-4*x)^(5/2), where c(x) = g.f. for Catalan numbers A000108.

A046658 Triangle related to A001700 and A000302 (powers of 4).

Original entry on oeis.org

1, 3, 1, 10, 7, 1, 35, 38, 11, 1, 126, 187, 82, 15, 1, 462, 874, 515, 142, 19, 1, 1716, 3958, 2934, 1083, 218, 23, 1, 6435, 17548, 15694, 7266, 1955, 310, 27, 1, 24310, 76627, 80324, 44758, 15086, 3195, 418, 31, 1, 92378, 330818, 397923, 259356, 105102, 27866, 4867, 542, 35, 1

Offset: 1

Wolfdieter Lang

			Triangle begins as:
      1;
      3,     1;
     10,     7,     1;
     35,    38,    11,     1;
    126,   187,    82,    15,     1;
    462,   874,   515,   142,    19,    1;
   1716,  3958,  2934,  1083,   218,   23,   1;
   6435, 17548, 15694,  7266,  1955,  310,  27,  1;
  24310, 76627, 80324, 44758, 15086, 3195, 418, 31, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Column sequences for m=1..6: A001700, A000531, A029887, A045724, A045492, A045530.
Row sums: A046885.
Cf. A000302.

A046658:= func< n,k | Binomial(n,k)*(Binomial(n+1,2)*Catalan(n )/Catalan(k-1) -4^(n-k+1)*Binomial(k,2))/(n*(n-k+1)) >;
[A046658(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 28 2024

T[n_, k_]:= (1/2)*Binomial[n,k-1]*(Binomial[2*n,n]/Binomial[2*(k-1), k -1] - 4^(n-k+1)*(k-1)/n);
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 28 2024 *)

def A046658(n,k): return (1/2)*binomial(n,k-1)*(binomial(2*n, n)/binomial(2*(k-1), k-1) - 4^(n-k+1)*(k-1)/n)
flatten([[A046658(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Jul 28 2024

T(n, k) = (1/2)*binomial(n, k-1)*( binomial(2*n, n)/binomial(2*(k-1), k-1) - 4^(n-k+1)*(k-1)/n ), n >= k >= 1.

G.f. for column k: x*c(x)*((x/(1-4*x))^(k-1))/sqrt(1-4*x), where c(x) is the g.f. for Catalan numbers (A000108).

A158109 G.f.: A(x) = exp(Sum_{n>=1} C(2n-1,n)L(n)x^n/n) such that Sum_{n>=1} L(n)x^n/n = log(1+xA(x)) where L(n) = A158259(n) and C(2n-1,n) = A001700(n-1).

Original entry on oeis.org

1, 1, 2, 15, 479, 58981, 27087299, 46407723445, 298505825690021, 7255847001783419768, 670260315103084510835973, 236409648316126537191063108559, 319643614642063671478190549232176669

Offset: 0

Paul D. Hanna, Mar 28 2009

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 15*x^3 + 479*x^4 + 58981*x^5 +...
log(1+x*A(x)) = x + 1*x^2/2 + 4*x^3/3 + 53*x^4/4 + 2321*x^5/5 +...
log(A(x)) = x + 3*x^2/2 + 40*x^3/3 + 1855*x^4/4 + 292446*x^5/5 +...
log(A(x)) = x + 3*1*x^2/2 + 10*4*x^3/3 + 35*53*x^4/4 + 126*2321*x^5/5 +...

Cf. A158259, A158257 (variant), A001700.

{a(n)=local(A=1+x);if(n==0,1,for(i=1,n,A=exp(sum(m=1,n,binomial(2*m-1,m)*x^m*polcoeff(log(1+x*A+x*O(x^m)),m))+x*O(x^n)));polcoeff(A,n))}

A158259 L.g.f.: exp(Sum_{n>=1} a(n)x^n/n) = 1 + xexp(Sum_{n>=1} C(2n-1,n)a(n)x^n/n) where C(2n-1,n) = A001700(n-1).

Original entry on oeis.org

1, 1, 4, 53, 2321, 351010, 189198136, 371045084781, 2686134761118382, 72555484959298332681, 7372783651816395650943931, 2836907736669733620359204710274, 4155363917021399525198623243750199333

Offset: 0

Paul D. Hanna, Mar 28 2009

nonn

			L.g.f.: A(x) = x + 1*x^2/2 + 4*x^3/3 + 53*x^4/4 + 2321*x^5/5 +...
exp(A(x)) = 1 + x + 2*x^2 + 15*x^3 + 479*x^4 + 58981*x^5 +...
exp(A(x)) = 1 + x*G(x) where G(x) is the g.f. of A158109 such that:
log(G(x)) = x + 3*1*x^2/2 + 10*4*x^3/3 + 35*53*x^4/4 + 126*2321*x^5/5 +...

Cf. A158109, A158258 (variant), A001700.

{a(n)=local(A=x+x^2);if(n==0,1,for(i=1,n-1,A=log(1+x*exp(sum(m=1,n,binomial(2*m-1,m)*x^m*polcoeff(A+x*O(x^m),m) )+x*O(x^n))));n*polcoeff(A,n))}

L.g.f.: exp(Sum_{n>=1} a(n)*x^n/n) = 1 + x*G(x) where G(x) = g.f. of A158109.

exp(Sum_{n>=1} a(n)*x^n/n) = [1 + Sum_{n>=1} C(2n-1,n)*a(n)*x^n]/[1 + Sum_{n>=1} (C(2n-1,n)-1)*a(n)*x^n].

A187366 One half of a trisection of A001700: binomial(6n+5,3(n+1))/2, n>=0.

Original entry on oeis.org

5, 231, 12155, 676039, 38779380, 2268783825, 134564468610, 8061900920775, 486734856412028, 29566145391215356, 1804857108504066435, 110628135069209194801, 6804253717299758003900, 419727621552972772561830, 25956855321888352842417780

Offset: 0

Wolfdieter Lang, Mar 10 2011

For trisection of a sequence see a comment and a reference under A187357.

Cf. A187364 binomial(2(3n)+1,3n+1),

A187365 binomial(2(3n+1)+1,(3n+1)+1)/3.

a(n)= binomial(2*(3*n+2)+1,(3*n+2)+1)/2 = binomial(6*n+5,3*(n+1))/2 , n>=0.
O.g.f.: (cb(x^(1/3)) - 3 + sqrt(2)*P(x^(1/3))*sqrt(1/P(x^(1/3)) + 1 + 2*x^(1/3)))/(12*x),
with cb(x):=1/sqrt(1-4*x) (o.g.f. of A000984) and P(x):=P(-1/2,4*x)=1/sqrt(1+4*x+16*x^2) (o.g.f. of A116091, with P(x,z)the o.g.f. of the Legendre polynomials).

cp's OEIS Frontend

A045720 3-fold convolution of A001700(n), n >= 0.

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A045894 4-fold convolution of A001700(n), n >= 0.

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A100033 Bisection of A001700.

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A205945 Triangle read by rows related to A001700.

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A246432 Convolution inverse of A001700.

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A035330 5-fold convolution of A001700(n), n >= 0.

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A046658 Triangle related to A001700 and A000302 (powers of 4).

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A158109 G.f.: A(x) = exp(Sum_{n>=1} C(2n-1,n)*L(n)*x^n/n) such that Sum_{n>=1} L(n)*x^n/n = log(1+x*A(x)) where L(n) = A158259(n) and C(2n-1,n) = A001700(n-1).

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A158109 G.f.: A(x) = exp(Sum_{n>=1} C(2n-1,n)L(n)x^n/n) such that Sum_{n>=1} L(n)x^n/n = log(1+xA(x)) where L(n) = A158259(n) and C(2n-1,n) = A001700(n-1).

A158259 L.g.f.: exp(Sum_{n>=1} a(n)x^n/n) = 1 + xexp(Sum_{n>=1} C(2n-1,n)a(n)x^n/n) where C(2n-1,n) = A001700(n-1).