A064062 -id:A064062 - cp's OEIS frontend

A115137 Second diagonal of triangle A113647 (called Y(2,1)).

1, 1, 7, 41, 247, 1545, 9975, 66057, 446455, 3067913, 21372919, 150618121, 1071841271, 7691763721, 55600938999, 404488323081, 2959189475319, 21757613309961, 160691417776119, 1191577871450121, 8868160862158839

Offset: 0

Wolfdieter Lang, Jan 13 2006

			41=a(3)= A062992(3) - 2*A062992(2) = 67 - 2*13.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

CoefficientList[Series[(1-2*x)*(2*(1-Sqrt[1-8*x])/(4*x)-1)/(1+x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

a(n)= b(n) - 2*b(n-1) with b(n):=A062992(n)= A064062(n+1), n>=1. a(0):=1.
G.f.: (1-2*x)*(2*c(2*x)-1)/(1+x) with c(x) g.f. of A000108 (Catalan).
a(n)= A113647(n, n), n>=1.
Recurrence: (n-2)*(n+1)*a(n) = (7*n^2-19*n+14)*a(n-1) + 4*(n-1)*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(3*n+2)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

A155761 Riordan array (c(2x^2), xc(2*x^2)) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 4, 0, 1, 8, 0, 6, 0, 1, 0, 20, 0, 8, 0, 1, 40, 0, 36, 0, 10, 0, 1, 0, 112, 0, 56, 0, 12, 0, 1, 224, 0, 224, 0, 80, 0, 14, 0, 1, 0, 672, 0, 384, 0, 108, 0, 16, 0, 1, 1344, 0, 1440, 0, 600, 0, 140, 0, 18, 0, 1

Offset: 0

Paul Barry, Jan 26 2009

Inverse of Riordan array (1/(1+2*x^2), x/(1+2*x^2)).

			Triangle begins:
    1;
    0,   1;
    2,   0,   1;
    0,   4,   0,  1;
    8,   0,   6,  0,  1;
    0,  20,   0,  8,  0,  1;
   40,   0,  36,  0, 10,  0,  1;
    0, 112,   0, 56,  0, 12,  0, 1;
  224,   0, 224,  0, 80,  0, 14, 0, 1;
Production matrix begins as:
  0, 1;
  2, 0, 1;
  0, 2, 0, 1;
  0, 0, 2, 0, 1;
  0, 0, 0, 2, 0, 1;
  0, 0, 0, 0, 2, 0, 1;
  0, 0, 0, 0, 0, 2, 0, 1;
  0, 0, 0, 0, 0, 0, 2, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 2, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A064062, A126087 (row sums).

T[n_, k_]:= (1+(-1)^(n-k))*2^((n-k-2)/2)*((k+1)/(n+1))*Binomial[n+1, (n-k)/2];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 06 2021 *)

def A155761(n,k): return (1+(-1)^(n-k))*2^((n-k-2)/2)*((k+1)/(n+1))*binomial(n+1, (n-k)/2)
flatten([[A155761(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 06 2021

T(n,k) = (1+(-1)^(n-k)) * ((k+1)/(n+1)) * binomial(n+1, (n-k)/2) * 2^((n-k-2)/2).
Sum_{k=0..n} T(n, k) = A126087(n).
T(n,k) = 2^((n-k)/2) * A053121(n,k). - Philippe Deléham, Feb 11 2009
Sum_{k=0..n} T(2*n-k, k) = A064062(n+1). - G. C. Greubel, Jun 06 2021

A115125 A sequence related to Catalan numbers A000108.

Original entry on oeis.org

1, 2, 4, 16, 80, 448, 2688, 16896, 109824, 732160, 4978688, 34398208, 240787456, 1704034304, 12171673600, 87636049920, 635361361920, 4634400522240, 33985603829760, 250420238745600, 1853109766717440, 13765958267043840, 102618961627054080, 767411365211013120

Offset: 0

Wolfdieter Lang, Jan 13 2006

Essentially identical to A025225.
The convolution of this sequence with the sequence {(-1)^n} is A064062 (see also A062992).
The sequence A064062 appears in the Derrida et al. 1992 reference (see A064094) for alpha=2, beta=1 (or alpha=1, beta=2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.

Cf. A000108, A025225, A062992, A064062, A064094.

[1] cat [2^n*Binomial(2*n-2, n-1)/n: n in [1..30]]; // G. C. Greubel, May 03 2018

a:= n-> `if`(n=0, 1, 2^n*binomial(2*n-2, n-1)/n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2022

a[0] = 1; a[n_] := 2^n*CatalanNumber[n - 1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 09 2013 *)

a(n)=if(n==0,1,polcoeff((1-sqrt(1-8*x+x*O(x^n)))/2,n)); \\ Joerg Arndt, May 14 2013

a(n) = C(n-1)*2^n, n>=1, a(0):=1, with C(n):=A000108(n) (Catalan).
G.f.: 1 + (2*x)*c(2*x) with c(x):=(1-sqrt(1-4*x))/(2*x), the o.g.f. of Catalan numbers A000108.
a(n) = A025225(n), n>0. - R. J. Mathar, Aug 11 2008
G.f.: (3 - sqrt(1-8*x))/2 = 2 - U(0) where U(k)=1 - 2*x/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 29 2012
G.f.: 2 - 1/Q(0), where Q(k)= 1 + (8*k+2)*x/(k+1 - x*(2*k+2)*(8*k+6)/(2*x*(8*k+6) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013

A116872 Subtriangle of generalized Catalan triangle CM(1,2) = A116880.

Original entry on oeis.org

1, 3, 7, 13, 29, 41, 67, 147, 195, 247, 381, 829, 1069, 1277, 1545, 2307, 4995, 6339, 7379, 8451, 9975, 14589, 31485, 39549, 45373, 50733, 56829, 66057, 95235, 205059, 255747, 290691, 320707, 351187, 388099

Offset: 1

Wolfdieter Lang, Mar 24 2006

This triangle a(n,m) appears for the unnormalized one-point function T(n,n+m-1) in the totally asymmetric exclusion process (see A067323 for the references) for the (unphysical) values alpha=1, beta=2.

W. Lang: First 10 rows.

Row sums give A116879.

a(n,m)=A116880(n,m-1), n>=m>=1.

G.f. for column m>=1: (x^m)*(-(C2(m) + ((2^2)/x^(m-1))*(c(m-1,2*x)-1)/(2*x)) + 2*(C2(m-1) + (2/x^(m-1))*c(m-2,2*x))*c(2*x))/(1+x) where C2(n):=A064062(n), c(m,x):=sum(C(k)*x^k,k=0..m) with C(k):=A000108(k) (Catalan numbers) and c(x) is the g.f. of A000108.

A157491 A050165*A130595 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, 0, -1, 2, 0, 2, -6, 5, 0, -5, 20, -28, 14, 0, 14, -70, 135, -120, 42, 0, -42, 252, -616, 770, -495, 132, 0, 132, -924, 2730, -4368, 4004, -2002, 429, 0, -429, 3432, -11880, 23100, -27300, 19656, -8008, 1430

Offset: 0

Philippe Deléham, Mar 01 2009

Triangle, read by rows, given by [0,-1,-1,-1,-1,-1,-1,...] DELTA [1,1,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938. Triangle related to k-regular trees.

			Triangle begins:
  1;
  0,  1;
  0, -1,  2;
  0,  2, -6,   5;
  0, -5, 20, -28, 14;
  ...

Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A000108, A062991, A094385.

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000012(n), A000984(n), A089022(n), A035610(n), A130976(n), A130977(n), A130978(n), A130979(n), A130980(n), A131521(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively.

Sum_{k=0..n} T(n,k)*x^(n-k) = A064093, A064092, A064091, A064090, A064089, A064088, A064087, A064063, A064062, A000108, A000012, A064310, A064311, A064325, A064326, A064327, A064328, A064329, A064330, A064331, A064332, A064333 for x = -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12 respectively. [Philippe Deléham, Mar 03 2009]

A244469 a(0) = 0, thereafter, a(n) = 2^(2n-1)( binomial((3n-1)/2,n) - binomial(3n/2, n)/3 ).

Original entry on oeis.org

0, 1, 7, 58, 515, 4746, 44758, 428772, 4154403, 40599130, 399429602, 3950996556, 39255152846, 391466112324, 3916110379020, 39281346256008, 394942611929379, 3978982062756090, 40160256911157610, 405995113593507900, 4110284071450416090, 41666530928566504620, 422876855107176561780

Offset: 0

N. J. A. Sloane, Jun 28 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656 [math.CO], 2014. See f_4(n).

Cf. A000108, A007297, A064062, A091527, A244039.

[n eq 0 select 0 else Round(2^(2*n-1)*(Gamma((3*n+1)/2)/Gamma((n+1)/2) - Gamma((3*n+2)/2)/(3*Gamma((n+2)/2)))/Factorial(n)): n in [0..30]]; // G. C. Greubel, Apr 17 2019

f4:=n->-(2^(2*n-1)/3)*binomial(3*n/2,n) + 2^(2*n-1)*binomial((3*n-1)/2,n);
[seq(f4(n),n=1..40)]; # then prepend f4(0)=0.

Join[{0}, Table[-(2^(2 n - 1)/3) Binomial[3 n/2, n] + 2^(2 n - 1) Binomial[(3 n - 1)/2, n], {n, 1, 30}]] (* Vincenzo Librandi, Jun 29 2014 *)

{a(n) = if(n==0,0, 2^(2*n-1)*(binomial((3*n-1)/2, n) - binomial(3*n/2, n)/3) )}; \\ G. C. Greubel, Apr 17 2019

def a(n):
   if n==0: return 0
   else: return 2^(2*n-1)*(binomial((3*n-1)/2, n) - binomial(3*n/2, n)/3)
[a(n) for n in (0..30)] # G. C. Greubel, Apr 17 2019

G.f.: g'(x)/g(x)-1, g(x)=(2*sqrt(9*x+1)*sin(arcsin((54*x^2+27*x+2)/(2*(9*x+1)^(3/2)))/3))/3-1/3. - Vladimir Kruchinin, Apr 14 2019
From Peter Bala, Mar 05 2022: (Start)
a(n) = (1/n)*Sum_{k = 0..n} k*2^(n-k)*binomial(n+k-1,k)*binomial(2*n-k-1,n-k) for n >= 1.
a(n) = [x^n] G(x)^n = [x^n] 1/(1 - x*C(2*x))^n, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108 and G(x) is the g.f. of A064062.
n*(n-1)*(6*n-7)*a(n) = - 18*(n-1)*a(n-1) + 12*(3*n-5)*(6*n-1)*(3*n-4)*a(n-2) with a(1) = 1 a(2) = 7.
exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 4*x^2 + 23*x^3 + 156*x^4 + ... is the g.f. of A007297.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)

A117505 Triangle of coefficients for polynomials used for the column g.f.s of triangle A116880, called CM(1,2).

Original entry on oeis.org

1, 2, 1, 2, 4, 3, 2, 4, 16, 13, 2, 4, 16, 80, 67, 2, 4, 16, 80, 448, 381, 2, 4, 16, 80, 448, 2688, 2307, 2, 4, 16, 80, 448, 2688, 16896, 14589, 2, 4, 16, 80, 448, 2688, 16896, 109824, 95235, 2, 4, 16, 80, 448, 2688

Offset: 0

Wolfdieter Lang, Apr 13 2006

The g.f. G(m,x) for column m=1,2,... of triangle A116880=CM(1,2) is x*(-sum(a(m,k)*x^(k-1),k=1..m) + sum(a(m,k)*x^k,k=0..m)*2*c(2*x))/(1+x), with the o.g.f. c(x) of A000108 (Catalan numbers).

			m=3: G(3,x)= x*(-(4+16*x+13*x^2) +
(2+4*x+16*x^2+13*x^3)*2*c(2*x))/(1+x).

W. Lang: First 10 rows.

a(m,m)= A064062(m) =:C(2;m), m>=0 and a(m,k)=2*A052701(k) = C(k)*2^(k+1), for k=1,...,m-1 and C(k):=A000108(k) (Catalan).

A157328 Expansion of 1/(1-2x*c(4x)) with c(x) g.f. of Catalan numbers (A000108).

Original entry on oeis.org

1, 2, 12, 104, 1072, 12192, 147648, 1867392, 24380160, 326105600, 4445965312, 61555599360, 863154221056, 12233140576256, 174954419109888, 2521749245558784, 36595543723671552, 534249057803698176

Offset: 0

Philippe Deléham, Feb 27 2009

nonn

Hankel transform is A122067.

Cf. A000108, A000079, A000984, A039599, A064062, A110520, A122067, A151374.

a(n) = 2^n*A064062(n).
From Paul Barry, Sep 15 2009: (Start)
a(n) = Sum_{k, 0<=k<=n} A039599(n,k)*(-2)^k*4^(n-k).
Integral representation: a(n) = (1/(2*Pi))*Integral(x^n*sqrt(x(16-x))/(x(2+x)),x,0,16). (End)
a(n) = upper left term in M^n, M = an infinite square production matrix as follows:
  2, 2, 0, 0, 0, 0, ...
  4, 4, 4, 0, 0, 0, ...
  4, 4, 4, 4, 0, 0, ...
  4, 4, 4, 4, 4, 0, ...
  4, 4, 4, 4, 4, 4, ...
  ...
- Gary W. Adamson, Jul 13 2011
Conjecture: n*a(n) +2*(12-7n)*a(n-1) +16*(3-2n)*a(n-2) = 0. - R. J. Mathar, Dec 14 2011
a(n) = (12*(-1)^n*2^(n - 1)*sqrt(Pi)*n! + 16^n*gamma(n - 1/2)*hypergeometric2F1([1, -n], [3/2 - n], -1/8))/(4*sqrt(Pi)*n!). - Karol A. Penson, Feb 04 2025

Entries corrected by R. J. Mathar, Dec 14 2011

cp's OEIS Frontend

A115137 Second diagonal of triangle A113647 (called Y(2,1)).

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A155761 Riordan array (c(2*x^2), x*c(2*x^2)) where c(x) is the g.f. of A000108.

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A115125 A sequence related to Catalan numbers A000108.

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A116872 Subtriangle of generalized Catalan triangle CM(1,2) = A116880.

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A157491 A050165*A130595 as infinite lower triangular matrices.

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A244469 a(0) = 0, thereafter, a(n) = 2^(2*n-1)*( binomial((3*n-1)/2,n) - binomial(3*n/2, n)/3 ).

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A117505 Triangle of coefficients for polynomials used for the column g.f.s of triangle A116880, called CM(1,2).

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A157328 Expansion of 1/(1-2x*c(4x)) with c(x) g.f. of Catalan numbers (A000108).

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A155761 Riordan array (c(2x^2), xc(2*x^2)) where c(x) is the g.f. of A000108.

A244469 a(0) = 0, thereafter, a(n) = 2^(2n-1)( binomial((3n-1)/2,n) - binomial(3n/2, n)/3 ).