A085455 -id:A085455 - cp's OEIS frontend

A085362 a(0)=1; for n>0, a(n) = 25^(n-1) - (1/2)Sum_{i=1..n-1} a(i)*a(n-i).

1, 2, 8, 34, 150, 678, 3116, 14494, 68032, 321590, 1528776, 7301142, 35003238, 168359754, 812041860, 3926147730, 19022666310, 92338836390, 448968093320, 2186194166950, 10659569748370, 52037098259090, 254308709196660

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jun 25 2003

Number of bilateral Schroeder paths (i.e. lattice paths consisting of steps U=(1,1), D=(1,-1) and H=(2,0)) from (0,0) to (2n,0) and with no H-steps at even (zero, positive or negative) levels. Example: a(2)=8 because we have UDUD, UUDD, UHD, UDDU and their reflections in the x-axis. First differences of A026375. - Emeric Deutsch, Jan 28 2004
From G. C. Greubel, May 22 2020: (Start)
This sequence is part of a class of sequences, for m >= 0, with the properties:
a(n) = 2*m*(4*m+1)^(n-1) - (1/2)*Sum_{k=1..n-1} a(k)*a(n-k).
a(n) = Sum_{k=0..n} m^k * binomial(n-1, n-k) * binomial(2*k, k).
a(n) = (2*m) * Hypergeometric2F1(-n+1, 3/2; 2; -4*m), for n > 0.
n*a(n) = 2*((2*m+1)*n - (m+1))*a(n-1) - (4*m+1)*(n-2)*a(n-2).
(4*m + 1)^n = Sum_{k=0..n} Sum_{j=0..k} a(j)*a(k-j).
G.f.: sqrt( (1 - t)/(1 - (4*m+1)*t) ).
This sequence is the case of m=1. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.

Bisection of A026392.
Cf. A000351 (5^n), A026375, A085363, A085364, A085455.
Cf. A110170, A162478, A359758, A360132.
Cf. A063886, A360317, A360318, A360319, A360321, A360322.
Essentially the same as A026387.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt((1-x)/(1-5*x)) )); // G. C. Greubel, May 23 2020

a := n -> `if`(n=0,1,2*hypergeom([3/2, 1-n], [2], -4)):
seq(simplify(a(n)), n=0..22); # Peter Luschny, Jan 30 2017

CoefficientList[Series[Sqrt[(1-x)/(1-5x)], {x, 0, 25}], x]

my(x='x+O('x^66)); Vec(sqrt((1-x)/(1-5*x))) \\ Joerg Arndt, May 10 2013

def A085362_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt((1-x)/(1-5*x)) ).list()
A085362_list(30) # G. C. Greubel, May 23 2020

G.f.: sqrt((1-x)/(1-5*x)).
Sum_{i=0..n} (Sum_{j=0..i} a(j)*a(i-j)) = 5^n.
D-finite with recurrence: a(n) = (2*(3*n-2)*a(n-1)-5*(n-2)*a(n-2))/n; a(0)=1, a(1)=2. - Emeric Deutsch, Jan 28 2004
a(n) ~ 2*5^(n-1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 14 2012
G.f.: G(0), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)* (2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 22 2013
a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(n-1,n-k). - Vladimir Kruchinin, May 30 2016
a(n) = 2*hypergeom([3/2, 1-n], [2], -4) for n>0. - Peter Luschny, Jan 30 2017
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n+k) * a(k). - Seiichi Manyama, Mar 28 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A025565 a(n) = T(n,n-1), where T is array defined in A025564.

Original entry on oeis.org

1, 2, 4, 10, 26, 70, 192, 534, 1500, 4246, 12092, 34606, 99442, 286730, 829168, 2403834, 6984234, 20331558, 59287740, 173149662, 506376222, 1482730098, 4346486256, 12754363650, 37461564504, 110125172682, 323990062452, 953883382354

Offset: 1

Clark Kimberling

nonn

a(n+1) is the number of UDU-free paths of n upsteps (U) and n downsteps (D), n>=0. - David Callan, Aug 19 2004
Hankel transform is A120580. - Paul Barry, Mar 26 2010
If interpreted with offset 0, the inverse binomial transform of A006134 - Gary W. Adamson, Nov 10 2007
Also the number of different integer sets { k_1, k_2, ..., k_(i+1) } with Sum_{j=1..i+1} k_j = i and k_j >= 0, see the "central binomial coefficients" (A000984), without all sets in which any two successive k_j and k_(j+1) are zero. See the partition problem eq. 3.12 on p. 19 in my dissertation below. - Eva Kalinowski, Oct 18 2012

			G.f. = x + 2*x^2 + 4*x^3 + 10*x^4 + 26*x^5 + 70*x^6 + 192*x^7 + 534*x^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Kassie Archer and Christina Graves, A new statistic on Dyck paths for counting 3-dimensional Catalan words, arXiv:2205.09686 [math.CO], 2022.
Andrei Asinowski and Cyril Banderier, On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16.
D. Baccherini, D. Merlini, and R. Sprugnoli, Binary words excluding a pattern and proper Riordan arrays, Discrete Math. 307 (2007), no. 9-10, 1021--1037. MR2292531 (2008a:05003). See page 1034. - _N. J. A. Sloane_, Mar 25 2014
J. L. Jacobsen and J. Salas, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions, J. Stat. Phys. 122 (2006) 705-760; arXiv:cond-mat/0407444, 2004-2006. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014
Eva Kalinowski, Mott-Hubbard-Isolator in hoher Dimension, Dissertation, Marburg: Fachbereich Physik der Philipps-Universität, 2002.

Cf. A005773, A025564.
First column of A097692.
Partial sums of A105696.

a025565 n = a025565_list !! (n-1)
a025565_list = 1 : f a001006_list [1] where
   f (x:xs) ys = y : f xs (y : ys) where
     y = x + sum (zipWith (*) a001006_list ys)
-- Reinhard Zumkeller, Mar 30 2012

seq( add(binomial(i-2, k)*(binomial(i-k, k+1)), k=0..floor(i/2)), i=1..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
# Alternatively:
a := n -> `if`(n=1,1,2*(-1)^n*hypergeom([3/2, 2-n], [2], 4)):
seq(simplify(a(n)),n=1..28); # Peter Luschny, Jan 30 2017

T[, 0] = 1; T[1, 1] = 2; T[n, k_] /; 0 <= k <= 2n := T[n, k] = T[n-1, k-2] + T[n-1, k-1] + T[n-1, k]; T[, ] = 0;
a[n_] := T[n-1, n-1];
Array[a, 30] (* Jean-François Alcover, Jul 30 2018 *)

def A():
    a, b, n  = 1, 1, 1
    yield a
    while True:
        yield a + b
        n += 1
        a, b = b, ((3*(n-1))*a+(2*n-1)*b)//n
A025565 = A()
print([next(A025565) for  in range(28)]) # _Peter Luschny, Jan 30 2017

G.f.: x*sqrt((1+x)/(1-3*x)).
a(n) = 2*A005773(n-1) for n > 1.
a(n) = |A085455(n-1)| = A025577(n) - A025577(n-1) = A002426(n) + A002426(n-1).
Sum_{i=0..n} Sum_{j=0..i} (-1)^(n-i)*a(j)*a(i-j) = 3^n. - Mario Catalani (mario.catalani(AT)unito.it), Jul 02 2003
a(1) = 1, a(n) = M(n-1) + Sum_{k=1..n-1} M(k-1)*a(n-k) with M=A001006, the Motzkin Numbers. - Reinhard Zumkeller, Mar 30 2012
D-finite with recurrence: (-n+1)*a(n) +2*(n-1)*a(n-1) +3*(n-3)*a(n-2)=0. - R. J. Mathar, Dec 02 2012
G.f.: G(0), where G(k) = 1 + 4*x*(4*k+1)/( (1+x)*(4*k+2) - x*(1+x)*(4*k+2)*(4*k+3)/(x*(4*k+3) + (1+x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013
a(n) = n*hypergeom([2-n, 1/2-n/2, 1-n/2], [2, -n], 4). - Peter Luschny, Jul 12 2016
a(n) = (-1)^n*2*hypergeom([3/2, 2-n], [2], 4) for n > 1. - Peter Luschny, Jan 30 2017

Incorrect statement related to A000984 (see A002426) and duplicate of the g.f. removed by R. J. Mathar, Oct 16 2009

Edited by R. J. Mathar, Aug 09 2010

A085456 Sum_{i=0..n} Sum_{j=0..i} a(j) * a(i-j) = (-7)^n.

Original entry on oeis.org

1, -4, 20, -116, 708, -4452, 28532, -185300, 1215268, -8030404, 53381844, -356577588, 2391430020, -16092704292, 108605848116, -734783381652, 4982063186916, -33844621986180, 230306722637204, -1569571734301172, 10711405584991300, -73188920628617956, 500643475619050740

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jul 01 2003

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A085455, A085457, A085458.

CoefficientList[Series[Sqrt[(1-x)/(1+7 x)], {x, 0, 30}], x]

a(n) = sum(k=0, n, (-2)^k*binomial(n-1, n-k)*binomial(2*k, k)); \\ Seiichi Manyama, Feb 03 2023

G.f.: A(x)=Sqrt((1-x)/(1+7x)).
From Seiichi Manyama, Feb 03 2023: (Start)
a(n) = Sum_{k=0..n} (-2)^k * binomial(n-1,n-k) * binomial(2*k,k).
n*a(n) = -2*(3*n-1)*a(n-1) + 7*(n-2)*a(n-2). (End)

A098494 Triangle read by rows: coefficients of polynomials E(n,x) related to partitions with parts occurring at most thrice.

Original entry on oeis.org

1, 1, -1, 1, -5, 4, 1, -12, 35, -30, 1, -22, 143, -362, 312, 1, -35, 405, -2065, 4814, -4200, 1, -51, 925, -7965, 35434, -78744, 69120, 1, -70, 1834, -24010, 173929, -709240, 1525236, -1345680, 1, -92, 3290, -61040, 655529, -4235588, 16255420, -34148400, 30240000

Offset: 0

Ralf Stephan, Sep 12 2004

The polynomials generate (-1)^k*n! times the diagonals of A098493.

			E(0,x) = 1
E(1,x) = x - 1
E(2,x) = x^2 - 5*x + 4
E(3,x) = x^3 - 12*x^2 + 35*x - 30
E(4,x) = x^4 - 22*x^3 + 143*x^2 - 362*x + 312
E(5,x) = x^5 - 35*x^4 + 405*x^3 - 2065*x^2 + 4814*x - 4200

Seiichi Manyama, Rows n = 0..139, flattened
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.

Columns include -A000326.
Constant terms E(n, 0) = -E(n-1, -1) = n!/2*A085455 = (-1)^n*n!*A005773.
Row sums are E(n, 1) = (-1)^n*n!*A005774(n-2). [corrected by Seiichi Manyama, Feb 04 2023]

E(n+1,x+1) - E(n+1,x) = (n+1) * ( E(n,x) - n * E(n-1,x-1) ).

A085457 Sum_{i=0..n} Sum_{j=0..i} a(j) * a(i-j) = (-11)^n.

Original entry on oeis.org

1, -6, 48, -438, 4206, -41586, 418980, -4277130, 44089320, -457891170, 4783741248, -50218890738, 529300238574, -5597562756894, 59366869030668, -631200956847558, 6725615443683870, -71800018913609970, 767806202604650880, -8223081959016322530, 88187484604146004506

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jul 01 2003

Seiichi Manyama, Table of n, a(n) for n = 0..960

Cf. A085455, A085456.

CoefficientList[Series[Sqrt[(1-x)/(1+11 x)], {x, 0, 20}], x]

a(n) = sum(k=0, n, (-3)^k*binomial(n-1, n-k)*binomial(2*k, k)); \\ Seiichi Manyama, Feb 03 2023

G.f.: A(x)=Sqrt((1-x)/(1+11x)).
From Seiichi Manyama, Feb 03 2023: (Start)
a(n) = Sum_{k=0..n} (-3)^k * binomial(n-1,n-k) * binomial(2*k,k).
n*a(n) = -2*(5*n-2)*a(n-1) + 11*(n-2)*a(n-2). (End)

cp's OEIS Frontend

A085362 a(0)=1; for n>0, a(n) = 2*5^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).

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A025565 a(n) = T(n,n-1), where T is array defined in A025564.

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A085456 Sum_{i=0..n} Sum_{j=0..i} a(j) * a(i-j) = (-7)^n.

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A098494 Triangle read by rows: coefficients of polynomials E(n,x) related to partitions with parts occurring at most thrice.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A085457 Sum_{i=0..n} Sum_{j=0..i} a(j) * a(i-j) = (-11)^n.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A085362 a(0)=1; for n>0, a(n) = 25^(n-1) - (1/2)Sum_{i=1..n-1} a(i)*a(n-i).