A097692 -id:A097692 - cp's OEIS frontend

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

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

A274708 A statistic on orbital systems over n sectors: the number of orbitals with k peaks.

Original entry on oeis.org

1, 1, 2, 4, 2, 4, 2, 12, 15, 3, 10, 8, 2, 38, 68, 30, 4, 26, 30, 12, 2, 121, 272, 183, 49, 5, 70, 104, 60, 16, 2, 384, 1026, 912, 372, 72, 6, 192, 350, 260, 100, 20, 2, 1214, 3727, 4095, 2220, 650, 99, 7, 534, 1152, 1050, 520, 150, 24, 2, 3822, 13200, 17178, 11600, 4510, 1032, 130, 8

Offset: 0

Peter Luschny, Jul 10 2016

The definition of an orbital system is given in A232500 (see also the illustration there). The number of orbitals over n sectors is counted by the swinging factorial A056040.
An orbital w has a 'peak' at i+1 when signum(w[i]) < signum(w[i+1]) and signum(w[i+1]) > signum(w[i+2]).
A097692 is a subtriangle.

			Triangle read by rows, n>=0. The length of row n is floor((n+1)/2) for n>=1.
[ n] [k=0,1,2,...]               [row sum]
[ 0] [  1]                           1
[ 1] [  1]                           1
[ 2] [  2]                           2
[ 3] [  4,    2]                     6
[ 4] [  4,    2]                     6
[ 5] [ 12,   15,   3]               30
[ 6] [ 10,    8,   2]               20
[ 7] [ 38,   68,  30,   4]         140
[ 8] [ 26,   30,  12,   2]          70
[ 9] [121,  272, 183,  49,  5]     630
[10] [ 70,  104,  60,  16,  2]     252
[11] [384, 1026, 912, 372, 72, 6] 2772
[12] [192,  350, 260, 100, 20, 2]  924
T(6, 2) = 2 because the two orbitals [-1, 1, -1, 1, -1, 1] and [1, -1, 1, -1, 1, -1] have 2 peaks.

Peter Luschny, Orbitals

Cf. A025565 (even col. 0), A056040 (row sum), A097692, A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

# uses[unit_orbitals from A274709]
# Brute force counting
def orbital_peaks(n):
    if n == 0: return [1]
    S = [0]*((n+1)//2)
    for u in unit_orbitals(n):
        L = [1 if sgn(u[i]) < sgn(u[i+1]) and sgn(u[i+1]) > sgn(u[i+2]) else 0 for i in (0..n-3)]
        S[sum(L)] += 1
    return S
for n in (0..12): print(orbital_peaks(n))

A171651 Triangle T, read by rows : T(n,k) = A007318(n,k)*A005773(n+1-k).

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 13, 15, 6, 1, 35, 52, 30, 8, 1, 96, 175, 130, 50, 10, 1, 267, 576, 525, 260, 75, 12, 1, 750, 1869, 2016, 1225, 455, 105, 14, 1, 2123, 6000, 7476, 5376, 2450, 728, 140, 16, 1, 6046, 19107, 27000, 22428, 12096, 4410, 1092, 180, 18, 1

Offset: 0

Philippe Deléham, Dec 14 2009

			Triangle begins:
   1;
   2,   1;
   5,   4,  1;
  13,  15,  6, 1;
  35,  52, 30, 8, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A097692.
Cf. A007318, A005773.
Cf. A168491, A099323, A001405, A005773, A001700, A026378, A005573, A122898.

b:= proc(u, d, t) option remember; `if`(u=0 and d=0, 1/2,
      expand(`if`(u=0, 0, b(u-1, d, 2)*`if`(t=3, x, 1))
      +`if`(d=0, 0, b(u, d-1, `if`(t=2, 3, 1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n+1$2, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Apr 29 2015
# second program:
A171651:= (n, k)-> binomial(n,k)*add((-1)^(n-k-j)*binomial(n-k,j)*binomial(2*j+1,j+1),j=0..n-k): seq(print(seq(A171651(n, k), k=0..n)), n=0..9);  # Mélika Tebni, Dec 16 2023

b[u_, d_, t_] := b[u, d, t] = If[u == 0 && d == 0, 1/2, Expand[If[u == 0, 0, b[u-1, d, 2]*If[t == 3, x, 1]] + If[d == 0, 0, b[u, d-1, If[t == 2, 3, 1]]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n+1, n+1, 1] ];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 21 2016, after Alois P. Heinz *)

Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -3, -2, -1, 0, 1, 2, 3, 4 respectively.

E.g.f. of column k: exp(x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 16 2023

Corrected by Philippe Deléham, Dec 18 2009

cp's OEIS Frontend

A097577 Duplicate of A097692.

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

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A274708 A statistic on orbital systems over n sectors: the number of orbitals with k peaks.

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Keywords

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Programs

Sage

A171651 Triangle T, read by rows : T(n,k) = A007318(n,k)*A005773(n+1-k).

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Programs

Maple

Mathematica

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Extensions