Sergey Kirgizov - cp's OEIS frontend

A343773 Excess of the number of even Motzkin n-paths (A107587) over the odd ones (A343386).

1, 1, 0, -2, -3, 1, 11, 15, -13, -77, -86, 144, 595, 495, -1520, -4810, -2485, 15675, 39560, 6290, -159105, -324805, 87075, 1592843, 2616757, -2136539, -15726114, -20247800, 32296693, 152909577, 145139491, -417959049, -1460704685, -885536173, 4997618808, 13658704994

Offset: 0

Gennady Eremin, Sergey Kirgizov, Apr 29 2021

All terms a(n), n >= 0, are contained in both A100223 and A214649, as well as in A007440 (if the signs of integers are not taken into account). So these sequences form a cluster, the base of which is the current sequence.
The Motzkin number A001006(n) is split into two parts A107587(n) and A343386(n) (see A343386). The value a(n), the difference between A107587(n) and A343386(n), can be called the "shadow" of A001006(n). This is clearly seen if we compare the g.f. for the Motzkin numbers M(x) = 1 + x*M(x) + x^2*M(x)^2 and the current g.f. A(x) = 1 + x*A(x) - x^2*A(x)^2.
Binomial transform of 1, 0, -1, 0, 2, 0, -5, 0, 14, 0, -42, 0, ... (see A000108). - Gennady Eremin, Jul 14 2021
The Toeplitz Number Wall of a sequence a(0), a(1), ... is defined to be the 2D array A(m, n) of determinants of Toeplitz matrices a(i-j+n){i,j=1..m} where a(i) = 0 if i<0. Thus A(0, n) = 1, A(1, n) = a(n), A(2, n) = a(n)^2 - a(n+1)*a(n-1), A(m, n)^2 = A(m-1, n)*A(m+1, n) + A(m, n-1)*A(m, n+1) in general. For this sequence A(n, n) = F(n+1), A(n+1, n) = 1 and it is the unique sequence with this property. - _Michael Somos, Oct 27 2024

			G.f. = 1 + x - 2*x^3 - 3*x^4 + x^5 + 11*x^6 + 15*x^7 - 13*x^8 - 77*x^9 - 86*x^10 + 144*x^11 + ...

Gennady Eremin, Table of n, a(n) for n = 0..800
Gennady Eremin, Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers, arXiv:2108.10676 [math.CO], 2021.
W. F. Lunnon, The number-wall algorithm: an LFSR cookbook, Journal of Integer Sequences 4 (2001), no. 1, 01.1.1.

Cf. A107587, A343386, A100223, A214649, A007440, A001006, A000108, A039834.

Cf. A085143.

With[{$MaxExtraPrecision = 1000}, CoefficientList[Series[(-1 + x + Sqrt[1 - 2 x + 5 x^2])/(2 x^2), {x, 0, 36}], x] ] (* Michael De Vlieger, May 01 2021 *)
a[n_] := Hypergeometric2F1[(1 - n)/2, -n/2, 2, -4];
Table[a[n], {n, 0, 35}] (* Peter Luschny, May 30 2021 *)
a[ n_] := If[n<0, 0, SeriesCoefficient[Nest[1 + x*# - (x*#)^2&, 1 + O[x], n], {x, 0, n}]]; (* Michael Somos, Oct 27 2024 *)
a[ n_] := SeriesCoefficient[2/(1 - x + (1 - 2*x + 5*x^2)^(1/2)), {x, 0, n}]; (* Michael Somos, Oct 27 2024 *)

{a(n) = my(y = 1 + O(x)); for(i = 1, n, y = 1 + x*y - x^2*y^2); polcoeff(y, n)}; /* Michael Somos, Oct 27 2024 */

{a(n) = polcoeff( 2/(1 - x + (1 - 2*x + 5*x^2 + x*O(x^n))^(1/2)), n)}; /* Michael Somos, Oct 27 2024 */

A343773 = [1, 1]
for n in range(2, 801):
    A343773.append(((2*n+1)*A343773[-1]
      - 5*(n-1)*A343773[-2]) // (n+2))

a(n) = A107587(n) - A343386(n), n>=0.
a(n) = A100223(n+2) = A214649(n+1), n>=0.
a(n) = (-1)^n * A007440(n+1), n>=0.
D-finite with recurrence a(n) = ((2*n+1)*a(n-1) - 5*(n-1)*a(n-2))/(n+2), n>1.
G.f.: (-1 + x + sqrt(1 - 2*x + 5*x^2))/(2*x^2).
G.f. A(x) satisfies A(x) = 1 + x*A(x) - x^2*A(x)^2.
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n, 2*k) * A000108(k).
a(n) = 2*A107587(n) - A001006(n) = A001006(n) - 2*A343386(n).
Limit_{n->oo} a(n)/A001006(n) = 0.
a(n) = hypergeom([(1 - n)/2, -n/2], [2], -4). - Peter Luschny, May 30 2021
G.f. A(x) with offset 1 is the reversion of g.f. for signed Fibonacci numbers 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, ... (see A039834 starting at offset 1). - Gennady Eremin, Jul 15 2021

A343386 Number of odd Motzkin n-paths, i.e., Motzkin n-paths with an odd number of up steps.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 20, 56, 168, 456, 1137, 2827, 7458, 20670, 57577, 157691, 427976, 1170552, 3248411, 9096497, 25505562, 71436182, 200338074, 564083786, 1595055520, 4522769520, 12842772295, 36514010301, 103995490758, 296794937626, 848620165860, 2430089817720

Offset: 0

Gennady Eremin, Sergey Kirgizov, Apr 13 2021

a(n) is the number of Motzkin n-paths with an odd number of U-steps (see A001006). For example, there are 9 Motzkin 4-paths, of which six have one U-step each, namely: 00UD, 0U0D, 0UD0, U00D, U0D0, and UD00. So a(4) = 6.

Number of Motzkin n-paths that, after removing the horizontal steps, are converted to Dyck (2m)-paths, where 2m <= n and m is odd (see A024492).

			G.f. = x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 56*x^7 + 168*x^8 + ...

Gennady Eremin, Table of n, a(n) for n = 0..800
Gennady Eremin, Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers, arXiv:2108.10676 [math.CO], 2021.

Cf. A000108, A001006, A107587, A024492, A100223.

a[n_] := ((n - 1) n HypergeometricPFQ[{1/2 - n/4, 3/4 - n/4, 1 - n/4, 5/4 - n/4}, {3/2, 3/2, 2}, 16])/2;
Table[a[n], {n, 0, 31}] (* Peter Luschny, Sep 24 2021 *)

M = [4, 9]; E = [1, 1, 1, 1, 3];
A343386 = [0, 0, 1, 3, 6]
for n in range(5, 801):
    M.append(((2*n+1)*M[1]+(3*n-3)*M[0])//(n+2))
    E.append(((5*n**2+n-3)*E[4] - (10*n**2-16*n+3)*E[3]
      + (10*n**2-34*n+27)*E[2] + (11*n-5)*(n-3)*E[1]
      - 15*(n-3)*(n-4)*E[0]) // (n*n+2*n))
    A343386.append(M[-1] - E[-1])
    M.pop(0); E.pop(0)

a(n) = Sum_{k=0..n} binomial(n, 4*k+2) * A000108(2*k+1).
a(n) = A001006(n) - A107587(n).
G.f.: A(x) = (2 - 2*x - sqrt(1-2*x-3*x^2) - sqrt(1-2*x+5*x^2))/(4*x^2).
G.f. A(x) satisfies A(x) = x*A(x) + x^2*A(x)^2 + x^2*B(x)^2 where B(x) is the g.f. of A107587.
a(n) = A107587(n) - A100223(n+2). - R. J. Mathar, Apr 16 2021
D-finite with recurrence: n*(n+2)*a(n) + (-5*n^2-n+3)*a(n-1) + (10*n^2-16*n+3)*a(n-2) + (-10*n^2+34*n-27)*a(n-3) - (11*n-5)*(n-3)*a(n-4)  + 15*(n-3)*(n-4)*a(n-5) = 0, n >= 5. - R. J. Mathar, Apr 17 2021
D-finite with recurrence: n*(n-2)*(n+2)*a(n) - (2*n-1)*(2*n^2-2*n-3)*a(n-1) + 3*(n-1)*(2*n^2-4*n+1)*a(n-2) - 2*(n-1)*(n-2)*(2*n-3)*a(n-3) - 15*(n-1)*(n-2)*(n-3)*a(n-4) = 0, n >= 4. - R. J. Mathar, Apr 17 2021
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * A000108(k) * (k mod 2). - Gennady Eremin, May 03 2021 [after Paul Barry (A107587)]
a(n) = ((n-1)*n*hypergeom([1/2-n/4, 3/4-n/4, 1-n/4, 5/4-n/4], [3/2, 3/2, 2], 16))/2. - Peter Luschny, Sep 24 2021
a(n) ~ 3^(n + 3/2) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 27 2024

A340569 Total number of consecutive triples matching the pattern 123 in all faro permutations of length n.

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 24, 53, 116, 246, 520, 1082, 2248, 4628, 9520, 19469, 39796, 81022, 164904, 334670, 679064, 1374924, 2783440, 5625666, 11368904, 22945820, 46307664, 93358228, 188202256, 379078952, 763506784, 1536708413, 3092806516, 6220970702, 12512656744, 25154958278

Offset: 0

Sergey Kirgizov, Jan 12 2021

nonn

Faro permutations are permutations avoiding the three consecutive patterns 231, 321 and 312. They are obtained by a perfect faro shuffle of two nondecreasing words of lengths differing by at most one.

			For n = 4, there are 6 faro permutations: 1234, 1243, 1324, 2134, 2143, 3142. They contain in total 4 consecutive patterns 123.

Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020. See Table 1.

A001405 counts faro permutations of length n.

Cf. A107373, A340567, A340568.

Table[SeriesCoefficient[x*(1+2*x)*(1-Sqrt[1-4*x^2])/((1-2*x) * (1+Sqrt[1-4*x^2])),{x,0,n}],{n,0,35}] (* Stefano Spezia, Jan 12 2021 *)

G.f.: x * (1+2*x) * (1-sqrt(1-4*x^2)) / ((1-2*x) * (1+sqrt(1-4*x^2))).

A340568 Total number of consecutive triples matching the pattern 132 in all faro permutations of length n.

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 28, 61, 152, 318, 748, 1538, 3496, 7124, 15832, 32093, 70192, 141814, 306508, 617878, 1323272, 2663340, 5662600, 11383986, 24061264, 48330540, 101653368, 204049636, 427414672, 857503784, 1789891888, 3589478621, 7469802592, 14974962854, 31081371148

Offset: 0

Sergey Kirgizov, Jan 11 2021

nonn

Faro permutations are permutations avoiding the three consecutive patterns 231, 321 and 312. They are obtained by a perfect faro shuffle of two nondecreasing words of lengths differing by at most one. Also the popularity of consecutive pattern 213.

			For n = 4, there are 6 faro permutations: 1234, 1243, 1324, 2134, 2143, 3142. They contain in total 4 consecutive patterns 132 and also 4 consecutive patterns 213.

Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020. See Table 1.

A001405 counts faro permutations of length n.

Cf. A107373, A340567, A340569.

seq(n)={my(t=sqrt(1-4*x^2+O(x^n))); Vec(x*(-1+4*x^2+2*x+(1-2*x)*t) / ((1-2*x)*(1+t)*t), -(1+n))} \\ Andrew Howroyd, Jan 11 2021

G.f.: x*(-1+4*x^2+2*x+(1-2*x)*sqrt(1-4*x^2)) / ((1-2*x)*(1+sqrt(1-4*x^2))*sqrt(1-4*x^2)).

A340567 Total number of ascents in all faro permutations of length n.

Original entry on oeis.org

0, 0, 1, 4, 11, 26, 62, 134, 303, 634, 1394, 2872, 6206, 12676, 27068, 54994, 116423, 235706, 495722, 1001168, 2094714, 4223020, 8798756, 17715084, 36782246, 73980516, 153161332, 307808464, 635675228, 1276699336, 2630957432, 5281304554, 10863149303, 21797013946

Offset: 0

Sergey Kirgizov, Jan 11 2021

nonn

Faro permutations are permutations avoiding the three consecutive patterns 231, 321 and 312. They are obtained by a perfect faro shuffle of two nondecreasing words of lengths differing by at most one.

			For n = 3 there are 3 faro permutations, namely 123, 213, 132. They contain 4 ascents (12, 23, 13 and 13) in total.

Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020. See Table 1.

A001405 counts faro permutations of length n.

Cf. A107373 (descents), A340568, A340569.

seq(n)={my(t=sqrt(1-4*x^2+O(x^n))); Vec(2*x*(4*x^2 + x + t - 1)/((1 - 2*x)*t*(t + 1)), -(1+n))} \\ Andrew Howroyd, Jan 11 2021

G.f.: 2*x*(4*x^2 + x + sqrt(1 - 4*x^2) - 1)/((1 - 2*x)*sqrt(1 - 4*x^2)*(sqrt(1 - 4*x^2) + 1)).

A335203 a(n) is the packing chromatic number of n-hypercube graph.

Original entry on oeis.org

2, 3, 5, 7, 15, 25, 49, 95

Offset: 1

Sergey Kirgizov, May 26 2020

A packing coloring of a graph associates an integer color to every graph vertex in such a way that for any k > 0 if two different vertices share the same color k, they must be at distance at least k+1. a(n) is the minimal number of colors (1,2,3,...) needed to perform a packing coloration of an n-dimensional hypercube graph. Only the first terms, up to n = 8, are known. In contrast, the ordinary chromatic number of any hypercube is always equals 2, since any hypercube is a bipartite graph.

There are no known exact formulas or recurrence relations. Some asymptotic results and bounds are given in the Formula section.

			Hypercube of dimension 1 needs 2 colors:
  1 --- 2
Hypercube of dimension 2 needs 3 colors:
  1 --- 2
  |     |
  |     |
  3 --- 1
Hypercube of dimension 3 needs 5 colors:
  1 ----------- 2
  | \         / |
  |  \       /  |
  |   4 --- 1   |
  |   |     |   |
  |   |     |   |
  |   2 --- 5   |
  |  /       \  |
  | /         \ |
  3 ----------- 1

B. Brešar, J. Ferme, S. Klavžar, and D. F. Rall, Survey on packing colorings, Discussiones Mathematicae Graph Theory, to appear (2020).
W. Goddard, S. M. Hedetniemi, S. T. Hedetniemi, J. M. Harris, and R. F. Rall, Broadcast chromatic numbers of graphs, Ars Combinatoria, 86 (2008) 33-49.
P. Torres and M. Valencia-Pabon, The packing chromatic number of hypercubes, Discrete Applied Mathematics, 190 (2015), 127-140.

a(n) ~ (1/2 - O(1/k)) * 2^k  (Proposition 7.3 from Goddard et al.).
a(n) >= 2*a(n-1) - (n-1) (Corollary 1 from Torres and Valencia-Pabon).
a(n) <= 3 + 2^n * (1/2 - 1/(2^ceiling(log_2(n)))) - 2 * floor((n-4)/2) (Thm. 1 from Torres and Valencia-Pabon).

A302483 Number of FF-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths.

Original entry on oeis.org

1, 1, 2, 2, 5, 9, 17, 32, 59, 107, 192, 342, 606, 1070, 1885, 3316, 5828, 10237, 17975, 31555, 55387, 97210, 170605, 299405, 525434, 922088, 1618168, 2839704, 4983351, 8745190, 15346758, 26931703, 47261865, 82938813, 145547493, 255418068, 448227487, 786584431

Offset: 0

Sergey Kirgizov, Apr 08 2018

nonn

Number of FF-equivalence classes of Łukasiewicz paths. A Łukasiewicz path of length n is a lattice path from (0,0) to (n,0) using up steps U_{k} = (1,k) for any positive integer k, flat steps F = (1,0) and down steps D = (1,-1). Łukasiewicz paths are alpha-equivalent whenever the positions of occurrences of pattern alpha are identical on these paths.

			There are 14 Łukasiewicz of length 4 divided in the 5 following FF-equivalence classes: {FFFF}, {FFU_{1}D}, {U_{1}DFF}, {U_{1}FFD}, {FU_{1}DF, FU_{1}FD, FU_{2}DD, U_{1}DU_{1}D, U_{1}FDF, U_{1}U_{1}DD, U_{2}DDF, U_{2}DFD, U_{2}FDD, U_{3}DDD}.

Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.

Cf. A001405, A191385, A000045, A005251, A000325, A011782, A001006, A023431, A292460, A004148 enumerates the numbers of P-equivalence classes of Łukasiewicz paths for other values of P.

CoefficientList[Series[(1 - 3 x + 4 x^2 - 5 x^3 + 7 x^4 - 7 x^5 + 6 x^6 - 3 x^7 + x^8)/((1 - 2 x + x^2 - x^3) (1 - x)^2), {x, 0, 32}], x] (* Michael De Vlieger, Apr 12 2018 *)

x='x+O('x^99); Vec((1-3*x+4*x^2-5*x^3+7*x^4-7*x^5+6*x^6-3*x^7+x^8)/((1-2*x+x^2-x^3)*(1-x)^2)) \\ Altug Alkan, Apr 12 2018

G.f.: (1 - 3*x + 4*x^2 - 5*x^3 + 7*x^4 - 7*x^5 + 6*x^6 - 3*x^7 + x^8) / ((1-2*x+x^2-x^3) * (1-x)^2).

A278679 Popularity of left children in treeshelves avoiding pattern T213.

Original entry on oeis.org

1, 5, 24, 128, 770, 5190, 38864, 320704, 2894544, 28382800, 300575968, 3419882304, 41612735632, 539295974000, 7417120846080, 107904105986048, 1655634186628352, 26721851169634560, 452587550053179392, 8026445538106839040, 148751109541600495104

Offset: 2

Sergey Kirgizov, Nov 26 2016

nonn

Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. Classical Françon's bijection maps bijectively treeshelves into permutations. Pattern T213 illustrated below corresponds to a treeshelf constructed from permutation 213. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n.

			Treeshelves of size 3:
      1  1          1    1       1        1
     /    \        /      \     / \      / \
    2      2      /        \   2   \    /   2
   /        \    2          2       3  3
  3          3    \        /
                   3      3
Pattern T213:
    1
   / \
  2   \
       3
Treeshelves of size 3 that avoid pattern T213:
      1  1          1    1        1
     /    \        /      \      / \
    2      2      /        \    /   2
   /        \    2          2  3
  3          3    \        /
                   3      3
Popularity of left children is 5.

Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.
J. Françon, Arbres binaires de recherche : propriétés combinatoires et applications, Revue française d'automatique informatique recherche opérationnelle, Informatique théorique, 10 no. 3 (1976), p. 35-50.

Cf. A000110, A000111, A000142, A001286, A008292, A131178, A278677, A278678.

terms = 21;
egf = (E^(Sqrt[2] z)(4z - 4) - (Sqrt[2] - 2) E^(2 Sqrt[2] z) + Sqrt[2] + 2)/((Sqrt[2] - 2) E^(Sqrt[2] z) + 2 + Sqrt[2])^2;
CoefficientList[egf + O[z]^(terms + 2), z]*Range[0, terms + 1]! // Round // Drop[#, 2]& (* Jean-François Alcover, Jan 26 2019 *)

## by Taylor expansion
from sympy import *
from sympy.abc import z
h = (exp(sqrt(2)*z) * (4*z-4) - (sqrt(2)-2)*exp(2*sqrt(2)*z) + sqrt(2) + 2) / ((sqrt(2)-2)*exp(sqrt(2)*z) + 2 + sqrt(2))**2
NUMBER_OF_COEFFS = 20
coeffs = Poly(series(h,n = NUMBER_OF_COEFFS)).coeffs()
coeffs.reverse()
## and remove first coefficient 1 that corresponds to O(n**k)
coeffs.pop(0)
print([coeffs[n]*factorial(n+2) for n in range(len(coeffs))])

E.g.f.: (e^(sqrt(2)*z) * (4*z-4) - (sqrt(2)-2)*e^(2*sqrt(2)*z) + sqrt(2) + 2) / ((sqrt(2)-2)*e^(sqrt(2)*z) + 2 + sqrt(2))^2.

Asymptotic: n * (sqrt(2) / log(2*sqrt(2)+3) )^(n+1).

More terms from Alois P. Heinz, Oct 27 2017

A278678 Popularity of left children in treeshelves avoiding pattern T321.

Original entry on oeis.org

1, 4, 19, 94, 519, 3144, 20903, 151418, 1188947, 10064924, 91426347, 887296422, 9164847535, 100398851344, 1162831155151, 14198949045106, 182317628906283, 2455925711626404, 34632584722468115, 510251350142181470, 7840215226100517191, 125427339735162102104

Offset: 2

Sergey Kirgizov, Nov 26 2016

nonn

Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. Classical Françon's bijection maps bijectively treeshelves into permutations. Pattern T321 illustrated below corresponds to a treeshelf constructed from permutation 321. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n.

			Treeshelves of size 3:
      1  1          1    1       1        1
     /    \        /      \     / \      / \
    2      2      /        \   2   \    /   2
   /        \    2          2       3  3
  3          3    \        /
                   3      3
Pattern T321:
      1
     /
    2
   /
  3
Treeshelves of size 3 that avoid pattern T321:
  1          1    1       1        1
   \        /      \     / \      / \
    2      /        \   2   \    /   2
     \    2          2       3  3
      3    \        /
            3      3
Popularity of left children is 4.

Alois P. Heinz, Table of n, a(n) for n = 2..483
Jean-Luc Baril, Sergey Kirgizov, Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.
J. Françon, Arbres binaires de recherche : propriétés combinatoires et applications, Revue française d'automatique informatique recherche opérationnelle, Informatique théorique, 10 no. 3 (1976), pp. 35-50

Cf. A000110, A000111, A000142, A001286, A008292, A131178, A278677, A278679.

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> (n+1)*b(n+1, 0)-b(n+2, 0):
seq(a(n), n=2..25);  # Alois P. Heinz, Oct 27 2017

b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]];
a[n_] := (n+1)*b[n+1, 0] - b[n+2, 0];
Table[a[n], {n, 2, 25}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

# by Taylor expansion
from sympy import *
from sympy.abc import z
h = (-sin(z) + 1 + (z-1)*cos(z))/ (1-sin(z))**2
NUMBER_OF_COEFFS = 20
coeffs = Poly(series(h,n = NUMBER_OF_COEFFS)).coeffs()
coeffs.reverse()
# and remove first coefficient 1 that corresponds to O(n**k)
coeffs.pop(0)
print([coeffs[n]*factorial(n+2) for n in range(len(coeffs))])

E.g.f.: (-sin(z) + 1 + (z-1)*cos(z))/ (1-sin(z))^2.
a(n) = (n+1)*e(n) - e(n+1), where e(n) is the n-th Euler number (see A000111).
Asymptotic: a(n) ~ 8*(Pi-2) / Pi^3 * n^2 * (2/Pi)^n.

A278677 a(n) = Sum_{k=0..n} A011971(n, k)*(k + 1). The Aitken-Bell triangle considered as a linear transform applied to the positive numbers.

Original entry on oeis.org

1, 5, 23, 109, 544, 2876, 16113, 95495, 597155, 3929243, 27132324, 196122796, 1480531285, 11647194573, 95297546695, 809490850313, 7126717111964, 64930685865768, 611337506786061, 5940420217001199, 59502456129204083, 613689271227219015, 6510381400140132872

Offset: 0

Sergey Kirgizov, Nov 26 2016

nonn

Original name: Popularity of left children in treeshelves avoiding pattern T231 (with offset 2).
Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. Classical Françon's bijection maps bijectively treeshelves into permutations. Pattern T231 illustrated below corresponds to a treeshelf constructed from permutation 231. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n.
a(n) is also the sum of the last entries in all blocks of all set partitions of [n-1]. a(4) = 23 because the sum of the last entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 3+5+5+4+6  = 23. - Alois P. Heinz, Apr 24 2017
a(n-2) is the number of lines that rhyme (with at least one earlier line) across all rhyme schemes counted by A000110. - Martin Fuller, Apr 20 2025

			Treeshelves of size 3:
      1  1          1    1       1        1
     /    \        /      \     / \      / \
    2      2      /        \   2   \    /   2
   /        \    2          2       3  3
  3          3    \        /
                   3      3
Pattern T231:
     1
    /
   /
  2
   \
    3
Treeshelves of size 3 that avoid pattern T231:
      1  1      1       1        1
     /    \      \     / \      / \
    2      2      \   2   \    /   2
   /        \      2       3  3
  3          3    /
                 3
Popularity of left children here is 5.

Alois P. Heinz, Table of n, a(n) for n = 0..572
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.
J. Françon, Arbres binaires de recherche : propriétés combinatoires et applications, Revue française d'automatique informatique recherche opérationnelle, Informatique théorique, 10 no. 3 (1976), pp. 35-50.

Cf. A000110, A000111, A000142, A001286, A008292, A011971, A131178, A278678, A278679, A285595, A286897, A367955.

b:= proc(n, m) option remember; `if`(n=0, [1, 0],
     (p-> p+[0, p[1]*n])(b(n-1, m+1))+m*b(n-1, m))
    end:
a:= n-> b(n+1, 0)[2]:
seq(a(n), n=0..22);  # Alois P. Heinz, Dec 15 2023
# Using the generating function:
gf := ((exp(z + exp(z)-1)*(z-1)) + exp(exp(z)-1))/z^2: ser := series(gf, z, 25):
seq((n+2)!*coeff(ser, z, n), n=0..22);  # Peter Luschny, Feb 01 2025

a[n_] := (n+3) BellB[n+2] - BellB[n+3];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Dec 01 2018 *)

from sympy import bell
HOW_MANY = 30
print([(n + 3) * bell(n+2) - bell(n + 3) for n in range(HOW_MANY)])

E.g.f.: ((exp(z + exp(z)-1)*(z-1)) + exp(exp(z)-1))/z^2.
a(n) = (n + 3)*b(n + 2) - b(n + 3) where b(n) is the n-th Bell number (see A000110).
Asymptotics: a(n) ~ n*b(n).
a(n) = Sum_{k=1..n+1} A285595(n+1,k)/k. - Alois P. Heinz, Apr 24 2017
a(n) = Sum_{k=0..n} Stirling2(n+2, k+1) * (n+1-k). - Ilya Gutkovskiy, Apr 06 2021
a(n) ~ n*Bell(n)*(1 - 1/LambertW(n)). - Vaclav Kotesovec, Jul 28 2021
a(n) = Sum_{k=n+1..(n+1)*(n+2)/2} k * A367955(n+1,k). - Alois P. Heinz, Dec 11 2023

New name and offset 0 by Peter Luschny, Feb 01 2025

cp's OEIS Frontend

User: Sergey Kirgizov

A343773 Excess of the number of even Motzkin n-paths (A107587) over the odd ones (A343386).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

A343386 Number of odd Motzkin n-paths, i.e., Motzkin n-paths with an odd number of up steps.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A340569 Total number of consecutive triples matching the pattern 123 in all faro permutations of length n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A340568 Total number of consecutive triples matching the pattern 132 in all faro permutations of length n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A340567 Total number of ascents in all faro permutations of length n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A335203 a(n) is the packing chromatic number of n-hypercube graph.

Views

Author

Keywords

Comments

Examples

Links

Formula

A302483 Number of FF-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI