A001879 -id:A001879 - cp's OEIS frontend

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A288953 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except after the last branch node on level 0.

1, 1, 3, 10, 51, 280, 1995, 15120, 138075, 1330560, 14812875, 172972800, 2271359475, 31135104000, 471038042475, 7410154752000, 126906349444875, 2252687044608000, 43078308695296875, 851515702861824000, 17984171447178811875, 391697223316439040000

Offset: 0

Michael Wallner, Jun 20 2017

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. A branch node is a node with a left and right edge (no pointer). See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where in the growth process induced by the labels maximal young leaves and non-maximal young leaves alternate except for a sequence of maximal young leaves at the beginning. A young leaf is a leaf with no left sibling. A maximal young leaf is a young leaf with maximal label. See the Wallner link. - Michael Wallner, Apr 20 2017

			Denote by L the leaf and by o nodes. Every node has exactly two out-going edges or pointers. Internal edges are denoted by - or |. Pointers are omitted and may point to any node further right. The root is at level 0 at the very left.
The general structure is
  L-o-o-o-o-o-o-o-o
          | | | | |
          o o o o o.
For n=0 the a(0)=1 solution is L.
For n=1 the a(1)=1 solution is L-o.
For n=2 the a(2)=3 solutions are
L-o-o     L-o
            |
            o
  2    +   1    solutions of this shape with pointers.

Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017

Cf. A288954 (variation with additional initial sequence).
Cf. A177145 (variation without final sequence).
Cf. A001147 (relaxed compacted binary trees of right height at most one).
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A000032, A000246, A001879, A051577, A213527, A288950, A288952, A288954 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of relaxed compacted binary trees of right height at most one, see the Wallner link).

E.g.f.: (2-z)/(3*(1-z)^2) + 1/(3*sqrt(1-z^2)).

A288954 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except before the first and after the last branch node on level 0.

Original entry on oeis.org

1, 1, 3, 13, 79, 555, 4605, 42315, 436275, 4894155, 60125625, 794437875, 11325612375, 172141044075, 2793834368325, 48009995908875, 874143494098875, 16757439016192875, 338309837281040625, 7157757510792763875, 158706419654857449375, 3673441093896736036875

Offset: 2

Michael Wallner, Jun 20 2017

nonn

a(n) is the number of plane increasing trees with n+1 nodes where in the growth process induced by the labels a maximal young leaves and non-maximal young leaves alternate except for a sequence of maximal young leaves at the begininning and at the end. A young leaf is a leaf with no left sibling. A maximal young leaf is a young leaf with maximal label. See the Wallner link. - Michael Wallner, Apr 20 2017

			See A288950 and A288953.

Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A288953 (variation without initial sequence).
Cf. A177145 (variation without initial and final sequence).
Cf. A001147 (relaxed compacted binary trees of right height at most one).
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A000032, A000246, A001879, A051577, A213527, A288950, A288952 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of relaxed compacted binary trees of right height at most one, see the Wallner link).

terms = 22; egf = 1/(3(1-z))(1/Sqrt[1-z^2] + (3z^3 - z^2 - 2z + 2)/((1-z)(1-z^2))) + O[z]^terms;
CoefficientList[egf, z] Range[0, terms-1]! (* Jean-François Alcover, Dec 13 2018 *)

E.g.f.: 1/(3*(1-z))*( 1/sqrt(1-z^2) + (3*z^3-z^2-2*z+2)/((1-z)*(1-z^2)) ).

A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2z)!/(2^zz!).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 3, 1, 0, 10, 0, 15, 0, 1, 0, 15, 0, 45, 0, 15, 1, 0, 21, 0, 105, 0, 105, 0, 1, 0, 28, 0, 210, 0, 420, 0, 105, 1, 0, 36, 0, 378, 0, 1260, 0, 945, 0, 1, 0, 45, 0, 630, 0, 3150, 0, 4725, 0, 945, 1, 0, 55, 0, 990, 0, 6930, 0, 17325, 0, 10395, 0

Offset: 0

Peter Luschny, Jan 13 2023

The Kummer numbers K(n, k) are a refinement of the oddfactorial numbers (A001147) in the sense that they are the coefficients of polynomials K(n, x) = Sum_{n..k} K(n, k) * x^k that take the value oddfactorial(n) at x = 1. The coefficients of x^n are the aerated oddfactorial numbers A123023.

These numbers appear in many different versions (see the crossrefs). They are the coefficients of the Chebyshev-Hermite polynomials in signed form when ordered in decreasing powers. Our exposition is based on the seminal paper by Kummer, which preceded the work of Chebyshev and Hermite for more than 20 years. They are also referred to as Bessel numbers of the second kind (Mansour et al.) when the odd powers are omitted.

			Triangle K(n, k) starts:
 [0] 1;
 [1] 1, 0;
 [2] 1, 0,  1;
 [3] 1, 0,  3, 0;
 [4] 1, 0,  6, 0,   3;
 [5] 1, 0, 10, 0,  15, 0;
 [6] 1, 0, 15, 0,  45, 0,   15;
 [7] 1, 0, 21, 0, 105, 0,  105, 0;
 [8] 1, 0, 28, 0, 210, 0,  420, 0, 105;
 [9] 1, 0, 36, 0, 378, 0, 1260, 0, 945, 0;

John Riordan, Introduction to Combinatorial Analysis, Dover (2002), pp. 85-86.

Pierre Humbert, Monographie des polynômes de Kummer, Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Serie 5, Volume 1 (1922), pp. 81-92.
E. E. Kummer, Über die hypergeometrische Reihe, Journal für die reine und angewandte Mathematik 15 (1836): 39-83.
T. Mansour, M. Schork and M. Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
Ladislav Truksa, Hypergeometric orthogonal systems of polynomials III, Aktuárské vědy, Vol. 2 (1931), No. 4, 177-203, (see p.200).

Variants: Signed version: A073278. Other variants are the irregular triangle A100861 with zeros deleted, A066325 and A099174 with reversed rows, A111924, A144299, A104556.
Cf. A000085, A001147, A123023, A047974, A115329, A002522, A056107, A359739,
A001879, A001464, A005425, A344501, A123022, A359761.

oddfactorial := proc(z) (2*z)! / (2^z*z!) end:
K := (n, k) -> ifelse(irem(k, 2) = 1, 0, binomial(n, k) * oddfactorial(k/2)):
seq(seq(K(n, k), k = 0..n), n = 0..11);
# Alternative, as coefficients of polynomials:
p := (n, x) -> 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)):
seq(print(seq(coeff(simplify(p(n, x)), x, k), k = 0..n)), n = 0 ..9);
# Using the exponential generating function:
egf := exp(x + (t*x)^2 / 2): ser := series(egf, x, 12):
seq(print(seq(coeff(n! * coeff(ser, x, n), t, k), k = 0..n)), n = 0..9);

K[n_, k_] := K[n, k] = Which[OddQ[k], 0, k == 0, 1, n == k, K[n - 1, n - 2], True, K[n - 1, k] n/(n - k)];
Table[K[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 25 2023 *)

from functools import cache
@cache
def K(n: int, k: int) -> int:
    if k %  2: return 0
    if n <  3: return 1
    if n == k: return K(n - 1, n - 2)
    return (K(n - 1, k) * n) // (n - k)
for n in range(10): print([K(n, k) for k in range(n + 1)])

Let p(n, x) = 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)).
p(n, 1) = A000085(n); p(n, sqrt(2)) = A047974(n); p(n, 2) = A115329(n);
p(2, n) = A002522(n) (n >= 1); p(3, n) = A056107(n) (n >= 1);
p(n, n) = A359739(n) (n >= 1); 2^n*p(n, 1/2) = A005425(n).
K(n, k) = [x^k] p(n, x).
K(n, k) = [t^k] (n! * [x^n] exp(x + (t*x)^2 / 2)).
K(n, n) = A123023(n).
K(n, n-1) = A123023(n + 1).
K(2*n, 2*n) = A001147(n).
K(4*n, 2*n) = A359761, the central terms without zeros.
K(2*n+2, 2*n) = A001879.
Sum_{k=0..n} (-1)^n * i^k * K(n, k) = A001464(n), ((the number of even involutions) - (the number of odd involutions) in the symmetric group S_n (Robert Israel)).
Sum_{k=0..n} Sum_{j=0..k} K(n, j) = A000085(n + 1).
For a recursion see the Python program.

A227528 a(n) = numerator of r(n) = 2(3n+2)!/((2n)!2^n), n>=0.

Original entry on oeis.org

4, 60, 840, 13860, 270270, 6126120, 158722200, 4633467300, 150587687250, 5394582443250, 211240491462000, 8977720887135000, 411608985890602500, 20251162105817643000, 1064311075116860571000, 59509669251792738478500, 3527387653231263127556250, 220942735734212754080568750

Offset: 0

Karol A. Penson, Jul 14 2013

The first values with denominators > 1 occur at n = {84, 148, 164, 168, 169, 276, 292, 296, 297, ...}. - G. C. Greubel, Jul 04 2017

G. C. Greubel, Table of n, a(n) for n = 0..370

Cf. A001879.

seq(numer(4*(3*n+2)!/((2*n)!*2^(n+1))), n=0..14);

Table[Numerator[2(3n + 2)!/((2n)! 2^n)], {n, 0, 84}]  (* G. C. Greubel, Jul 04 2017 *)

for(n=0,50, print1(numerator(2*(3*n + 2)!/((2*n)!*2^n)), ", ")) \\ G. C. Greubel, Jul 04 2017

In Maple notation,
E.g.f. of r: ((135/8)*z+4)*cos((2/3)*arcsin((3/4)*sqrt(6)*sqrt(z)))/(1-(27/8)*z)^2+(3/8)*sqrt(z)*((135/4)*z+17)*sin((2/3)*arcsin((3/4)*sqrt(6)*sqrt(z)))*sqrt(6)/(1-(27/8)*z)^(5/2).
E.g.f. of r: 4*hypergeom([4/3,5/3],[1/2],27*z/8).
Integral representation as n-th moment of a signed function w(x) of bounded variation on (0,infinity),
  w(x)=(8/3)*sqrt(3)*((1/36) *(128*x^2/81-40*x/3+20) *exp(-4*x/27) *BesselK(1/3,(4/27)*x)/Pi +(2/81)*x *(-5+16*x/9) *exp(-4*x/27) *BesselK(4/3,4*x/27)/Pi);
  w(x)=-(14/243)*16^(2/3)*x^(2/3)*3 *hypergeom([13/6], [4/3], -(8/27)*x)/GAMMA(2/3)-(10/243)*sqrt(3)*16^(1/3)*x^(1/3)*9*GAMMA(2/3)*hypergeom([11/6], [2/3], -(8/27)*x)/Pi.
  For x>3.32, w(x)>0.
  w(0)=w(3.32)=limit(w(x),x=infinity)=0.
  For x<3.32, w(x)<0.
r(n) = int(x^n*w(x), x=0..infinity), n>=0.
Asymptotics: r(n)->(1/1152)*sqrt(6)*(10368*n^2+10224*n+2161)*(27/8)^n*exp(-n)*(n^n), for n->infinity.
The rational values are given by 4*(-2*n+1)*r(n) + 3*(3*n+2)*(3*n+1) * r(n-1)=0. - R. J. Mathar, Jul 20 2013

Erroneous definition corrected by G. C. Greubel, Jul 04 2017

A261065 Second column of A086872.

Original entry on oeis.org

1, 8, 75, 840, 11025, 166320, 2837835, 54054000, 1137161025, 26189163000, 655383804075, 17709112020600, 513880482740625, 15938200818540000, 526174085058496875, 18422283260401020000, 681816379418800250625, 26597171457203972625000, 1090705672840839577396875

Offset: 1

R. J. Mathar, Aug 08 2015

Cf. A086872.

(n-1)*(n+1)*a(n) -n*(n+2)*(2*n-1)*a(n-1)=0.
G.f. x*3F1(3/2,4,2; 3; 2x).
a(n) = A001879(n-1)+2*A001880(n+2).

A334823 Triangle, read by rows, of Lambert's denominator polynomials related to convergents of tan(x).

Original entry on oeis.org

1, 1, 0, 3, 0, -1, 15, 0, -6, 0, 105, 0, -45, 0, 1, 945, 0, -420, 0, 15, 0, 10395, 0, -4725, 0, 210, 0, -1, 135135, 0, -62370, 0, 3150, 0, -28, 0, 2027025, 0, -945945, 0, 51975, 0, -630, 0, 1, 34459425, 0, -16216200, 0, 945945, 0, -13860, 0, 45, 0, 654729075, 0, -310134825, 0, 18918900, 0, -315315, 0, 1485, 0, -1

Offset: 0

G. C. Greubel, May 12 2020, following a suggestion from Michel Marcus

Lambert's numerator polynomials related to convergents of tan(x), g(n, x), are given in A334824.

			Polynomials:
f(0, x) = 1;
f(1, x) = x;
f(2, x) = 3*x^2 - 1;
f(3, x) = 15*x^3 - 6*x;
f(4, x) = 105*x^4 - 45*x^2 + 1;
f(5, x) = 945*x^5 - 420*x^3 + 15*x;
f(6, x) = 10395*x^6 - 4725*x^4 + 210*x^2 - 1;
f(7, x) = 135135*x^7 - 62370*x^5 + 3150*x^3 - 28*x;
f(8, x) = 2027025*x^8 - 945945*x^6 + 51975*x^4 - 630*x^2 + 1.
Triangle of coefficients begins as:
        1;
        1, 0;
        3, 0,      -1;
       15, 0,      -6, 0;
      105, 0,     -45, 0,     1;
      945, 0,    -420, 0,    15, 0;
    10395, 0,   -4725, 0,   210, 0,   -1;
   135135, 0,  -62370, 0,  3150, 0,  -28, 0;
  2027025, 0, -945945, 0, 51975, 0, -630, 0, 1.

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
J.-H. Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendantes et logarithmiques (Memoir on some properties that can be traced from circular transcendent and logarithmic quantities), Histoire de l’Académie royale des sciences et belles-lettres (1761), Berlin. See also.

Cf. A001497, A001498, A053983, A053984, A053987, A053988, A094674, A334824.

Columns k: A001147 (k=0), A001879 (k=2), A001880 (k=4), A038121 (k=6).

C := ComplexField();
T:= func< n, k| Round( i^k*Factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*Factorial(k)*Factorial(n-k)) ) >;
[T(n,k): k in [0..n], n in [0..10]];

Maple

T:= (n, k) -> I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!); seq(seq(T(n, k), k = 0 .. n), n = 0 .. 10);

Mathematica

(* First program *) y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x]; f[n_, k_]:= Coefficient[((-I)^n/2)*(y[n, I*x] + (-1)^n*y[n, -I*x]), x, k]; Table[f[n, k], {n,0,10}, {k,n,0,-1}]//Flatten (* Second program *) Table[ I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), {n,0,10}, {k,0,n}]//Flatten

Sage

[[ i^k*factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*factorial(k)*factorial(n-k)) for k in (0..n)] for n in (0..10)]

Formula

Equals the coefficients of the polynomials, f(n, x), defined by: (Start)

f(n, x) = Sum_{k=0..floor(n/2)} ((-1)^k*(2*n-2*k)!/((2*k)!*(n-2*k)!))*(x/2)^(n-2*k).

f(n, x) = ((2*n)!/n!)*(x/2)^n*Hypergeometric2F3(-n/2, (1-n)/2; 1/2, -n, -n+1/2; -1/x^2).

f(n, x) = ((-i)^n/2)*(y(n, i*x) + (-1)^n*y(n, -i*x)), where y(n, x) are the Bessel Polynomials.

f(n, x) = (2*n-1)*x*f(n-1, x) - f(n-2, x).

E.g.f. of f(n, x): cos((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).

f(n, 1) = (-1)^n*f(n, -1) = A053983(n) = (-1)^(n+1)*A053984(-n-1) = (-1)^(n+1) * g(-n-1, 1).

f(n, 2) = (-1)^n*f(n, -2) = A053988(n+1). (End)

As a number triangle:

T(n, k) = i^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), where i = sqrt(-1).

T(n, 0) = A001147(n).

A227624 a(n) = numerator of r(n), where r(n) = (4*n+2)!/((3*n)!*2^n).

Original entry on oeis.org

2, 60, 1260, 30030, 835380, 26860680, 984233250, 40560770250, 1858741384500, 93814013878200, 5172710627284200, 309424040950649625, 19960884210345828750, 1381474908065669917500, 102111212412024699633750, 8028503070893011778321250

Offset: 0

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Author

Karol A. Penson, Jul 18 2013

Keywords

nonn
easy

Comments

The first values with denominators > 1 occur at n = (43, 86, 87, 91, 107, 171, 172, ...). - G. C. Greubel, Jul 04 2017

Links

G. C. Greubel, Table of n, a(n) for n = 0..355

Crossrefs

Cf. A001879, A227528.

Programs

Maple

seq(numer(2*(4*n+2)!/((3*n)!*2^(n+1)), n=0..15)

Mathematica

Table[(4 n + 2)!/((3 n)!*2^n), {n, 0, 15}] (* Michael De Vlieger, Oct 08 2016 *)(* generates values of r(n) *) Table[Numerator[(4*n + 2)!/((3*n)! *2^n)], {n, 0, 50}] (* G. C. Greubel, Jul 04 2017 *)

PARI

for(n=0,50, print1(numerator((4*n + 2)!/((3*n)! *2^n)), ", ")) \\ G. C. Greubel, Jul 05 2017

Formula

In Maple notation,

E.g.f. of r(n): 2*hypergeom([3/4, 5/4, 3/2], [1/3, 2/3], (128/27)*z).

Integral representation as n-th moment of a signed function w(x) of bounded variation on (0,infinity),

w(x) = (5/64)*2^(3/4)*hypergeom([13/12, 17/12], [1/4, 1/2], -(27/128)*x)/(GAMMA(3/4)*x^(1/4))-(231/512)*2^(3/4)*GAMMA(3/4)*x^(1/4)*hypergeom([19/12, 23/12], [3/4, 3/2], -(27/128)*x)/Pi-(35/64)*sqrt(2)*sqrt(x)*hypergeom([11/6, 13/6], [5/4, 7/4], -(27/128)*x)/sqrt(Pi);

For x>5.723, w(x)>0.

w(0)=w(5.723)=limit(w(x),x=infinity)=0.

For x<5.723, w(x)<0.

r(n) = int(x^n*w(x), x=0..infinity), n>=0.

Asymptotics: r(n)-> (1/3888)*sqrt(3)*(41472*n^2+30816*n+4969)*(128/27)^n*exp(-n)*(n)^(n), for n->infinity.

3*(3*n-1)*(3*n-2)*r(n) -4*(4*n+1)*(2*n+1)*(4*n-1)*r(n-1)=0. - R. J. Mathar, Oct 08 2016

A316666 Number of simple relaxed compacted binary trees of right height at most one with no sequences on level 1 and no final sequences on level 0.

Original entry on oeis.org

1, 0, 1, 3, 15, 87, 597, 4701, 41787, 413691, 4512993, 53779833, 695000919, 9680369943, 144560191149, 2303928046437, 39031251610227, 700394126116851, 13270625547477177, 264748979672169681, 5547121478845459983, 121784530649198053263, 2795749225338111831429, 66981491857058929294653

Offset: 0

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Author

Michael Wallner, Jul 10 2018

Keywords

nonn

Comments

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and at most n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. It is called simple if for nodes with two pointers both point to the same node. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. See the Wallner link.

a(n) is one of two "basis" sequences for sequences of the form a(0)=a, a(1)=b, a(n) = n*a(n-1) + (n-1)*a(n-2), the second basis sequence being A096654 (with 0 appended as a(0)). The sum of these sequences is listed as A000255. - Gary Detlefs, Dec 11 2018

Links

G. C. Greubel, Table of n, a(n) for n = 0..448

Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.

Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Crossrefs

Cf. A000032, A000246, A001879, A051577, A213527, A288950, A288952, A288953 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).

Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of simple relaxed compacted binary trees of right height at most one, see the Wallner link).

Cf. A000255, A096654.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (3*Exp(-x) + x-2)/(1-x)^2 )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Dec 12 2018

Maple

aseq := n-> 3*round((n+2)*n!/exp(1))-(n+2)*n!: bseq := n-> (n+2)*n!- 2* round((n+2)*n!/exp(1)): s := (a,b,n)-> a*aseq(n) + b*bseq( n): seq(s(1,0,n),n = 0..20); # Gary Detlefs, Dec 11 2018

Mathematica

terms = 24; CoefficientList[(3E^-z+z-2)/(1-z)^2 + O[z]^terms, z] Range[0, terms-1]! (* Jean-François Alcover, Sep 14 2018 *)

PARI

Vec(serlaplace((3*exp(-x + O(x^25)) + x - 2)/(1 - x)^2)) \\ Andrew Howroyd, Jul 10 2018

Formula

E.g.f.: (3*exp(-z)+z-2)/(1-z)^2.

a(n) ~ (3*exp(-1) - 1) * n * n!. - Vaclav Kotesovec, Jul 12 2018

a(n) = 3*round((n+2)*n!/e) - (n+2)*n!. - Gary Detlefs, Dec 11 2018

From Seiichi Manyama, Apr 25 2025: (Start)

a(n) = 3 * A000255(n) - n! - (n+1)!.

a(0) = 1, a(1) = 0; a(n) = n*a(n-1) + (n-1)*a(n-2). (End)

A363935 Triangle of coefficients of the primitive Eulerian polynomials of type D T(n,k) (n >= 2, 1 <= k <= n) read by rows.

Original entry on oeis.org

0, 1, 1, 4, 1, 4, 20, 20, 1, 11, 116, 216, 76, 1, 26, 632, 2072, 1732, 262, 1, 57, 3158, 18404, 28064, 11824, 862, 1, 120, 14800, 151104, 386640, 317200, 73320, 2760, 1, 247, 66424, 1158040, 4777264, 6608800, 3169168, 427576, 8680, 1

Offset: 2

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Author

Jose Bastidas, Jun 28 2023

Keywords

nonn
tabf

Comments

Row n counts even signed permutations w in D_n such that (i.) w(1) != -n and (ii.) all right-to-left maxima of |w| are negative in w, by their type D descent number. For example, T(3,2) = 4 counts the signed permutations (1,-3,-2), (-2,1,-3), (2,-1,-3), (2,-3,-1).

Examples

0, 1; 1, 4, 1; 4, 20, 20, 1; 11, 116, 216, 76, 1; 26, 632, 2072, 1732, 262, 1; 57, 3158, 18404, 28064, 11824, 862, 1; 120, 14800, 151104, 386640, 317200, 73320, 2760, 1; ...

Links

Jose Bastidas, Table of n, a(n) for n = 2..1225

J. Bastidas, C. Hohlweg and F. Saliola, The Primitive Eulerian polynomial, arXiv:2306.15556 [math.CO], 2023. Table 3.

Crossrefs

Row sums are A001879(n-2).

A161127 Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k descents (n>=1; 1<=k<2n-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 7, 3, 1, 1, 6, 27, 37, 27, 6, 1, 1, 10, 76, 220, 331, 220, 76, 10, 1, 1, 15, 176, 897, 2438, 3341, 2438, 897, 176, 15, 1, 1, 21, 357, 2885, 12825, 30807, 41343, 30807, 12825, 2885, 357, 21, 1, 1, 28, 658, 7871, 53312, 203927, 452931, 589569

Offset: 1

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Author

Emeric Deutsch, Jun 09 2009

Keywords

nonn
tabf

Comments

Row n contains 2n-1 entries.

Sum of entries in row n = (2n-1)!! = A001147(n).

Sum_{k=1..2n-1} k*T(n,k) = A001879(n-1).

Examples

T(3,2)=3 because we have 215634, 341265, and 351624. Triangle starts: 1; 1,1,1; 1,3,7,3,1; 1,6,27,37,27,6,1; 1,10,76,220,331,220,76,10,1; ...

Links

J. Désarménien and D. Foata, Fonctions symetriques et series hypergeometriques basiques multivariees, Bull. Soc. Math. France, 113, 1985, 3-22.

I. M. Gessel and C. Reutenauer, Counting permutations with given cycle structure and descent set, J. Combin. Theory, Ser. A, 64, 1993, 189-215.

V. J. W. Guo and J. Zeng, The Eulerian distribution on involutions is indeed unimodal, J. Combin. Theory, Ser. A, 113, 2006, 1061-1071.

Crossrefs

Cf. A001147, A001879.

Programs

Maple

T := proc (n, k) if k <= 0 or n <= 0 then 0 elif n = 1 and k = 1 then 1 elif 2*n <= k then 0 else ((1/2)*(k*(k+1)+2*n-2)*T(n-1, k)+(1/2)*(2*(k-1)*(2*n-k-1)+2)*T(n-1, k-1)+(1/2)*((2*n-k)*(2*n-k+1)+2*n-2)*T(n-1, k-2))/n end if end proc: for n to 8 do seq(T(n, k), k = 1 .. 2*n-1) end do; # end of program for n to 8 do P[n] := sort(expand(simplify((1-t)^(2*n+1)*(sum(binomial((1/2)*i*(i+1)+n-1, n)*t^i, i = 0 .. infinity))))) end do: for n to 8 do seq(coeff(P[n], t, j), j = 1 .. 2*n-1) end do; # end of program

Mathematica

T[n_, k_] := Which[k <= 0 || n <= 0, 0, n == 1 && k == 1, 1, 2n <= k, 0, True, ((1/2)*(k*(k + 1) + 2n - 2)*T[n - 1, k] + (1/2)*(2*(k - 1)*(2n - k - 1) + 2)*T[n - 1, k - 1] + (1/2)*((2n - k)*(2n - k + 1) + 2n - 2)*T[n - 1, k - 2])/n]; Table[T[n, k], {n, 1, 8}, {k, 1, 2n - 1}] // Flatten (* Jean-François Alcover, Sep 05 2024, after first Maple program *)

Formula

Recurrence: 2nT(n,k) = [k(k+1)+2n-2]T(n-1,k)+2[(k-1)(2n-k-1)+1]T(n-1,k-1)+[(2n-k)(2n-k+1)+2n-2]T(n-1,k-2) (k>=1, n>=2) (see Eq. (2.1) in the Guo-Zeng paper; see first Maple program).

Generating polynomial of row n is P(n,t) = (1 - t)^{2n+1}*Sum(C(j(j+1)/2 + n - 1, n)*t^j, j=0..oo) (see Eq. (2.2) in the Guo-Zeng paper; see 2nd Maple program).

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cp's OEIS Frontend

A288953 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except after the last branch node on level 0.

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A288954 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except before the first and after the last branch node on level 0.

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Mathematica

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A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2*z)!/(2^z*z!).

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Python

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A227528 a(n) = numerator of r(n) = 2*(3*n+2)!/((2*n)!*2^n), n>=0.

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Extensions

A261065 Second column of A086872.

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A334823 Triangle, read by rows, of Lambert's denominator polynomials related to convergents of tan(x).

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Sage

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A227624 a(n) = numerator of r(n), where r(n) = (4*n+2)!/((3*n)!*2^n).

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A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2z)!/(2^zz!).

A227528 a(n) = numerator of r(n) = 2(3n+2)!/((2n)!2^n), n>=0.

A227624 a(n) = numerator of r(n), where r(n) = (4n+2)!/((3n)!*2^n).