Alexander Burstein - cp's OEIS frontend

A370276 Self-convolution of A138020.

1, 4, 16, 72, 352, 1816, 9728, 53584, 301568, 1726488, 10022912, 58864240, 349102080, 2087772784, 12576358400, 76237953440, 464736354304, 2847019090712, 17518413479936, 108224749140784, 670996707147776, 4173817417204944, 26040046909915136, 162905940337309792, 1021700454913933312

Offset: 0

Alexander Burstein, Feb 13 2024

nonn

Cf. A138020, A000984.

A370276 := proc(n)
    add( A138020(i)*A138020(n-i),i=0..n) ;
end proc:
seq(A370276(n),n=0..80) ; # R. J. Mathar, Sep 27 2024

CoefficientList[(InverseSeries[Series[x Sqrt[(1-2x)/(1+2x)],{x,0,25}]])^2/x^2,x]

G.f.: A(x) = F(x)^2, where F(x) is the g.f. of A138020.
G.f.: (A(x)-1)/(A(x)+1) = 2*x*sqrt(A(x)) = 2*x*F(x).
G.f.: A(-x*A(x)) = 1/A(x).
G.f.: A(x) = 1 + 4*x*A(x)*B(x^2*A(x)), where B(x) is the g.f. of the central binomial coefficients A000984.
D-finite with recurrence (n-1)*(n+2)*(5*n-12)*a(n) +4*(-55*n^3+242*n^2-316*n+120)*a(n-2) -16*(n-3)*(n-4)*(5*n-2)*a(n-4)=0. - R. J. Mathar, Sep 27 2024

A349562 Number of labeled rooted forests with 2-colored leaves.

Original entry on oeis.org

1, 2, 8, 56, 576, 7872, 134656, 2771456, 66744320, 1842237440, 57354338304, 1988721131520, 76015173369856, 3175757373243392, 143980934947930112, 7040807787705663488, 369414622819764928512, 20700889684976244621312, 1233951687316746828513280, 77963762014950356953333760

Offset: 0

Alexander Burstein, Nov 22 2021

nonn

a(n) is the number of labeled trees on vertices 0,1,...,n rooted at 0, where all leaves have 2 colors (except the singleton tree 0 has only 1 color).

			a(2)=8 counts trees 0-1-2B, 0-1-2R, 0-2-1B, 0-2-1R, 1B-0-2B, 1B-0-2R, 1R-0-2B, 1R-0-2R (where B and R stand for colors Blue and Red).

Seiichi Manyama, Table of n, a(n) for n = 0..372
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A000007, A007889, A216857, A038049, A201595,

CoefficientList[u/.AsymptoticSolve[u-E^(x(1+u))==0,u->1,{x,0,24}][[1]],x]Factorial/@Range[0,24]
nmax = 20; CoefficientList[Series[-LambertW[-x*Exp[x]]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 25 2021 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k+1)^(k-1)*x^k/(1-(k+1)*x)^(k+1))) \\ Seiichi Manyama, Nov 26 2021

a(n) = Sum_{k=0..n} binomial(n,k)*(k+1)^(n-1).
a(n) = A216857(n+1)/(n+1).
a(n) = A038049(n+1)/(n+1) for n>=1, and a(0) = A038049(1)/2.
a(n) = 2*A201595(n) - A000007(n).
E.g.f. satisfies: A(x) = e^(x*(1 + A(x))).
E.g.f. satisfies: A(-x*A(x)) = 1/A(x).
From Vaclav Kotesovec, Nov 25 2021: (Start)
E.g.f.: -LambertW(-x*exp(x))/x.
a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n) * LambertW(exp(-1))^(n+1)).
(End)
From Seiichi Manyama, Nov 26 2021: (Start)
G.f.: Sum_{k>=0} (k+1)^(k-1) * x^k/(1 - (k+1)*x)^(k+1).
a(n) = 2^n * A007889(n). (End)

A349648 Expansion of g.f.: Catalan(x)/Catalan(-x).

Original entry on oeis.org

1, 2, 2, 8, 14, 64, 132, 640, 1430, 7168, 16796, 86016, 208012, 1081344, 2674440, 14057472, 35357670, 187432960, 477638700, 2549088256, 6564120420, 35223764992, 91482563640, 493132709888, 1289904147324, 6979724509184, 18367353072152, 99710350131200

Offset: 0

Alexander Burstein, Nov 23 2021

Cf. A000108, A001622, A048990 (bijection), A052707 (bijection), A006318, A079489, A246062, A333564.

gf:= (c-> c(x)/c(-x))(x-> hypergeom([1/2, 1], [2], 4*x)):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 23 2021

CoefficientList[Series[(1-Sqrt[1-4x])/(Sqrt[1+4x]-1),{x,0,24}],x]

a(2*n) = A048990(n) = A000108(2*n), n>=0.
a(2*n+1) = A052707(n+1) = 2^(2*n+1)*A000108(n), n>=0.
G.f.: A(x) = C(x)/C(-x) = (1 - sqrt(1 - 4*x))/(sqrt(1 + 4*x) - 1), where C(x) is the g.f. of A000108.
G.f.: A(x) = F(x^2) + 2*x*F(x^2)^2 = (C(x) + C(-x))/2 + 2*x*C(4*x^2), where F(x) is the g.f. of A048990.
G.f.: A(-x) = 1/A(x).
G.f.: A(x) = R(x*C(-x)^2) = 1/R(-x*C(x)^2), where R(x) is the g.f. of A006318.
G.f.: A(x) = (1 + x*C(x)*C(-x))/(1 - x*C(x)*C(-x)), see A079489 for the expansion of C(x)*C(-x).
D-finite with recurrence n*(n-1)*(n+1)*a(n) -4*(n-1)*(8*n^2-32*n+35)*a(n-2) +64*(2*n-5)*(2*n-7)*(n-4)*a(n-4)=0. - R. J. Mathar, Mar 06 2022
Sum_{n>=0} 1/a(n) = 28/15 + 2*Pi/(9*sqrt(3)) + 64*arcsin(1/4)/(75*sqrt(15)) - 12*log(phi)/(25*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Apr 20 2023
G.f.: A(x) = exp( Sum_{n >= 1} binomial(4*n-2,2*n-1)*x^(2*n-1)/(2*n-1) ). - Peter Bala, Apr 28 2023

A347953 G.f.: A(x) = 1/C(-xT(x)^3), where C(x) = 1 + xC(x)^2 is the g.f. of A000108 and T(x) = 1 + x*T(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 2, 8, 35, 171, 882, 4744, 26286, 149045, 860596, 5042968, 29913676, 179270434, 1083794310, 6601817952, 40479778395, 249646876065, 1547539929810, 9637085582640, 60260786147261, 378212395786511, 2381767469829332, 15045137488662048, 95304451461770250

Offset: 0

Alexander Burstein, Nov 02 2021

nonn

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Cf. A000108, A001764, A166135.

cx := (1-sqrt(1-4*x))/2/x ;
tx := 2/sqrt(3*x)*sin( 1/3*arcsin(sqrt(27*x/4))) ;
gf := 1/subs(x=-x*tx^3,cx) ;
taylor(%,x=0,40) ;
gfun[seriestolist](%) ; # R. J. Mathar, Jul 20 2023

CoefficientList[y/.AsymptoticSolve[y-1-x(1-y+y^2)^3/y==0,y->1,{x,0,24}][[1]],x]

seq(n) = {Vec(1/subst((1 - sqrt(1 - 4*x + O(x^2*x^n))) / (2*x), x, -serreverse(x / (1+x)^3 + O(x*x^n))))} \\ Andrew Howroyd, Nov 22 2021

G.f.: A(-x*A(x)^3) = 1/A(x).
G.f.: The series reversion of x*A(x)^3 is x*A(-x)^3.
G.f.: A(x) satisfies A(x) = 1 + x*(1 - A(x) + A(x)^2)^3/A(x).
D-finite with recurrence +4*n*(4*n-1)*(4*n+1)*a(n) +6*(-342*n^3+1233*n^2-1453*n+542)*a(n-1) +243*(n-2)*(33*n^2-123*n+112)*a(n-2) +2187*(n-3)*(3*n-4)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Jul 20 2023

A348197 Composition of the g.f. of A086246 with itself.

Original entry on oeis.org

0, 1, 2, 4, 10, 28, 84, 264, 860, 2880, 9862, 34392, 121770, 436688, 1583146, 5793216, 21370806, 79391536, 296760222, 1115327844, 4212125662, 15976390684, 60833679424, 232452408632, 891060970152, 3425639505624, 13204738280326, 51024408662932, 197607503526934

Offset: 0

Alexander Burstein, Oct 06 2021

nonn

G.f.: A(x) is the pseudo-involutory Riordan companion of 2*M(x)-1, where M(x) is the g.f. of A001006.
For 1 <= n <= 7, a(n) coincides with A068875(n-1).
Conjecture: a(n) > A068875(n-1) for n > 7 (equivalently, a(n) > 2*A000108(n-1) for n > 7).

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Cf. A086246, A001006.

gf:= (f-> f(f(x)))(x->(1+x-sqrt(1-2*x-3*x^2))/2):
a:= n-> coeff(series(gf,x,n+1),x,n):
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 06 2021

f[x_] := (1 + x - Sqrt[1 - 2*x - 3*x^2])/2; a[n_] := SeriesCoefficient[f[f[x]], {x, 0, n}]; Array[a, 25, 0] (* Amiram Eldar, Oct 06 2021 *)

f(x) = (1+x-sqrt(1-2*x-3*x^2))/2;
my(x='x+O('x^30)); concat(0, Vec(f(f(x)))) \\ Michel Marcus, Oct 06 2021

G.f.: A(x) = F(F(x)), where F(x) is the g.f. of A086246.
Let G(x) = 2*M(x) - 1, where M(x) is the g.f. of A001006 (equivalently, x*G(x) is the g.f. of A007971). Then G(-A(x)) = 1/G(x).
A(-A(x)) = -x.
a(n) ~ ((1 + sqrt(3))^(n - 1/2) * 3^(n - 1/2)) / (sqrt(Pi) * n^(3/2) * 2^n). - Vaclav Kotesovec, Oct 07 2021

A348189 Pseudo-involutory Riordan companion of 1 + 2xM(x), where M(x) is the g.f. of A001006.

Original entry on oeis.org

1, 0, 0, 2, 0, 6, 8, 24, 60, 148, 396, 1026, 2744, 7350, 19872, 54102, 148104, 407682, 1127328, 3130542, 8726256, 24407634, 68482776, 192698124, 543642476, 1537443024, 4357677516, 12376868254, 35221087656, 100409367690, 286730523104, 820078634232, 2348966799132

Offset: 1

Alexander Burstein, Oct 06 2021

nonn

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.

Cf. A001006, A007971, A086246.

a[n_] := SeriesCoefficient[(1 - Sqrt[1-2*x-3*x^2])/(x * (2 + x - Sqrt[1-2*x-3*x^2])), {x, 0, n}]; Array[a, 33, 0] (* Amiram Eldar, Oct 06 2021 *)

my(x='x+O('x^35)); Vec((1-sqrt(1-2*x-3*x^2))/(x*(2+x-sqrt(1-2*x-3*x^2)))) \\ Michel Marcus, Oct 06 2021

G.f.: A(x) = (1 - sqrt(1 - 2*x - 3*x^2))/(x*(2 + x - sqrt(1 - 2*x - 3*x^2))).
If M(x) is the g.f. of A001006, then A(x) = (1 + 2*x*M(x))/(1 + 2*x + 2*x^2*M(x)).
Let M(x) be the g.f. of A001006 and F(x) = 1 + 2*x*M(x) (equivalently, x*F(x) = g.f. of A007971). Then F(-x*A(x)) = 1/F(x).
A(-x*A(x)) = 1/A(x).
G.f.: Let F(x) be the g.f. of A107264, then A(x) = 1 + 2*x^3*A(x)^2*F(x^2*A(x)). - Alexander Burstein, Feb 14 2022

A344623 Pseudo-involution companion for the Fibonacci generating function.

Original entry on oeis.org

1, 3, 9, 32, 126, 538, 2429, 11412, 55201, 272993, 1373784, 7011297, 36201841, 188761743, 992491049, 5256244537, 28013213196, 150128293038, 808543940999, 4373798584407, 23753913152691, 129469596050953, 707969244301884, 3882857013894482, 21353585584100401

Offset: 0

Alexander Burstein, May 24 2021

nonn

a(n) is the number of colored Schröder paths of semilength n with steps U=(1,0) and D=(1,-1) of 1 color and H=(2,0) of 2 colors, red and blue, where H does not follow D, and no two red H steps are consecutive.

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Cf. A000045, A000108, A200031.

G.f.: A(x) satisfies A(-x*A(x)) = 1/A(x) and F(-x*A(x)) = 1/F(x), where x*F(x) is g.f. of A000045. I.e., the Riordan array (F(x), x*A(x)) is a pseudo-involution.
G.f.: A(x) = (F(x) - 1)*C(F(x) - 1)/x, where C(x) is the g.f. of A000108 and x*F(x) is g.f. of A000045.
G.f.: A(x) = (1 - sqrt((1 - 5*x - 5*x^2)/(1 - x - x^2)))/(2x).
G.f.: Let B(x) = 2 + g.f.(A200031(n)), then A(x) = 1 + x*A(x)*B(x^2*A(x)).
a(n) ~ sqrt(3) * 5^(n/2 + 1) * phi^(2*n + 1) / (8*sqrt(Pi)*n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 25 2021

A343795 Number of Dumont permutations of the fourth kind of length 2n avoiding the pattern 312.

Original entry on oeis.org

1, 1, 3, 10, 39, 174, 872, 4805, 28474, 178099, 1160173, 7803860, 53924841, 381640934, 2761331130, 20400560942, 153738854242, 1180631743440, 9229687049249, 73372263658451, 592476077260123, 4854377724124700, 40315729803287046, 339065862485375334, 2885324166565733641

Offset: 0

Alexander Burstein and Opel Jones, Apr 29 2021

Dumont permutations of the fourth kind are permutations of even length where all deficiencies (drops) are even values at even positions.

			For n=2, a(2)=3 counts the permutations 1234, 1342, 1432.

O. Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, Ph.D. thesis, Howard University, 2019.

A. Burstein and O. Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, arXiv:2002.12189 [math.CO], 2020.
A. Burstein, M. Josuat-Vergès, and W. Stromquist, New Dumont permutations, Pure Math. Appl. (Pu.M.A.) 21 (2010), no. 2, 177-206.

Cf. A000108 (permutations avoiding 312), A024492, A048990, A110501 (length 2n Dumont permutations of 4th kind).

seq(n)={my(h=sqrt(1-16*x + O(x*x^n)), F=sqrt((1-h)/(8*x)), G=(1-sqrt((1+h)/2))/(2*x), A=O(1)); forstep(k=n\3, 0, -1, my(f=Pol(F + O(x*x^k))); A = f/((1 - x*Pol(G + O(x^k)))^2 - x*f/(1 - x*Pol(G + O(x*x^k)) - x*f^2/(1 - x*A))) ); Vec(A + O(x*x^n))} \\ Andrew Howroyd, Apr 29 2021

Let F_k(x) be the truncation of the g.f. of A048990 to a polynomial of degree k. Let G_k(x) be the truncation of the g.f. of A024492 to a polynomial of degree k. Let G_{-1}(x) = 0. For k>=0, define A_k(x) recursively as follows: A_k(x) = F_k(x)/((1-x*G_{k-1}(x))^2-x*F_k(x)/(1-x*G_k(x)-x*F_k(x)^2/(1-x*A_{k+1}(x)))). Then A_0(x) is the g.f. of this sequence.

Terms a(12) and beyond from Andrew Howroyd, Apr 29 2021

A308750 Number of Dumont permutations of the first kind of length 2n avoiding pattern 2143 (or pattern 3421).

Original entry on oeis.org

1, 1, 2, 7, 36, 239, 1892, 17015, 168503, 1799272, 20409644

Offset: 0

Alexander Burstein and Opel Jones, Jun 21 2019

Conjecture: The number of Dumont permutations of the first kind avoiding pattern 2143 equals the number of Dumont permutations of the first kind avoiding pattern 3421 for all n >= 0.

Data for n=7,8,9,10 is due to Michael Albert.

			For n=3, the 7 Dumont permutations of the first kind avoiding pattern 2143 are 356421, 364215, 435621, 563421, 564213, 634215, 642135, and the 7 Dumont permutations of the first kind avoiding pattern 3421 are 214365, 216435, 421365, 421563, 421635, 621435, 642135.

O. Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, Ph.D. thesis, Howard University, 2019.

D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.

Cf. A001469, A110501.

A033820 Triangle read by rows: T(k,j) = ((2j+1)/(k+1))binomial(2j,j)binomial(2k-2j,k-j).

Original entry on oeis.org

1, 1, 3, 2, 4, 10, 5, 9, 15, 35, 14, 24, 36, 56, 126, 42, 70, 100, 140, 210, 462, 132, 216, 300, 400, 540, 792, 1716, 429, 693, 945, 1225, 1575, 2079, 3003, 6435, 1430, 2288, 3080, 3920, 4900, 6160, 8008, 11440, 24310, 4862, 7722, 10296, 12936, 15876, 19404

Offset: 0

Alexander Burstein

f(n,k) = 2^{n-2(k-2)}sum(T(k-2,j)*binomial(n+2*(k-2-j),2*(k-2-j)),j=0..k-2) is the number of length n k-ary strings (k >= 2) which avoid a rising triple (pattern 123) or any other given 3-letter permutation pattern.

Row sums are the powers of 4. This is explained by a simple statistic on the 4^n lattice paths of length 2n formed from upsteps U=(1,1) and downsteps D=(1,-1). For such a path, define X = number of upsteps that lie above ground level (GL), the horizontal line through the initial vertex, and before the last vertex at GL. For UDDUUUUDDU for instance, the last vertex at GL follows the fourth step, and so X = 1. T(n,k) is the number of these paths with X=n-k. For example, T(2,1)=4 counts UDUU, UDDU, UDDD, DUUD because each has n-k=1 upsteps above GL and before the last vertex at GL. - David Callan, Nov 21 2011

			{1},
{1, 3},
{2, 4, 10},
{5, 9, 15, 35},
{14, 24, 36, 56, 126},
{42, 70, 100, 140, 210, 462},
{132, 216, 300, 400, 540, 792, 1716},
...

Alexander Burstein, Enumeration of words with forbidden patterns, Ph.D. thesis, University of Pennsylvania, 1998.
Ira Gessel, Super ballot numbers, J. Symbolic Computation 14 (1992), 179-194.
Walter Shur, Two Game-Set Inequalities, J. Integer Seqs., Vol. 6, 2003.

Cf. A000108, A000984, A000302.

Essentially a reflected version of A078817.

T(k,0) = binomial(2*k, k)/(k+1), the k-th Catalan number; T(k,k) = binomial(2*(k+1),k+1)/2, half the (k+1)-st central binomial coefficient sum of entries in row k (over j) = 2^{2*(k-1)}
T(k,j) = sum(C(k-i)D(i), i=0..j), C(i) = binomial(2*i,i)/(i+1), D(i) = binomial(2*i,i).
G.f.: 2/(1-4*x*y+sqrt((1-4*x)*(1-4*x*y))). - Vladeta Jovovic, Dec 14 2003

More terms from Vladeta Jovovic, Dec 10 2003

cp's OEIS Frontend

User: Alexander Burstein

A370276 Self-convolution of A138020.

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A349562 Number of labeled rooted forests with 2-colored leaves.

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A349648 Expansion of g.f.: Catalan(x)/Catalan(-x).

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A347953 G.f.: A(x) = 1/C(-x*T(x)^3), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108 and T(x) = 1 + x*T(x)^3 is the g.f. of A001764.

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A348197 Composition of the g.f. of A086246 with itself.

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A348189 Pseudo-involutory Riordan companion of 1 + 2*x*M(x), where M(x) is the g.f. of A001006.

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A344623 Pseudo-involution companion for the Fibonacci generating function.

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A343795 Number of Dumont permutations of the fourth kind of length 2n avoiding the pattern 312.

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A347953 G.f.: A(x) = 1/C(-xT(x)^3), where C(x) = 1 + xC(x)^2 is the g.f. of A000108 and T(x) = 1 + x*T(x)^3 is the g.f. of A001764.

A348189 Pseudo-involutory Riordan companion of 1 + 2xM(x), where M(x) is the g.f. of A001006.

A033820 Triangle read by rows: T(k,j) = ((2j+1)/(k+1))binomial(2j,j)binomial(2k-2j,k-j).