A357588 -id:A357588 - cp's OEIS frontend

A357368 Triangle read by rows. Convolution triangle of the prime indicator sequence A089026.

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 1, 10, 6, 1, 0, 5, 14, 21, 8, 1, 0, 1, 23, 47, 36, 10, 1, 0, 7, 28, 90, 108, 55, 12, 1, 0, 1, 49, 147, 258, 205, 78, 14, 1, 0, 1, 46, 249, 520, 595, 346, 105, 16, 1, 0, 1, 75, 360, 978, 1437, 1185, 539, 136, 18, 1

Offset: 0

Peter Luschny, Oct 03 2022

To clarify our terminology: We say T is the convolution triangle of a (or T is the partition transform of a), if a is a sequence of integers defined for n >= 1, and the transformation, as defined by the Maple function below, applied to a, returns T. In the generated lower triangular matrix T, i.e., in the convolution triangle of a, T(n, 1) = a(n) for n >= 1.
For instance, let a(n) = Bell(n), then we call A357583 the convolution triangle of the Bell numbers, but not A205574, which is also called the "Bell convolution triangle" in the comments but leads to a different triangle. Similarly, if a(n) = n!, then A090238 is the convolution triangle of the factorial numbers, not A084938. Third example: A128899 is the convolution triangle of the Catalan numbers in our setup, not A059365. More generally, we recommend that when computing the transform of a 0-based sequence to use only the terms for n >= 1 and not to shift the sequence.
Note that T is (0, 0)-based and the first column of a convolution triangle always is 1, 0, 0, 0, ... and the main diagonal is 1, 1, 1, 1, ... if a(1) = 1. The (1, 1)-based subtriangle of a genuine convolution triangle, i.e., a convolution triangle without column 0 and row 0, is often used in place of the convolution triangle but does not fit well into some applications of the convolution triangles.

			Triangle T(n, k) starts:
[0] 1;
[1] 0, 1;
[2] 0, 2,  1;
[3] 0, 3,  4,   1;
[4] 0, 1, 10,   6,   1;
[5] 0, 5, 14,  21,   8,   1;
[6] 0, 1, 23,  47,  36,  10,   1;
[7] 0, 7, 28,  90, 108,  55,  12,   1;
[8] 0, 1, 49, 147, 258, 205,  78,  14,  1;
[9] 0, 1, 46, 249, 520, 595, 346, 105, 16, 1;

Donald E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA]; Mathematica J. 2.1 (1992), no. 4, 67-78.
Peter Luschny, The P-transform.
Cormac O'Sullivan, De Moivre and Bell polynomials, arXiv:2203.02868 [math.CO], 2022.

Cf. A089026, A340991, A357588.

PMatrix := proc(dim, a) local n, k, m, g, M, A;
   if n = 0 then return [1] fi;
   A := [seq(a(i), i = 1..dim-1)];
   M := Matrix(dim, shape=triangular[lower]); M[1, 1] := 1;
   for m from 2 to dim do
       M[m, m] := M[m - 1, m - 1] * A[1];
       for k from m-1 by -1 to 2 do
           M[m, k] := add(A[i]*M[m-i, k-1], i = 1..m-k+1)
od od; M end:
a := n -> if isprime(n) then n else 1 fi: PMatrix(10, a);
# Alternatively, as the coefficients of row polynomials:
P := proc(n, x, a) option remember; ifelse(n = 0, 1,
    x*add(a(n - k)*P(k, x, a), k = 0..n-1)) end:
Pcoeffs := proc(n, a) seq(coeff(P(n, x, a), x, k), k=0..n) end:
seq(Pcoeffs(n, a), n = 0..9);
# Alternatively, term by term:
T := proc(n, k, a) option remember; # Alois P. Heinz style
    `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, a(n)),
    (q->add(T(j, q, a)*T(n-j, k-q, a), j=0..n))(iquo(k, 2)))) end:
seq(seq(T(n, k, a), k=0..n), n=0..9);

PMatrix[dim_, a_] := Module[{n, k, m, g, M, A}, If[n == 0, Return[1]]; A = Array[a, dim-1]; M = Array[0&, {dim, dim}]; M[[1, 1]] = 1; For[m = 2, m <= dim, m++, M[[m, m]] = M[[m-1, m-1]]*A[[1]]; For[k = m-1, k >= 2, k--, M[[m, k]] = Sum[A[[i]]*M[[m-i, k-1]], {i, 1, m-k+1}]]]; M];
a[n_] :=  If[PrimeQ[n], n, 1];
nmax = 10;
PM = PMatrix[nmax+1, a];
T[n_, k_] := PM[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 21 2022 *)

def ConvTriangle(dim: int, a) -> list[list[int]]:
    if callable(a): # Cache the input sequence.
        A = [a(i) for i in range(1, dim)]
    else:
        A = a
    print("In:", A)
    C = [[0 for k in range(m + 1)] for m in range(dim)]
    C[0][0] = 1
    for m in range(1, dim):
        C[m][m] = C[m - 1][m - 1] * A[0]
        for k in range(m - 1, 0, -1):
            C[m][k] = sum(A[i] * C[m - i - 1][k - 1] for i in range(m - k + 1))
    return C
from sympy import isprime, flatten
def a(n): return n if isprime(n) else 1
print(flatten(ConvTriangle(10, a)))

A126120 Catalan numbers (A000108) interpolated with 0's.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0, 16796, 0, 58786, 0, 208012, 0, 742900, 0, 2674440, 0, 9694845, 0, 35357670, 0, 129644790, 0, 477638700, 0, 1767263190, 0, 6564120420, 0, 24466267020, 0, 91482563640, 0, 343059613650, 0

Offset: 0

Philippe Deléham, Mar 06 2007

Inverse binomial transform of A001006.
The Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...].
Counts returning walks (excursions) of length n on a 1-d integer lattice with step set {+1,-1} which stay in the chamber x >= 0. - Andrew V. Sutherland, Feb 29 2008
Moment sequence of the trace of a random matrix in G=USp(2)=SU(2). If X=tr(A) is a random variable (A distributed according to the Haar measure on G) then a(n) = E[X^n]. - Andrew V. Sutherland, Feb 29 2008
Essentially the same as A097331. - R. J. Mathar, Jun 15 2008
Number of distinct proper binary trees with n nodes. - Chris R. Sims (chris.r.sims(AT)gmail.com), Jun 30 2010
-a(n-1), with a(-1):=0, n>=0, is the Z-sequence for the Riordan array A049310 (Chebyshev S). For the definition see that triangle. - Wolfdieter Lang, Nov 04 2011
See A180874 (also A238390 and A097610) and A263916 for relations to the general Bell A036040, cycle index A036039, and cumulant expansion polynomials A127671 through the Faber polynomials. - Tom Copeland, Jan 26 2016
A signed version is generated by evaluating polynomials in A126216 that are essentially the face polynomials of the associahedra. This entry's sequence is related to an inversion relation on p. 34 of Mizera, related to Feynman diagrams. - Tom Copeland, Dec 09 2019

			G.f. = 1 + x^2 + 2*x^4 + 5*x^6 + 14*x^8 + 42*x^10 + 132*x^12 + 429*x^14 + ...
From _Gus Wiseman_, Nov 14 2022: (Start)
The a(0) = 1 through a(8) = 14 ordered binary rooted trees with n + 1 nodes (ranked by A358375):
  o  .  (oo)  .  ((oo)o)  .  (((oo)o)o)  .  ((((oo)o)o)o)
                 (o(oo))     ((o(oo))o)     (((o(oo))o)o)
                             ((oo)(oo))     (((oo)(oo))o)
                             (o((oo)o))     (((oo)o)(oo))
                             (o(o(oo)))     ((o((oo)o))o)
                                            ((o(o(oo)))o)
                                            ((o(oo))(oo))
                                            ((oo)((oo)o))
                                            ((oo)(o(oo)))
                                            (o(((oo)o)o))
                                            (o((o(oo))o))
                                            (o((oo)(oo)))
                                            (o(o((oo)o)))
                                            (o(o(o(oo))))
(End)

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Ch. 49, Hemisphere Publishing Corp., 1987.

G. C. Greubel, Table of n, a(n) for n = 0..1000
V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
Martin Aigner, Catalan and other numbers: a recurrent theme, in Algebraic Combinatorics and Computer Science, a Tribute to Gian-Carlo Rota, pp.347-390, Springer, 2001.
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.
Radica Bojicic, Marko D. Petkovic and Paul Barry, Hankel transform of a sequence obtained by series reversion II-aerating transforms, arXiv:1112.1656 [math.CO], 2011.
Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020.
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv:1110.6638 [math.NT], 2011.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Kiran S. Kedlaya and Andrew V. Sutherland, HyperellipticCurves, L-Polynomials, and Random Matrices. In: Arithmetic, Geometry, Cryptography, and Coding Theory: International Conference, November 5-9, 2007, CIRM, Marseilles, France. (Contemporary Mathematics; v.487)
S. Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations, arXiv:1706.08527 [hep-th], 2017.
Eric Rowland, Pattern avoidance in binary trees, J. Comb. Theory A 117 (6) (2010) 741-758, Sec. 3.1.
Yidong Sun and Fei Ma, Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths, arXiv:1305.2015 [math.CO], 2013.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 99.
Y. Wang and Z.-H. Zhang, Combinatorics of Generalized Motzkin Numbers, J. Int. Seq. 18 (2015) # 15.2.4.

Cf. A000108, A036039, A036040, A097610, A127671, A180874, A238390, A263916.
Cf. A126216.
The unordered version is A001190, ranked by A111299.
These trees (ordered binary rooted) are ranked by A358375.
Cf. A000081, A001263, A005043, A032027, A063895, A245824.

&cat [[Catalan(n), 0]: n in [0..30]]; // Vincenzo Librandi, Jul 28 2016

with(combstruct): grammar := { BB = Sequence(Prod(a,BB,b)), a = Atom, b = Atom }: seq(count([BB,grammar], size=n),n=0..47); # Zerinvary Lajos, Apr 25 2007
BB := {E=Prod(Z,Z), S=Union(Epsilon,Prod(S,S,E))}: ZL:=[S,BB,unlabeled]: seq(count(ZL, size=n), n=0..45); # Zerinvary Lajos, Apr 22 2007
BB := [T,{T=Prod(Z,Z,Z,F,F), F=Sequence(B), B=Prod(F,Z,Z)}, unlabeled]: seq(count(BB, size=n+1), n=0..45); # valid for n> 0. # Zerinvary Lajos, Apr 22 2007
seq(n!*coeff(series(hypergeom([],[2],x^2),x,n+2),x,n),n=0..45); # Peter Luschny, Jan 31 2015
# Using function CompInv from A357588.
CompInv(48, n -> ifelse(irem(n, 2) = 0, 0, (-1)^iquo(n-1, 2))); # Peter Luschny, Oct 07 2022

a[n_?EvenQ] := CatalanNumber[n/2]; a[n_] = 0; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Sep 10 2012 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ BesselI[ 1, 2 x] / x, {x, 0, n}]]; (* Michael Somos, Mar 19 2014 *)
bot[n_]:=If[n==1,{{}},Join@@Table[Tuples[bot/@c],{c,Table[{k,n-k-1},{k,n-1}]}]];
Table[Length[bot[n]],{n,10}] (* Gus Wiseman, Nov 14 2022 *)
Riffle[CatalanNumber[Range[0,50]],0,{2,-1,2}] (* Harvey P. Dale, May 28 2024 *)

from math import comb
def A126120(n): return 0 if n&1 else comb(n,m:=n>>1)//(m+1) # Chai Wah Wu, Apr 22 2024

def A126120_list(n) :
    D = [0]*(n+2); D[1] = 1
    b = True; h = 2; R = []
    for i in range(2*n-1) :
        if b :
            for k in range(h,0,-1) : D[k] -= D[k-1]
            h += 1; R.append(abs(D[1]))
        else :
            for k in range(1,h, 1) : D[k] += D[k+1]
        b = not b
    return R
A126120_list(46) # Peter Luschny, Jun 03 2012

a(2*n) = A000108(n), a(2*n+1) = 0.
a(n) = A053121(n,0).
(1/Pi) Integral_{0 .. Pi} (2*cos(x))^n *2*sin^2(x) dx. - Andrew V. Sutherland, Feb 29 2008
G.f.: (1 - sqrt(1 - 4*x^2)) / (2*x^2) = 1/(1-x^2/(1-x^2/(1-x^2/(1-x^2/(1-... (continued fraction). - Philippe Deléham, Nov 24 2009
G.f. A(x) satisfies A(x) = 1 + x^2*A(x)^2. - Vladimir Kruchinin, Feb 18 2011
E.g.f.: I_1(2x)/x Where I_n(x) is the modified Bessel function. - Benjamin Phillabaum, Mar 07 2011
Apart from the first term the e.g.f. is given by x*HyperGeom([1/2],[3/2,2], x^2). - Benjamin Phillabaum, Mar 07 2011
a(n) = Integral_{x=-2..2} x^n*sqrt((2-x)*(2+x))/(2*Pi) dx. - Peter Luschny, Sep 11 2011
E.g.f.: E(0)/(1-x) where E(k) = 1-x/(1-x/(x-(k+1)*(k+2)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013
G.f.: 3/2- sqrt(1-4*x^2)/2 = 1/x^2 + R(0)/x^2, where R(k) = 2*k-1 - x^2*(2*k-1)*(2*k+1)/R(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 28 2013 (warning: this is not the g.f. of this sequence, R. J. Mathar, Sep 23 2021)
G.f.: 1/Q(0), where Q(k) = 2*k+1 + x^2*(1-4*(k+1)^2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 09 2014
a(n) = n!*[x^n]hypergeom([],[2],x^2). - Peter Luschny, Jan 31 2015
a(n) = 2^n*hypergeom([3/2,-n],[3],2). - Peter Luschny, Feb 03 2015
a(n) = ((-1)^n+1)*2^(2*floor(n/2)-1)*Gamma(floor(n/2)+1/2)/(sqrt(Pi)* Gamma(floor(n/2)+2)). - Ilya Gutkovskiy, Jul 23 2016
D-finite with recurrence (n+2)*a(n) +4*(-n+1)*a(n-2)=0. - R. J. Mathar, Mar 21 2021
From Peter Bala, Feb 03 2024: (Start)
a(n) = 2^n * Sum_{k = 0..n} (-2)^(-k)*binomial(n, k)*Catalan(k+1).
G.f.: 1/(1 + 2*x) * c(x/(1 + 2*x))^2 = 1/(1 - 2*x) * c(-x/(1 - 2*x))^2 = c(x^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

An erroneous comment removed by Tom Copeland, Jul 23 2016

A007440 Reversion of g.f. for Fibonacci numbers 1, 1, 2, 3, 5, ....

Original entry on oeis.org

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: 1

N. J. A. Sloane, May 24 1994

sign

Binomial transform of A104565 (reversion of Pell numbers). - Paul Barry, Mar 15 2005
From Paul Barry, Nov 03 2008: (Start)
Hankel transform of a(n) (starting 0,1,-1,..) is F(n)*(-1)^C(n+1,2).
Hankel transform of a(n+1) is (-1)^C(n+1,2).
Hankel transform of a(n+2) is F(n+2)*(-1)^C(n+2,2).
(End)
The sequence 1,1,-1,0,2,... given by 0^n + Sum_{k=0..floor((n-1)/2)} binomial(n-1,2k)*A000108(k)*(-1)^(n-k-1) has Hankel transform F(n+2)*(-1)^binomial(n+1,2). - Paul Barry, Jan 13 2009
Apart from signs, essentially the same as A343773. For odd terms, a(n) = A343773(n-1), while a(n) = -A343773(n-1) if n is even. - Gennady Eremin, May 19 2021

			G.f. = x - x^2 + 2*x^4 - 3*x^5 - x^6 + 11*x^7 - 15*x^8 - 13*x^9 + 77*x^10 - 86*x^11 - 144*x^12 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gennady Eremin, Table of n, a(n) for n = 1..800 (first 300 terms from Vincenzo Librandi)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972 (p. 16, Reversion of Series 3.6.25).
Paul Barry, Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences , J. Int. Seq. 13 (2010) #10.7.2.
Paul Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3, page 7.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Gennady Eremin, Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers, arXiv:2108.10676 [math.CO], 2021.
Index entries for reversions of series

Cf. A000045, A000032, A214649, A291535, A343773.

A007440 := n -> (-1)^(n+1)*hypergeom([1 - n/2, 1/2 -n/2], [2], -4):
seq(simplify(A007440(n)), n=1..35); # Peter Luschny, Mar 19 2018, adapted to offset Jul 21 2023
# Using function CompInv from A357588.
CompInv(25, n -> combinat:-fibonacci(n)); # Peter Luschny, Oct 07 2022

a[1] = 1; a[2] = -1; a[n_] := a[n] = (-5*(n-2)*a[n-2] + (1-2*n)*a[n-1])/(n+1); Array[a, 36] (* Jean-François Alcover, Apr 18 2014 *)
Rest[CoefficientList[Series[(-1-x+Sqrt[1+2*x+5*x^2])/(2*x),{x,0,20}],x]] (* Vaclav Kotesovec, Apr 25 2015 *)

a(n)=polcoeff((-1-x+sqrt(1+2*x+5*x^2+x^2*O(x^n)))/(2*x),n)

Vec(serreverse(x/(1-x-x^2)+O(x^66))) /* Joerg Arndt, Aug 19 2012 */

A007440 = [0, 1, -1]
for n in range(3, 801):
    A007440.append( (-(2*n-1)*A007440[-1]
      - 5*(n-2)*A007440[-2])//(n+1) )
for n in range(1, 801):
    print(n, A007440[n])  # Gennady Eremin, May 10 2021

def A007440_list(len):
    T = [0]*(len+1); T[1] = 1; R = [1]
    for n in (1..len-1):
        a,b,c = 1,0,0
        for k in range(n,-1,-1):
            r = a - b - c
            if k < n : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u; R.append(u)
    return R
A007440_list(36) # Peter Luschny, Nov 01 2012

D-finite with recurrence (n+3)*a(n+2) = -(2*n + 3)*a(n+1) - 5*n*a(n), a(1) = 1, a(2) = -1.
G.f.: A(x) = (-1 - x + sqrt(1 + 2*x + 5*x^2))/(2*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*C(k)*(-1)^(n-k), where C(n) is A000108(n). - Paul Barry, May 16 2005
a(n) = (5^((n+1)/2)*LegendreP(n-1,-1/sqrt(5)) + 5^(n/2)*LegendreP(n,-1/sqrt(5)))/(2*n+2). - Mark van Hoeij, Jul 02 2010
a(n) = 2^(-n-1)*Sum_{k=floor((n-1)/2)..n} binomial(k+1,n-k)*5^(n-k)*(-1)^k*C(k), n > 0, where C(k) is A000108. - Vladimir Kruchinin, Sep 21 2010
G.f.: (G(0)-x-1)/(x^2) = 1/G(0) where G(k) = 1 + x + x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 25 2011
From Peter Bala, Jun 23 2015: (Start)
Lucas(n) = [x^n] (x/A(x))^n for n >= 1.
-1/A(-x) = 1/x - 1 + x + x^2 - 2*x^4 - 3*x^5 + x^6 + 11*x^7 + 15*x^8 - 13*x^9 + ... is the Laurent series generating function for A214649. (End)
a(n) = (-1)^n*hypergeom([1/2 - n/2, -n/2], [2], -4). - Peter Luschny, Mar 19 2018
From Gennady Eremin, May 09 2021: (Start)
a(n) = -(-1)^n * A343773(n-1), n > 0.
G.f.: A(x) = x*B(-x), where B(x) is the g.f. of A343773.
Limit_{n->infinity} a(n)/A001006(n) = 0. (End)
G.f. A(x) satisfies A(x) + 1 + x^-1 = 1/A(x). - Gennady Eremin, May 29 2021

Extended and signs added by Olivier Gérard

Second formula adapted to offset by Vaclav Kotesovec, Apr 25 2015

A007312 Reversion of g.f. (with constant term omitted) for partition numbers.

Original entry on oeis.org

1, -2, 5, -15, 52, -200, 825, -3565, 15900, -72532, 336539, -1582593, 7524705, -36111810, 174695712, -851020367, 4171156249, -20555470155, 101787990805, -506227992092, 2527493643612, -12663916942984, 63656297034920, -320914409885850, 1622205233276889

Offset: 1

N. J. A. Sloane, Mira Bernstein

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
Index entries for reversions of series

Cf. A000041, A050393, A066398, A334315.

# Using function CompInv from A357588.
CompInv(25, n -> combinat:-numbpart(n)); # Peter Luschny, Oct 05 2022

nmax = 30; Rest[CoefficientList[InverseSeries[Series[Sum[PartitionsP[n]*x^n, {n, 1, nmax}], {x, 0, nmax}]], x]] (* Vaclav Kotesovec, Nov 11 2017 *)
Rest[CoefficientList[InverseSeries[Series[-1 + 1/QPochhammer[x],{x,0,30}],x],x]] (* Vaclav Kotesovec, Jan 18 2024 *)
(* Calculation of constant d: *) Chop[1/r /. FindRoot[{(1 + r)*QPochhammer[s, s] == 1, Log[1 - s] + QPolyGamma[0, 1, s] - (1 + r)*s*Log[s] * Derivative[0, 1][QPochhammer][s, s] == 0}, {r, -1/5}, {s, -1/2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 18 2024 *)

From Vaclav Kotesovec, Nov 11 2017: (Start)
a(n) ~ -(-1)^n * c * d^n / n^(3/2), where
d = 5.379264118840884783404842050140885100801253519243086... and
c = 0.10697042824132534557642152089737206588353695053... (End)
G.f. A(x) satisfies: A(x) = 1 - (1/(1 + x)) * Product_{k>=2} 1/(1 - A(x)^k). - Ilya Gutkovskiy, Apr 23 2020

Signs corrected Dec 24 2001

A179848 Expansion of series reversion of generating function for triangular numbers.

Original entry on oeis.org

0, 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, -50067108, 300830572, -1822766520, 11124755664, -68328754959, 422030545335, -2619631042665, 16332922290300, -102240109897695, 642312451217745, -4048514844039120, 25594403741131680

Offset: 0

Michael Somos, Jan 10 2011

sign

			G.f. = x - 3*x^2 + 12*x^3 - 55*x^4 + 273*x^5 - 1428*x^6 + 7752*x^7 - 43263*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..375

Cf. A000217.

[n le 0 select 0 else (-1)^(n+1)*Factorial(3*n)/( Factorial(n)* Factorial(2*n+1)): n in [0..30]]; // G. C. Greubel, Aug 14 2018

a:= n-> coeff(series(RootOf(A=x*(1-A)^3, A), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2012
# Using function CompInv from A357588.
0, CompInv(23, n -> n*(n+1)/2); # Peter Luschny, Oct 05 2022

CoefficientList[Series[1 - Sinh[ArcSinh[Sqrt[27*x/4]]/3]/Sqrt[3*x/4], {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)

{a(n) = if( n<1, 0, -(-1)^n * (3*n)! / (n! * (2*n+1)!) )};

{a(n) = if( n<1, 0, polcoeff( serreverse( x / (1 - x)^3 + x * O(x^n) ), n))};

{a(n) = my(A); if( n<0, 0, A = O(x); for( k = 0, n, A = x * (1 - A)^3 ); polcoeff( A, n ))};

A001764(n) = 0^n - (-1)^n * a(n).
G.f. A(x) satisfies A(x) = x * (1 - A(x))^3.
G.f.: 1 - sinh( arcsinh( sqrt( 27*x/4 ) ) / 3 ) / sqrt( 3*x/4 ).
D-finite with recurrence +2*n*(2*n+1)*a(n) +3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Mar 24 2023

A091593 Reversion of Jacobsthal numbers A001045.

Original entry on oeis.org

1, -1, -1, 5, -3, -21, 51, 41, -391, 407, 1927, -6227, -2507, 49347, -71109, -236079, 966129, 9519, -7408497, 13685205, 32079981, -167077221, 60639939, 1209248505, -2761755543, -4457338681, 30629783831, -22124857219, -206064020315, 572040039283, 590258340811

Offset: 0

Paul Barry, Jan 23 2004

Hankel transform is (-2)^C(n+1,2). - Paul Barry, Apr 28 2009

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A001263, A154825.

a := n -> hypergeom([-n,-n-1], [2], -2);
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 22 2014
# Using function CompInv from A357588.
CompInv(25, n -> (2^n - (-1)^n)/3 ); # Peter Luschny, Oct 07 2022

a[n_] := Hypergeometric2F1[-n - 1, -n - 1, 2, -2] + (n + 1)*Hypergeometric2F1[-n, -n, 3, -2]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 03 2016, after Vladimir Kruchinin *)

a(n) := hypergeometric([ -n - 1, -n - 1 ], [ 2 ], -2) + (n + 1) * hypergeometric([ -n, -n ], [ 3 ], -2); /* Vladimir Kruchinin, Oct 12 2011 */

# Algorithm of L. Seidel (1877)
def A091593_list(n) :
    D = [0]*(n+2); D[1] = 1
    R = []; b = false; h = 1
    for i in range(2*n) :
        if b :
            for k in range(1, h, 1) : D[k] += -2*D[k+1]
            R.append(D[1])
        else :
            for k in range(h, 0, -1) : D[k] += D[k-1]
            h += 1
        b = not b
    return R
A091593_list(30)  # Peter Luschny, Oct 19 2012

G.f.: (-(1+x)+sqrt(1+2*x+9*x^2))/(4*x^2). - Corrected by Seiichi Manyama, May 12 2019
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*C(k)*(-1)^(n-k)*2^k, where C(n) is A000108(n). - Paul Barry, May 16 2005
G.f.: 1/(1+x+2x^2/(1+x+2x^2/(1+x+2x^2/(1+x+2x^2/(1+ ... (continued fraction). - Paul Barry, Apr 28 2009
a(n) = Sum_{i=0..n} (2^(i)*(-1)^(n-i)*binomial(n+1,i)^2*(n-i+1)/(i+1))/(n+1). - Vladimir Kruchinin, Oct 12 2011
Conjecture: (n+2)*a(n) +(2*n+1)*a(n-1) +9*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012
a(n) = (-1)^n*hypergeom([-n/2, (1-n)/2], [2], -8). - Peter Luschny, May 28 2014
R. J. Mathar's conjecture confirmed by Maple using this hypergeom form. - Robert Israel, Sep 22 2014
a(n) = Sum_{k = 0..n} (-2)^k * (1/(n+1))*binomial(n+1, k)*binomial(n+1, k+1) =  Sum_{k = 0..n} (-2)^k * N(n+1,k+1), where N(n,k) = A001263(n,k) are the Narayana numbers. - Peter Bala, Sep 01 2023

A108524 Number of ordered rooted trees with n generators.

Original entry on oeis.org

1, 2, 7, 32, 166, 926, 5419, 32816, 203902, 1292612, 8327254, 54358280, 358769152, 2390130038, 16051344307, 108548774240, 738563388214, 5052324028508, 34727816264050, 239733805643552, 1661351898336676, 11553558997057772, 80603609263563262, 563972937201926432

Offset: 1

Christian G. Bower, Jun 07 2005

nonn

A generator is a leaf or a node with just one child.
The Hankel transform of this sequence is 3^C(n+1,2). The Hankel transform of this sequence with 1 prepended (1,1,2,7,...) is 3^C(n,2). - Paul Barry, Jan 26 2011
a(n) is the number of Schroder paths of semilength n-1 in which the (2,0)-steps that are not on the horizontal axis come in 2 colors. Example: a(3)=7 because we have HH, UDUD, UUDD, HUD, UDH, UBD, and URD, where U=(1,1), H=(2,0), D=(1,-1), while B and R are, respectively, blue and red (2,0)-steps. - Emeric Deutsch, May 02 2011
Also the compositional inverse of A327767. - Peter Luschny, Oct 08 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
Shishuo Fu, Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
Index entries for sequences related to rooted trees

Cf. A108521-A108529, A108525, A000108, A001003, A327767.

# Using function CompInv from A357588.
CompInv(24, n -> [1, -2][irem(n-1, 2) + 1]); # Peter Luschny, Oct 08 2022

Rest[CoefficientList[Series[(Sqrt[4*x^2-8*x+1]-1)/(2*x-4), {x, 0, 20}], x]] (* Vaclav Kotesovec, Oct 18 2012 *)

a(n):=sum((i*binomial(n+1,i)*sum((-1)^j*2^(n-j)*binomial(n,j)*binomial(2*n-j-i-1,n-1),j,0,n-i))/2^i,i,1,n+1)/(n*(n+1)); /* Vladimir Kruchinin, May 10 2011 */

G.f.: (sqrt(4*x^2-8*x+1) - 1)/(2*x-4).
G.f.: 1/(1-x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-... (continued fraction). - Paul Barry, Feb 10 2009
a(n) = sum(i=1..n+1, (i*C(n+1,i)*sum(j=0..n-i, (-1)^j*2^(n-j)*C(n,j)*C(2*n-j-i-1,n-1)))/2^i)/(n*(n+1)). - Vladimir Kruchinin, May 10 2011
From Gary W. Adamson, Jul 11 2011: (Start)
a(n) is upper left term in the following infinite square production matrix:
  1, 1, 0, 0, 0, ...
  1, 1, 1, 0, 0, ...
  3, 3, 1, 1, 0, ...
  9, 9, 3, 1, 1, ...
  ...
where columns are (1, 1, 3, 9, 27, 81, ...) prefaced with (0,0,1,2,3,...) zeros. (End)
Conjecture: 2*n*a(n) +(24-17*n)*a(n-1) +4*(4*n-9)*a(n-2) +4*(3-n)*a(n-3)=0. - R. J. Mathar, Nov 14 2011
G.f.: A(x)=(sqrt(4*x^2-8*x+1) - 1)/x/(2*x-4) = 1/(G(0)-x); G(k) = 1 + 2*x - 3*x/G(k+1); (continued fraction, 1-step ). - Sergei N. Gladkovskii, Jan 05 2012
a(n) ~ 3^(1/4)*(3^(3/2)-5)*(4+2*sqrt(3))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012
From Peter Bala, Mar 13 2015: (Start)
The o.g.f. A(x) satisfies the differential equation (2 - 17*x + 16*x^2 - 4*x^3)A'(x) + (7 - 4*x)*A(x) = 2 - 2*x. Mathar's conjectural recurrence above follows from this.
The o.g.f. A(x) is the series reversion of the rational function x*(1 - 2*x)/(1 - x^2). (End)

A108623 G.f. satisfies x = A(x)*(1-A(x))/(1-A(x)-(A(x))^2).

Original entry on oeis.org

1, 0, -1, -1, 1, 4, 3, -8, -23, -10, 67, 153, 9, -586, -1081, 439, 5249, 7734, -7941, -47501, -53791, 105314, 430119, 343044, -1249799, -3866556, -1730017, 13996097, 34243897, 1947204, -150962373, -296101864, 121857185, 1582561870

Offset: 1

Christian G. Bower, Jun 12 2005

sign

Row sums of inverse of Riordan array (1/(1-x-x^2), x*(1-x)/(1-x-x^2)) (Cf. A053538). - Paul Barry, Nov 01 2006

			G.f. = x - x^3 - x^4 + x^5 + 4*x^6 + 3*x^7 - 8*x^8 - 23*x^9 - 10*x^10 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Except for signs, same as A108624.

Cf. A039978, A105523, A212804.

R:=PowerSeriesRing(Rationals(), 41);
Coefficients(R!( (1+x-Sqrt(1-2*x+5*x^2))/(2*(1-x)) )); // G. C. Greubel, Oct 20 2023

# Using function CompInv from A357588.
CompInv(34, n -> ifelse(n=-1, 1, combinat:-fibonacci(n-2))); # Peter Luschny, Oct 05 2022

CoefficientList[Series[(1+x-Sqrt[1-2*x+5*x^2])/(2*x*(1-x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)
a[ n_] := SeriesCoefficient[ (1 + x - Sqrt[1 - 2 x + 5 x^2]) / (2 (1 - x)), {x, 0, n}]; (* Michael Somos, May 19 2014 *)
a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ (x - x^2) / (1 - x - x^2), {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, May 19 2014 *)

{a(n) = if( n<0, 0, polcoeff( (1 + x - sqrt(1 - 2*x + 5*x^2 + x^2 * O(x^n))) / (2 * (1 - x)), n))}; /* Michael Somos, May 19 2014 */

{b(n) = if( n<1, 0, polcoeff( serreverse( (x - x^2) / (1 - x - x^2) + x * O(x^n)), n))}; /* Michael Somos, May 19 2014 */

def A108623_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x-sqrt(1-2*x+5*x^2))/(2*(1-x)) ).list()
a=A108623_list(41); a[1:] # G. C. Greubel, Oct 20 2023

Binomial transform of A105523. - Paul Barry, Nov 01 2006
G.f.: (1+x-sqrt(1-2*x+5*x^2))/(2*(1-x)). - Paul Barry, Nov 01 2006
Conjecture: n*a(n) +3*(1-n)*a(n-1) +(7*n-18)*a(n-2) +5*(3-n)*a(n-3)=0. - R. J. Mathar, Nov 15 2011
Lim sup_{n->infinity} |a(n)|^(1/n) = sqrt(5). - Vaclav Kotesovec, Feb 08 2014
Series reversion of g.f. of A212804. - Michael Somos, May 19 2014
G.f.: x / (1 - x + x /(1 - x / (1 - x + x / (1 - x / ...)))). - Michael Somos, May 19 2014
0 = a(n)*(25*a(n+1) - 50*a(n+2) + 45*a(n+3) - 20*a(n+4)) + a(n+1)*(-20*a(n+1) + 34*a(n+2) - 44*a(n+3) + 25*a(n+4)) + a(n+2)*(12*a(n+2) - 2*a(n+3) - 6*a(n+4)) + a(n+3)*(a(n+4)) if n>=0. - Michael Somos, May 19 2014

A176025 Series reversion of eta(-x) - 1.

Original entry on oeis.org

1, 1, 2, 5, 15, 49, 169, 603, 2205, 8217, 31095, 119185, 461790, 1805810, 7117865, 28250549, 112806534, 452862663, 1826705940, 7399893522, 30092189864, 122799412699, 502709227763, 2063939448400, 8496355807149, 35061664792175

Offset: 0

Paul D. Hanna, Apr 06 2010

nonn

eta(q) is the Dedekind eta function without the q^(1/24) factor (A010815).

			G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 49*x^6 +...
eta(-x)-1 = x - x^2 - x^5 - x^7 - x^12 + x^15 + x^22 + x^26 +...
eta(-x)-1 = Sum_{n>=1} (-1)^[n/2]*x^(n(3n-1)/2)*(1 + (-x)^n).

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A010815.

# Using function CompInv from A357588.
CompInv(26, proc(n) 24*n + 1; if issqr(%) then sqrt(%); (-1)^(n + irem(iquo(% + irem(%, 6), 6), 2)) else 0 fi end); # Peter Luschny, Oct 05 2022

-InverseSeries[Series[QPochhammer[x], {x, 0, 20}]][[3]] (* Vladimir Reshetnikov, Nov 21 2015 *)

a(n)=polcoeff(serreverse(-1+eta(-x+x*O(x^n))),n)

G.f. A(x) satisfies: eta(-A(x)) = 1 + x, or, equivalently:
x = Sum_{n>=1} (-1)^[n/2] * A(x)^(n(3n-1)/2) * (1 + (-A(x))^n).
a(n) ~ c * d^n / n^(3/2), where d = 4.37926411884088478340484205014088510... and c = 0.422672515444252849172886523421828... - Vaclav Kotesovec, Nov 11 2017

cp's OEIS Frontend

A357368 Triangle read by rows. Convolution triangle of the prime indicator sequence A089026.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A126120 Catalan numbers (A000108) interpolated with 0's.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

Sage

Formula

Extensions

A007440 Reversion of g.f. for Fibonacci numbers 1, 1, 2, 3, 5, ....

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Sage

Formula

Extensions

A007312 Reversion of g.f. (with constant term omitted) for partition numbers.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A179848 Expansion of series reversion of generating function for triangular numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

Formula

A091593 Reversion of Jacobsthal numbers A001045.

Views

Author

Keywords

Comments