A108524 -id:A108524 - cp's OEIS frontend

A357585 Triangle read by rows. Inverse of the convolution triangle of A108524, the number of ordered rooted trees with n generators.

1, 0, 1, 0, 2, 1, 0, 7, 4, 1, 0, 32, 18, 6, 1, 0, 166, 92, 33, 8, 1, 0, 926, 509, 188, 52, 10, 1, 0, 5419, 2964, 1113, 328, 75, 12, 1, 0, 32816, 17890, 6792, 2078, 520, 102, 14, 1, 0, 203902, 110896, 42436, 13312, 3520, 772, 133, 16, 1

Offset: 0

Peter Luschny, Oct 08 2022

Also the matrix inverse of the signed version of A105475 with 1, 0, 0, 0, ... as column 0.

			Triangle T(n, k) starts:
[0] 1;
[1] 0,      1;
[2] 0,      2,      1;
[3] 0,      7,      4,     1;
[4] 0,     32,     18,     6,     1;
[5] 0,    166,     92,    33,     8,    1;
[6] 0,    926,    509,   188,    52,   10,  1;
[7] 0,   5419,   2964,  1113,   328,   75,  12,   1;
[8] 0,  32816,  17890,  6792,  2078,  520, 102,  14,  1;
[9] 0, 203902, 110896, 42436, 13312, 3520, 772, 133, 16, 1;

Cf. A108524 (column 1), A047891 (row sums), A105475.

InvPMatrix := proc(dim, seqfun) local k, m, M, A;
    if dim < 1 then return [] fi;
    A := [seq(seqfun(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] := M[m - 1, k - 1] -
                add(A[i+1] * M[m, k + i], i = 1..m-k) / A[1]
od od; M end:
InvPMatrix(10, n -> [1, -2][irem(n-1, 2) + 1]);

A108529 Number of asymmetric mobiles (cycle rooted trees) with n generators.

Original entry on oeis.org

1, 1, 2, 5, 16, 51, 177, 621, 2246, 8245, 30783, 116257, 443945, 1710255, 6640939, 25961690, 102105115, 403701135, 1603721999, 6397931901, 25621989760, 102965680728, 415091909292, 1678226164646, 6803121058354, 27645628327636

Offset: 1

Christian G. Bower, Jun 07 2005

nonn

A generator is a leaf or a node with just one child.

Here CHK(A(x)) = 1 - Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)), i.e., the constant 1 is included in the definition of the CHK transform. For other sequences that involve the CHK transform, the 1 is sometimes dropped; e.g., see sequence A032171. We have CHK(A(x)) = x + x^2 + 3*x^3 + 8*x^4 + 27*x^5 + 86*x^6 + 303*x^7 + 1065*x^8 + 3871*x^9 + ... - Petros Hadjicostas, Dec 05 2017

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to mobiles

Cf. A108521, A108522, A108523, A108524, A108525, A108527, A108528, A032171.

CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=x); for(n=2, n, p += x^n*polcoef(x*p + CHK(p, n), n)); Vecrev(p/x)} \\ Andrew Howroyd, Aug 31 2018

G.f. satisfies: (2-x)*A(x) = x - 1 + CHK(A(x)).
From Petros Hadjicostas, Dec 05 2017: (Start)
a(n) = (1/2)*(a(n-1) + (1/n)*Sum_{d|n} mu(d)*c(n/d)) for n>=2, where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) and a(1) = c(1) = 1.
The g.f. satisfies (2-x)*A(x) = x - Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)). (This is just a rephrasing of C. Bower's equation above.)
The auxiliary sequence (c(n): n>=1} has g.f. C(x) = Sum_{n>=1} c(n)*x^n = x*(dA/dx)/(1-A(x)) = x + 3*x^2 + 10*x^3 + 35*x^4 + 136*x^5 + 528*x^6 + 2122*x^7 + ...
(End)

A347205 a(2n+1) = a(n) for n >= 0, a(2n) = a(n) + a(n - 2^A007814(n)) for n > 0 with a(0) = 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 6, 3, 4, 1, 5, 4, 7, 3, 9, 5, 7, 2, 10, 6, 9, 3, 10, 4, 5, 1, 6, 5, 9, 4, 12, 7, 10, 3, 14, 9, 14, 5, 16, 7, 9, 2, 15, 10, 16, 6, 19, 9, 12, 3, 20, 10, 14, 4, 15, 5, 6, 1, 7, 6, 11, 5, 15, 9, 13, 4, 18, 12, 19, 7, 22, 10, 13

Offset: 0

Mikhail Kurkov, Aug 23 2021

Scatter plot might be called "Cypress forest on a windy day". - Antti Karttunen, Nov 30 2021

Antti Karttunen, Table of n, a(n) for n = 0..16384
J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6.
Index entries for sequences with interesting graphs or plots

Cf. A000108, A006318, A007814, A059894, A108524, A115197, A116867.

Similar recurrences: A124758, A243499, A284005, A329369, A341392.

a[0] = 1; a[n_] := a[n] = If[OddQ[n], a[(n - 1)/2], a[n/2] + a[n/2 - 2^IntegerExponent[n/2, 2]]]; Array[a, 100, 0] (* Amiram Eldar, Sep 06 2021 *)

a(n) = if (n==0, 1, if (n%2, a(n\2), a(n/2) + a(n/2 - 2^valuation(n/2, 2)))); \\ Michel Marcus, Sep 09 2021

a(2n+1) = a(n) for n >= 0.
a(2n) = a(n) + a(n - 2^A007814(n)) = a(2*A059894(n)) for n > 0 with a(0) = 1.
Sum_{k=0..2^n - 1} a(k) = A000108(n+1) for n >= 0.
a((4^n - 1)/3) = A000108(n) for n >= 0.
a(2^m*(2^n - 1)) = binomial(n + m, n) for n >= 0, m >= 0.
Generalization:
b(2n+1, p, q) = b(n, p, q) for n >= 0.
b(2n, p, q) = p*b(n, p, q) + q*b(n - 2^A007814(n), p, q) = for n > 0 with b(0, p, q) = 1.
Conjectured formulas: (Start)
Sum_{k=0..2^n - 1} b(k, 2, 1) = A006318(n) for n >= 0.
Sum_{k=0..2^n - 1} b(k, 2, 2) = A115197(n) for n >= 0.
Sum_{k=0..2^n - 1} b(k, 3, 1) = A108524(n+1) for n >= 0.
Sum_{k=0..2^n - 1} b(k, 3, 3) = A116867(n) for n >= 0.
b((4^n - 1)/3, p, q) is generalized Catalan number C(p, q; n). (End)
Conjecture: a(n) = T(n, wt(n)+1), a(2n) = Sum_{k=1..wt(n)+1} T(n, k) where T(2n+1, k) = T(n, k) for 1 <= k <= wt(n)+1, T(2n+1, wt(n)+2) = T(n, wt(n)+1), T(2n, k) = Sum_{i=1..k} T(n, i) for 1 <= k <= wt(n)+1 with T(0, 1) = 1. - Mikhail Kurkov, Dec 13 2024

A121575 Riordan array (-sqrt(4x^2+8x+1)+2x+2, (sqrt(4x^2+8x+1)-2x-1)/2).

Original entry on oeis.org

1, -2, 1, 6, -5, 1, -24, 24, -8, 1, 114, -123, 51, -11, 1, -600, 672, -312, 87, -14, 1, 3372, -3858, 1914, -618, 132, -17, 1, -19824, 22992, -11904, 4218, -1068, 186, -20, 1, 120426, -140991, 75183, -28383, 8043, -1689, 249, -23, 1, -749976, 884112, -481704, 190347, -58398, 13929, -2508, 321, -26, 1

Offset: 0

Paul Barry, Aug 08 2006

First column is (-1)^n*A054872(n). Row sums are a signed version of A108524. Inverse of generalized Delannoy triangle A121574. Unsigned triangle is A121576.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -2, -1, -3, -1, -3, -1, -3, -1, -3, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 09 2006

			Triangle begins
     1;
    -2,    1;
     6,   -5,    1;
   -24,   24,   -8,   1;
   114, -123,   51, -11,   1;
  -600,  672, -312,  87, -14, 1;

G. C. Greubel, Rows n=0..100 of triangle, flattened

T:=Flat(List([0..9],n->List([0..n],k->(-1)^(n-k)*Sum([0..n-k],i->Binomial(n,i)*Binomial(2*n-k-i,n)*(4-9*i+3*i^2-6*(i-1)*n+2*n^2)/((n-i+2)*(n-i+1))*2^i)/2))); # Muniru A Asiru, Nov 02 2018

[[(-1)^(n-k)*(&+[ 2^j*Binomial(n,j)*Binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1))/2: j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 02 2018

Flatten[Table[(-1)^(n-k)*Sum[Binomial[n, i] Binomial[2*n-k-i, n]*(4-9*i + 3*i^2 -6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i, {i, 0, n-k}]/2, {n, 0, 10}, {k, 0, n}]] (* G. C. Greubel, Nov 02 2018 *)

for(n=0,10, for(k=0,n, print1((-1)^(n-k)*sum(j=0, n-k, 2^j*binomial(n,j) *binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1)))/2, ", "))) \\ G. C. Greubel, Nov 02 2018

T(n,k) = (-1)^(n-k)*(1/2)*Sum_{i=0..n-k} binomial(n,i) * binomial(2*n-k-i,n)*(4 - 9*i + 3*i^2 - 6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i. - G. C. Greubel, Nov 02 2018

A108528 Number of increasing mobiles (cycle rooted trees) with n generators.

Original entry on oeis.org

1, 2, 10, 92, 1216, 20792, 435520, 10793792, 308874016, 10021509632, 363509706880, 14576530558592, 640275236943616, 30573223563625472, 1576805482203235840, 87353392124392020992, 5173324070004374358016, 326160898887563325581312, 21810458629345555407462400

Offset: 1

Christian G. Bower, Jun 07 2005

nonn

In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for sequences related to mobiles

Cf. A108521, A108522, A108523, A108524, A108525, A108526, A108527, A108528, A108529, A029768.

Rest[CoefficientList[InverseSeries[Series[Log[(1+x)*Sqrt[1-x^2]], {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

{a(n)=n!*polcoeff(serreverse(log((1+x)*sqrt(1-x^2+O(x^(n+2))))),n)} \\ Paul D. Hanna, Sep 11 2010

E.g.f. satisfies 2*A(x) = x - 1 + A'(x) - log(1-A(x)).
From Paul D. Hanna, Sep 11 2010: (Start)
E.g.f. satisfies: (1+A(x))*sqrt(1-A(x)^2) = exp(x).
E.g.f.: A(x) = Series_Reversion[ log((1+x)*sqrt(1-x^2)) ]. (End)
a(n) ~ 2^(n-2) * sqrt(3) * n^(n-1) / (exp(n) * (log(27/16))^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014

A265435 Riordan array (1, x*f(x)) where f(x) is the g.f. of A007564.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 19, 9, 3, 1, 0, 100, 46, 15, 4, 1, 0, 562, 254, 82, 22, 5, 1, 0, 3304, 1476, 474, 128, 30, 6, 1, 0, 20071, 8893, 2847, 773, 185, 39, 7, 1, 0, 124996, 55046, 17587, 4796, 1165, 254, 49, 8, 1, 0, 793774, 347922, 111006, 30378, 7461, 1665, 336, 60, 9, 1

Offset: 0

Philippe Deléham, Dec 09 2015

Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1, 3, 1, 3, 1, 3, 1, 3, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

			Triangle begins:
  1
  0,   1
  0,   1,  1
  0,   4,  2,  1
  0,  19,  9,  3, 1
  0, 100, 46, 15, 4, 1
Production matrix begins:
  0,  1
  0,  1,  1
  0,  3,  1, 1
  0,  9,  3, 1, 1
  0, 27,  9, 3, 1, 1
  0, 81, 27, 9, 3, 1, 1

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

Cf. A007564, A108524 (row sums).

f[x_]:=(1+2*x-Sqrt[1-8*x+4*x^2])/(6*x); T[n_,k_]:=SeriesCoefficient[(x*f[x])^k,{x,0,n}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Feb 05 2025 *)

cp's OEIS Frontend

A357585 Triangle read by rows. Inverse of the convolution triangle of A108524, the number of ordered rooted trees with n generators.

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A108529 Number of asymmetric mobiles (cycle rooted trees) with n generators.

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A347205 a(2n+1) = a(n) for n >= 0, a(2n) = a(n) + a(n - 2^A007814(n)) for n > 0 with a(0) = 1.

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A121575 Riordan array (-sqrt(4*x^2+8*x+1)+2*x+2, (sqrt(4*x^2+8*x+1)-2*x-1)/2).

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Magma

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A108528 Number of increasing mobiles (cycle rooted trees) with n generators.

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A265435 Riordan array (1, x*f(x)) where f(x) is the g.f. of A007564.

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Mathematica

A121575 Riordan array (-sqrt(4x^2+8x+1)+2x+2, (sqrt(4x^2+8x+1)-2x-1)/2).