keyword:eigen - cp's OEIS frontend

A055340 Triangle read by rows: number of mobiles (circular rooted trees) with n nodes and k leaves.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 6, 8, 4, 1, 0, 1, 9, 19, 16, 5, 1, 0, 1, 12, 37, 46, 25, 6, 1, 0, 1, 16, 66, 118, 96, 40, 7, 1, 0, 1, 20, 110, 260, 300, 184, 56, 8, 1, 0, 1, 25, 172, 527, 811, 688, 318, 80, 9, 1, 0, 1, 30, 257, 985, 1951, 2178, 1408, 524, 105, 10, 1, 0

Offset: 1

Christian G. Bower, May 14 2000

			G.f. = x^(y + x*y + x^2*(y + y^2) + x^3*(y + 2*y^2 + y^3) + x^4*(y + 4*y^2 + 3*y^3 + y^4) + ...).
n\k 1  2  3  4  5  6  7  8
--:-- -- -- -- -- -- -- --
1:  1
2:  1  0
3:  1  1  0
4:  1  2  1  0
5:  1  4  3  1  0
6:  1  6  8  4  1  0
7:  1  9 19 16  5  1  0
8:  1 12 37 46 25  6  1  0

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
C. G. Bower, Transforms (2)
Index entries for sequences related to mobiles

Row sums give A032200.
Columns 2..8 are A002620(n-1), A055341, A055342, A055343, A055344, A055345, A055346.
Cf. A055347, A055348, A055349, A055356, A055363.

m = 13; A[, ] = 0;
Do[A[x_, y_] = x (y - Sum[EulerPhi[i]/i Log[1 - A[x^i, y^i]], {i, 1, m}]) + O[x]^m + O[y]^m // Normal, {m}];
Join[{1}, Append[CoefficientList[#/y, y], 0]& /@ Rest @ CoefficientList[ A[x, y]/x, x]] // Flatten (* Jean-François Alcover, Oct 02 2019 *)

{T(n, k) = my(A = O(x)); if(k<1 || k>n, 0, for(j=1, n, A = x*y - x*sum(i=1, j, eulerphi(i)/i * log(1 - subst( subst( A + x * O(x^min(j, n\i)), x, x^i), y, y^i) ) )); polcoeff( polcoeff(A, n), k))}; /* Michael Somos, Aug 24 2015 */

G.f. satisfies A(x, y)=xy+x*CIK(A(x, y))-x. Shifts up under CIK transform.
G.f. satisfies A(x, y) = x*(y - Sum_{i>0} phi(i)/i * log(1 - A(x^i, y^i))). - Michael Somos, Aug 24 2015
Sum_k T(n, k) = A032200(n). - Michael Somos, Aug 24 2015

A071673 Sequence a(n) obtained by setting a(0) = 0; then reading the table T(x,y)=a(x)+a(y)+1 in antidiagonal fashion.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 3, 4, 4, 5, 4, 4, 4, 5, 5, 5, 5, 4, 4, 5, 6, 5, 6, 5, 4, 4, 5, 6, 6, 6, 6, 5, 4, 5, 5, 6, 6, 7, 6, 6, 5, 5, 5, 6, 6, 6, 7, 7, 6, 6, 6, 5, 4, 6, 7, 6, 7, 7, 7, 6, 7, 6, 4, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 5, 5, 5, 6, 6, 7, 8, 7, 7, 7, 8, 7, 6, 6, 5, 6, 6, 7, 6, 8, 8, 7, 7, 8, 8, 6, 7, 6, 6

Offset: 0

Antti Karttunen, May 30 2002

The fixed point of RASTxx transformation. The repeated applications of RASTxx starting from A072643 seem to converge toward this sequence. Compare to A072768 from which this differs first time at the position n=37, where A072768(37) = 4, while A071673(37) = 5.
Each term k occurs A000108(k) times, and maximal position where k occurs is A072638(k).
The size of each Catalan structure encoded by the corresponding terms in triangles A071671 & A071672 (i.e., the number of digits / 2), as obtained with the global ranking/unranking scheme presented in A071651-A071654.

			The first 15 rows of this irregular triangular table:
               0,
               1,
              2, 2,
             3, 3, 3,
            3, 4, 4, 3,
           4, 4, 5, 4, 4,
          4, 5, 5, 5, 5, 4,
         4, 5, 6, 5, 6, 5, 4,
        4, 5, 6, 6, 6, 6, 5, 4,
       5, 5, 6, 6, 7, 6, 6, 5, 5,
      5, 6, 6, 6, 7, 7, 6, 6, 6, 5,
     4, 6, 7, 6, 7, 7, 7, 6, 7, 6, 4,
    5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 5, 5,
   5, 6, 6, 7, 8, 7, 7, 7, 8, 7, 6, 6, 5,
  6, 6, 7, 6, 8, 8, 7, 7, 8, 8, 6, 7, 6, 6
etc.
E.g., we have
  a(1) = T(0,0) = a(0) + a(0) + 1 = 1,
  a(2) = T(1,0) = a(1) + a(0) + 1 = 2,
  a(3) = T(0,1) = a(0) + a(1) + 1 = 2,
  a(4) = T(2,0) = a(2) + a(0) + 1 = 3, etc.

Antti Karttunen, Table of n, a(n) for n = 0..10440 (rows 0..144 of the triangle, flattened)
N. J. A. Sloane, Transforms (Maple code for RASTxx transform)

Same triangle computed modulo 2: A071674.
Permutations of this sequence include: A072643, A072644, A072645, A072660, A072768, A072789, A075167.
Cf. also A000108, A002260, A004736, A002262, A025581, A072638.

up_to = 105;
A002260(n) = (n-binomial((sqrtint(8*n)+1)\2, 2)); \\ From A002260
A004736(n) = (1-n+(n=sqrtint(8*n)\/2)*(n+1)\2); \\ From A004736
A071673list(up_to) = { my(v=vector(1+up_to)); v[1] = 0; for(n=1,up_to,v[1+n] = 1 + v[A004736(n)] + v[A002260(n)]); (v); };
v071673 = A071673list(up_to);
A071673(n) = v071673[1+n]; \\ Antti Karttunen, Aug 17 2021

(define (A071673 n) (cond ((zero? n) n) (else (+ 1 (A071673 (A025581 (-1+ n))) (A071673 (A002262 (-1+ n)))))))

a(0) = 0, a(n) = 1 + a(A025581(n-1)) + a(A002262(n-1)) = 1 + a(A004736(n)) + a(A002260(n)).

Self-referential definition added Jun 03 2002

Term a(0) = 0 prepended and the Example-section amended by Antti Karttunen, Aug 17 2021

A075271 a(0) = 1 and, for n >= 1, (BM)a(n) = 2*a(n-1), where BM is the BinomialMean transform.

Original entry on oeis.org

1, 3, 17, 211, 5793, 339491, 41326513, 10282961907, 5181436229441, 5258784071302723, 10717167529963833681, 43779339268428732008723, 358114286723184561034838497, 5862685570087914880854259126371, 192026370558313054275618817346778353

Offset: 0

John W. Layman, Sep 11 2002

The BinomialMean transform BM is defined by (BM)a(n) = (M^n)a(0) where (M)a(n) is the mean (a(n) + a(n+1))/2, or, alternatively, by (BM)a(n) = (Sum_{k=0..n} binomial(n,k)*a(k))/(2^n).

The BinomialMean transform of this sequence is given in A075272.

			Given that a(0)=1 and a(1)=3. Then (BM)a(2) = (1 + 2*3 + a(2))/4 = 2a(1) = 6, hence a(2)=17.

Alois P. Heinz, Table of n, a(n) for n = 0..50
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A075272, A048651.

iBM:= proc(p) proc(n) option remember; add(2^(k)*p(k)*(-1)^(n-k) *binomial(n, k), k=0..n) end end: a:= iBM(aa): aa:= n-> `if`(n=0, 1, 2*a(n-1)): seq(a(n), n=0..16);  # Alois P. Heinz, Sep 09 2008

iBM[p_] := Module[{proc}, proc[n_] := proc[n] = Sum[2^k*p[k]*(-1)^(n-k) * Binomial[n, k], {k, 0, n}]; proc]; a = iBM[aa]; aa[n_] := If[n == 0, 1, 2*a[n-1]]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 08 2015, after Alois P. Heinz *)
Table[Sum[QFactorial[k, 2] Binomial[n + 1, k]/2, {k, 0, n + 1}], {n, 0, 15}] (* Vladimir Reshetnikov, Oct 16 2016 *)

O.g.f. as a continued fraction: A(x) = 1/(1 + x - 2^2*x/(1 - 2*(2 - 1)^2*x/(1 + x - 2^4*x/(1 - 2*(2^2 - 1)^2*x/(1 + x - 2^6*x/(1 - 2*(2^3 - 1)^2*x/(1 + x - 2^8*x/(1 - 2*(2^4 - 1)^2*x/(1 + x - ... ))))))))). Cf. A075272. - Peter Bala, Nov 10 2017

a(n) ~ A048651 * 2^(n*(n+3)/2). - Vaclav Kotesovec, Jun 09 2025

More terms from Alois P. Heinz, Sep 09 2008

A088717 G.f. satisfies: A(x) = 1 + xA(x)^2A(x*A(x)^2).

Original entry on oeis.org

1, 1, 3, 14, 84, 596, 4785, 42349, 406287, 4176971, 45640572, 526788153, 6392402793, 81247489335, 1078331283648, 14907041720241, 214187010762831, 3192620516380376, 49287883925072010, 786925082232918304, 12976244331714379149, 220728563512663520510

Offset: 0

Paul D. Hanna, Oct 12 2003 and Mar 10 2007

			G.f.: A(x) = 1 + x + 3*x^2 + 14*x^3 + 84*x^4 + 596*x^5 + 4785*x^6 +...
G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x*A*B;
B = A*(1 + x*B*C);
C = B*(1 + x*C*D);
D = C*(1 + x*D*E);
E = D*(1 + x*E*F); ...
where B(x) = A(x)*A(x*A(x)^2), C(x) = A(x)*B(x*A(x)^2),  D(x) = A(x)*C(x*A(x)^2), ...
Expansions of a few of the functions described above begin:
B(x) = 1 + 2*x + 9*x^2 + 55*x^3 + 402*x^4 + 3328*x^5 + 30312*x^6 +...
C(x) = 1 + 3*x + 18*x^2 + 138*x^3 + 1218*x^4 + 11856*x^5 + 124467*x^6 +...
D(x) = 1 + 4*x + 30*x^2 + 278*x^3 + 2901*x^4 + 32846*x^5 + 395913*x^6 +...
ALTERNATE GENERATING METHOD.
Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy:
A = 1 + x*A^2*B,
B = 1 + x*(A*B)^2*C,
C = 1 + x*(A*B*C)^2*D,
D = 1 + x*(A*B*C*D)^2*E, etc.,
then B(x) = A(x*A(x)^2), C(x) = B(x*A(x)^2), D(x) = C(x*A(x)^2), etc.,
where A(x) = 1 + x*A(x)^2*A(x*A(x)^2) is the g.f. of this sequence.
Expansions of a few of the functions described above begin:
B(x) = 1 + x + 5*x^2 + 33*x^3 + 256*x^4 + 2223*x^5 + 21058*x^6 +...
C(x) = 1 + x + 7*x^2 + 60*x^3 + 578*x^4 + 6045*x^5 + 67421*x^6 +...
D(x) = 1 + x + 9*x^2 + 95*x^3 + 1098*x^4 + 13526*x^5 + 175176*x^6 +...

Cf. A215505, A215506, A215507.

Cf. A182224, A002449, A030266, A087949, A088714, A091713, A120971.

m = 22; A[] = 0; Do[A[x] = 1 + x A[x]^2 A[x A[x]^2] + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 07 2019 *)

{a(n)=local(A=1+x);for(i=0,n,A=1+x*A^2*subst(A,x,x*A^2+x*O(x^n)));polcoeff(A,n)}

/* a(n) = [x^n] (1+x*A(x))^(2*n+1)/(2*n+1): */
{a(n)=local(A=1+x); for(i=0, n, A=sum(m=0,n,polcoeff((1+x*A+x*O(x^m))^(2*m+1)/(2*m+1),m)*x^m)+x*O(x^n));polcoeff(A,n)}

{a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(2*n+m, k)/(2*n+m)*a(n-k, k))))}

a(n) = coefficient of x^n in (1+x*A(x))^(2*n+1)/(2*n+1) where A(x) = Sum_{n=0} a(n)*x^n.
Recurrence:
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then
a(n,m) = Sum_{k=0..n} m*C(2n+m,k)/(2n+m) * a(n-k,k). [Paul D. Hanna, Dec 16 2010]
G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n)*F(x,n+1)) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007

A144042 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Euler transform applied k times.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 8, 9, 1, 1, 5, 13, 25, 20, 1, 1, 6, 19, 51, 77, 48, 1, 1, 7, 26, 89, 197, 258, 115, 1, 1, 8, 34, 141, 410, 828, 871, 286, 1, 1, 9, 43, 209, 751, 2052, 3526, 3049, 719, 1, 1, 10, 53, 295, 1260, 4337, 10440, 15538, 10834, 1842, 1, 1, 11, 64

Offset: 1

Alois P. Heinz, Sep 08 2008

			Square array begins:
    1,   1,    1,     1,     1,     1,      1,      1, ...
    1,   1,    1,     1,     1,     1,      1,      1, ...
    2,   3,    4,     5,     6,     7,      8,      9, ...
    4,   8,   13,    19,    26,    34,     43,     53, ...
    9,  25,   51,    89,   141,   209,    295,    401, ...
   20,  77,  197,   410,   751,  1260,   1982,   2967, ...
   48, 258,  828,  2052,  4337,  8219,  14379,  23659, ...
  115, 871, 3526, 10440, 25512, 54677, 106464, 192615, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Columns k=1-10 give: A000081, A007563, A144035, A144036, A144037, A144038, A144039, A144040, A144041, A305727.
Rows n=2-4 give: A000012, A000027, A034856.
Main diagonal gives A305725.
Cf. A316101.

etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1,
        add(add(d*p(d), d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
      end end:
g:= proc(k) option remember; local b, t; b[0]:= j->
      `if`(j<2, j, b[k](j-1)); for t to k do
       b[t]:= etr(b[t-1]) od: eval(b[0])
    end:
A:= (n, k)-> g(k)(n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # revised Alois P. Heinz, Aug 27 2018

etr[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; A[n_, k_] := Module[{a, b, t}, b[1] = etr[a]; For[t = 2, t <= k, t++, b[t] = etr[b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m-1]]]; a[n]]; Table[Table[A[n, 1 + d-n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Dec 20 2013, translated from Maple *)

A000360 Distribution of nonempty triangles inside a fractal rep-4-tile.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 0, 2, 2, 2, 1, 3, 1, 2, 1, 2, 2, 4, 1, 4, 3, 3, 1, 4, 2, 4, 2, 3, 2, 3, 0, 3, 3, 4, 2, 6, 3, 5, 2, 5, 4, 7, 2, 6, 4, 4, 1, 5, 3, 6, 3, 6, 4, 6, 1, 5, 4, 5, 2, 5, 2, 3, 1, 3, 3, 6, 2, 7, 5, 6, 2, 8, 5, 9, 4, 8, 5, 7, 1, 7, 6, 9, 4, 11, 6, 9, 3, 8, 6, 10, 3, 8, 5, 5, 1, 6, 4, 8, 4, 9, 6, 9, 2

Offset: 0

M. Jeremie Lafitte (Levitas)

a(n) = Running count of congruent nonempty triangles along lines perpendicular to the base of the Gosper-Lafitte triangle.
Also, a(n) = Sum of the coefficients of the terms with an even exponent in the Stern polynomial B(n+1,t), or in other words, the sum of the even-indexed terms (the leftmost is at index 0) of the irregular triangle A125184, starting from its second row. - Antti Karttunen, Apr 20 2017
Back in May 1995, it was proved that a(n) = modulo 3 mapping, (+1,-1,+0)/2, of the Stern-Brocot sequence A002487, dropping its 1st term. - M. Jeremie Lafitte (Levitas), Apr 23 2017

M. J. Lafitte, Sur l'Effet Noah en Géométrie, rapport à l'INPI, mars 1995.

T. D. Noe, Table of n, a(n) for n = 0..10000
S. Klavzar, U. Milutinovic and C. Petr, Stern polynomials, Adv. Appl. Math. 39 (2007) 86-95.
M. J. Lafitte, Ensembles Auto-Similaires d'Intérieur Non-Vide, Preprint Hiver 1997, Chaire de Géometrie, Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Switzerland. [Cached copy, with permission]
M. J. Lafitte, Fractal triangle underlying A000360, A000361, A000876
M. J. Lafitte, Notes on A000360, A000361, A000876 [Based on a latex file sent by M. Jeremie Lafitte (Levitas) to NJAS in 1995 - see file of emails below]
M. J. Lafitte, Latex source for the pdf file [Sent by MJL to NJAS in 1995 - see file of emails below]
M. J. Lafitte and N. J. A. Sloane, Emails, 1995-2007 (The three sequences mentioned in this correspondence are now A000360, A000361, A000876)

Cf. A002487, A000361, A000876.
Cf. A001222, A057078, A125184, A284553, A284556, A284565 (bisection).
Cf. also mutual recurrence pair A287729, A287730.

import Data.List (transpose)
a000360 n = a000360_list !! n
a000360_list = 1 : concat (transpose
   [zipWith (+) a000360_list $ drop 2 a057078_list,
    zipWith (+) a000360_list $ tail a000360_list])
-- Reinhard Zumkeller, Mar 22 2013
(Scheme, with memoization-macro definec):
(define (A000360 n) (A000360with_prep_0 (+ 1 n)))
(definec (A000360with_prep_0 n) (cond ((<= n 1) n) ((even? n) (A284556 (/ n 2))) (else (+ (A000360with_prep_0 (/ (- n 1) 2)) (A000360with_prep_0 (/ (+ n 1) 2))))))
(definec (A284556 n) (cond ((<= n 1) 0) ((even? n) (A000360with_prep_0 (/ n 2))) (else (+ (A284556 (/ (- n 1) 2)) (A284556 (/ (+ n 1) 2))))))
;; Antti Karttunen, Apr 07 2017

a[0] = 1; a[n_?EvenQ] := a[n] = a[n/2] + a[n/2-1]; a[n_?OddQ] := a[n] = a[(n-1)/2] - Mod[(n-1)/2-1, 3] + 1; Table[a[n], {n, 0, 103}] (* Jean-François Alcover, Jan 20 2015, after Ralf Stephan *)

a(n) = if(n==0, 1, if(n%2, a((n - 1)/2) - ((n - 1)/2 - 1)%3 + 1, a(n/2) + a(n/2 - 1))); \\ Indranil Ghosh, Apr 20 2017

a(3n) = (A002487(3n+1) + 1)/2, a(3n+1) = (A002487(3n+2) - 1)/2, a(3n+2) = A002487(3n+3)/2. - M. Jeremie Lafitte (Levitas), Apr 23 2017
a(0) = 1, a(2n) = a(n) + a(n-1), a(2n+1) = a(n) + 1 - (n-1 mod 3). - Ralf Stephan, Oct 05 2003; Note: according to Ralf Stephan, this formula was found empirically. It follows from that found for the Stern-Brocot sequence A002487 and the first formula. - Antti Karttunen, Apr 21 2017, M. Jeremie Lafitte (Levitas), Apr 23 2017
From Antti Karttunen, Apr 07 2017: (Start)
Ultimately equivalent to the above formulae, we have:
a(n) = A001222(A284553(1+n)).
a(n) = A002487(1+n) - A284556(1+n).
a(n) = b(1+n), with b from a mutual recurrence pair: b(0) = 0, b(1) = 1, b(2n) = c(n), b(2n+1) = b(n) + b(n+1), c(0) = c(1) = 0, c(2n) = b(n), c(2n+1) = c(n) + c(n+1). [c(n) = A284556(n), b(n)+c(n) = A002487(n).]
(End)

More terms from David W. Wilson, Aug 30 2000

Original relation to the Stern-Brocot sequence A002487 reformulated by M. Jeremie Lafitte (Levitas), Apr 23 2017

A001005 Number of ways of partitioning n points on a circle into subsets only of sizes 2 and 3.

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 8, 21, 42, 96, 222, 495, 1177, 2717, 6435, 15288, 36374, 87516, 210494, 509694, 1237736, 3014882, 7370860, 18059899, 44379535, 109298070, 269766655, 667224480, 1653266565, 4103910930, 10203669285, 25408828065, 63364046190, 158229645720, 395632288590, 990419552730

Offset: 0

N. J. A. Sloane

a(n) is also the number of rooted trees on n nodes such that each node has 0, 2, or 3 children. - Patrick Devlin, Mar 04 2012

a(n) is the number of Motzkin paths that have no flatsteps (F) at ground level and avoid a factor of the form FMF with M a Motzkin path (possibly empty). For example, a(5) = 5 counts UDUFD, UFDUD, UFUDD, UUDFD, UUFDD but not UFFFD. Proof: Such a path can have at most one flatstep at height 1 before the first return to ground level or else the first component will contain an FMF. Hence, with a dot denoting concatenation, such a path is either empty or has the form U.P1.D.P2 or the form U.P1.F.P2.D.P3 where P1, P2, P3 are all paths of the type being counted. Hence the gf F(x) = 1 + x^2 + x^3 + 2*x^4 + ... satisfies F = 1 + x^2*F^2 + x^3*F^3. - David Callan, Nov 21 2021

			a(7)=21: 7 rotations of [12][34][567], 7 rotations of [12][45][367], 7 rotations of [12][37][456]. - _Len Smiley_, Jun 18 2005
From _Wolfdieter Lang_, Nov 05 2018: (Start)
a(7) = b(8)/8, where b(8) = (d^7/dx^7)((1 + x^2 + x^3)^8)/7! evaluated for x = 0, which is 168, and 168/8 = 21.
a(7) =(1/8)*8!/((8-(2+1))!*2!*1!) =(1/8)*8!/(5!*2!)= 168/8 = 21, from the only solution [e2, e3] = [2, 1] of 2*e2 + 3*e3 = 7. (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 396
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
Len Smiley, a(7) and a(8)
Index entries for sequences related to rooted trees

Cf. A321201.

a:=proc(n::nonnegint) local k,j; a(n):=0; for k from 0 to floor(n/2) do for j from 0 to floor(n/3) do if (2*k+3*j=n) then a(n):=a(n)+(n)!/(k!*j!*(n-k-j+1)!) fi od od; print(a(n)) end proc; seq(a(i),i=0..30); # Len Smiley, Jun 18 2005
A001005 := n -> ifelse(n=0, 1, add(binomial(n, k-1)*binomial(k, n-2*k)/k, k = 1 + iquo(n-1,3)..iquo(n,2))): seq(A001005(n), n=0..35); # Peter Luschny, Oct 18 2022

Table[Sum[(n)!/(k!*j!*(n - k - j + 1)!) * KroneckerDelta[2*k + 3*j - n], {k, 0, Floor[n/2]}, {j, 0, Floor[n/3]}], {n, 0, 20}] (* Ricardo Bittencourt, Jun 09 2013 *)
CoefficientList[ InverseSeries[x/(1+x^2+x^3) + O[x]^66]/x, x] (* Jean-François Alcover, Feb 15 2016, after Joerg Arndt*)

Vec(serreverse(x/(1+x^2+x^3)+O(x^66))/x) /* Joerg Arndt, Aug 19 2012 */

G.f. for a(n-1), with a(-1) = 0, satisfies A(x)=x*(1+A(x)^2+A(x)^3). - Christian G. Bower, Dec 15 1999
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..floor(n/3)} n!/(k!*j!*(n-k-j+1)!)*[2*k+3*j=n]. - Len Smiley, Jun 18 2005
Recurrence: 2*(n+1)*(2*n+3)*(26*n+1)*a(n) = -(n-1)*(26*n^2 + 53*n + 18)*a(n-1) + 6*(n-1)*(78*n^2 + 42*n - 25)*a(n-2) + 31*(n-2)*(n-1)*(26*n+27)*a(n-3). - Vaclav Kotesovec, Aug 14 2013
a(n) ~ c*d^n/n^(3/2), where d = ((6371-624*sqrt(78))^(1/3)+(6371+624*sqrt(78))^(1/3)-1)/12 = 2.610718613276039349818649... is the root of the equation 4d^3 + d^2 - 18d - 31 = 0 and c = d^2 / (2*sqrt(Pi)*sqrt(1 + 3*d + sqrt(1 + 3*d))) = 0.559628309722556021604897336422272... - Vaclav Kotesovec, Aug 14 2013, updated Jun 27 2018
a(n) = Sum_{k=1..floor(n/2)} C(n,k-1)*C(k,n-2k)/k, n > 0. - Michael D. Weiner, Mar 02 2015
From Wolfdieter Lang, Nov 05 2018: (Start)
The o.g.f of a(n) is G(x) = F^[-1](x)/x, where F^[-1](x) is the compositional inverse of F(y) = y/(1 + y^2 + y^3), that is F(F^[-1](x)) = x, identically. (Compare this with the g.f. given above, and see the Pari and Mathematica programs below.)
a(n) = b(n+1)/(n+1), for n >= 0, where b(n+1) is the coefficient of x^n of (1 + x^2 + x^3)^(n+1). This follows from the Lagrange inversion series for G(x) = F^[-1](x)/x.
a(n) = (1/(n+1))*(Sum_{2*e2 + 3*e3 = n} (n+1)!/(n+1 - (e2 + e3))!*e2!*e3!) (from the multinomial formula for (x1 + x2 + x3)^(n+1)). For the solutions of 2*e2 + 3*e3 = n see the array A321201.
(End)

More terms from Christian G. Bower, Dec 15 1999

A006964 Number of directed rooted trees with n nodes.

Original entry on oeis.org

1, 3, 15, 82, 495, 3144, 20875, 142773, 1000131, 7136812, 51702231, 379234623, 2810874950, 21020047557, 158398829121, 1201617201230, 9169060501023, 70329406653879, 541949364313821, 4193569906262874, 32571403998781956, 253842927519362734, 1984442128649393178

Offset: 1

Simon Plouffe

Also rooted trees with n nodes and 3-colored non-root nodes. - Christian G. Bower, Apr 15 1998

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

Alois P. Heinz, Table of n, a(n) for n = 1..500
Loïc Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA], 2018.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 439
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Mathh
. Québec, Vol. 16, No. 1 (1992) pp. 53-80.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A038059.

Column k=3 of A242249.

with(numtheory): a:= proc(n) option remember; `if`(n<2, n, (add(add(d*a(d), d=divisors(j)) *a(n-j)*3, j=1..n-1))/(n-1)) end: seq(a(n), n=1..30); # Alois P. Heinz, Sep 06 2008

a[n_] := a[n] = If[n<2, n, (Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j]*3, {j, 1, n-1}])/(n-1)]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)

a(n+1) has g.f.: Product_{n>=1} (1 - x^3*a(n))^-1.
a(n) ~ c * d^n / n^(3/2), where d = 8.356026879295995368276069578708912..., c = 0.13645899548680457355557420025756... . - Vaclav Kotesovec, Aug 20 2014
G.f. A(x) satisfies: A(x) = x*exp(3*Sum_{k>=1} A(x^k)/k). - Ilya Gutkovskiy, Mar 19 2018

Extended by Christian G. Bower, Apr 15 1998

A010739 Shifts 2 places left under inverse binomial transform.

Original entry on oeis.org

1, 2, 1, 1, -2, 3, -7, 22, -71, 231, -794, 2945, -11679, 48770, -212823, 969221, -4605674, 22802431, -117322423, 625743878, -3452893503, 19684083947, -115787084242, 701935339725, -4380330298815, 28105726916034, -185229395693615, 1252696143653513

Offset: 0

N. J. A. Sloane, Jonas Wallgren

Alois P. Heinz, Table of n, a(n) for n = 0..650
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A010741, A010743, A010745, A010747.

a:= proc(n) option remember; (m-> `if`(m<0, 2^n,
      add(a(m-j)*binomial(m, j)*(-1)^j, j=0..m)))(n-2)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 02 2022

a[n_] := a[n] = Function[m, If[m<0, 2^n,
   Sum[a[m-j]*Binomial[m, j]*(-1)^j, {j, 0, m}]]][n-2];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jul 24 2022, after Alois P. Heinz *)

G.f. A(x) satisfies: A(x) = 1 + 2*x + x^2*A(x/(1 + x))/(1 + x). - Ilya Gutkovskiy, Feb 02 2022

A027826 Inverse binomial transform of a_0 = 1, a_1, a_2, etc. is a_0, 0, a_1, 0, a_2, 0, etc.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 50, 120, 290, 706, 1732, 4280, 10644, 26612, 66824, 168384, 425481, 1077529, 2733746, 6945812, 17669149, 44994345, 114682042, 292544200, 746831570, 1907983346, 4877966628, 12479883736, 31951158024, 81858610968, 209865391600, 538408691456

Offset: 0

Allan C. Wechsler

The self-convolution equals A051163. - Paul D. Hanna, Nov 23 2004

Equals row sums of triangle A152193. - Gary W. Adamson, Nov 28 2008

			Array of successive differences (col. 1 is the inverse binomial transform):
1, 1,  2,  4,  9, 21, 50, ...
0, 1,  2,  5, 12, 29, 70, ...
1, 1,  3,  7, 17, 41, ...
0, 2,  4, 10, 24, 59, ...
2, 2,  6, 14, 35, 87, ...
0, 4,  8, 21, 52, ...
4, 4, 13, 31, 79, ...
0, 9, 18, 48, ...
9, 9, 30, ...
...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A051163.

Cf. A152193. - Gary W. Adamson, Nov 28 2008

a:= proc(n) option remember; add(`if`(k=0, 1,
      `if`(k::odd, 0, a(k/2)))*binomial(n, k), k=0..n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 08 2015

a[n_] := a[n] = Sum[If[k == 0, 1, If[OddQ[k], 0, a[k/2]]]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 20 2017, translated from Maple *)

a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=2; A=subst(A,x,(x/(1-x))^2)/(1-x)); polcoeff(A,n))

a=List();for(n=1,100,listput(a,sum(i=1,n\2,a[i]*binomial(n,2*i),1))) \\ M. F. Hasler, Aug 19 2015

G.f. A(x) satisfies A(x^2) = A(x/(1+x))/(1+x) and A(x) = A(x^2/(1-x)^2)/(1-x).
The recursive formula A[n+1] = A[n](x^2/(1-x)^2)/(1-x), A[0]=1, yields exactly 2^n terms after n iterations: A(x) - A[n](x) = x^(2^n) + (2^n+1)*x^(2^n+1) + O(x^(2^n+2)). For example, A[4] = (1-x)^3*(1-2*x-x^2)/((1-2*x)(1-4*x+4*x^2-2*x^4)) = A(x) - x^16 - 17*x^17 + O(x^18). - M. F. Hasler, Aug 19 2015
E.g.f.: exp(x) * Sum_{n>=0} a(n) * x^(2*n) / (2*n)!. - Ilya Gutkovskiy, Feb 26 2022
The expansion of exp(Sum_{n >= 1} a(n)*(2*x)^n/n!) = 1 + 2*x + 6*x^2 + 20*x^3 + 74*x^4 + 292*x^5 + 1204*x^6 + ... appears to have integer coefficients. Equivalently, the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for positive integers k and n and all primes p >= 3. - Peter Bala, Jan 11 2023

Incorrect g.f. and formulas removed by R. J. Mathar, Oct 02 2012

Incorrect g.f.s removed by Peter Bala, Jul 07 2015

cp's OEIS Frontend

A055340 Triangle read by rows: number of mobiles (circular rooted trees) with n nodes and k leaves.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A071673 Sequence a(n) obtained by setting a(0) = 0; then reading the table T(x,y)=a(x)+a(y)+1 in antidiagonal fashion.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Scheme

Formula

Extensions

A075271 a(0) = 1 and, for n >= 1, (BM)a(n) = 2*a(n-1), where BM is the BinomialMean transform.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A088717 G.f. satisfies: A(x) = 1 + x*A(x)^2*A(x*A(x)^2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A144042 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Euler transform applied k times.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A000360 Distribution of nonempty triangles inside a fractal rep-4-tile.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A001005 Number of ways of partitioning n points on a circle into subsets only of sizes 2 and 3.

Views

Author

Keywords

Comments

Examples

A088717 G.f. satisfies: A(x) = 1 + xA(x)^2A(x*A(x)^2).