A122888 -id:A122888 - cp's OEIS frontend

A135080 Triangle, read by rows, that transforms diagonals in the table of coefficients in the successive iterations of x+x^2 (cf. A122888).

1, 1, 1, 2, 2, 1, 8, 7, 3, 1, 50, 40, 15, 4, 1, 436, 326, 112, 26, 5, 1, 4912, 3492, 1128, 240, 40, 6, 1, 68098, 46558, 14373, 2881, 440, 57, 7, 1, 1122952, 744320, 221952, 42604, 6135, 728, 77, 8, 1, 21488640, 13889080, 4029915, 748548, 103326, 11565, 1120, 100

Offset: 0

Paul D. Hanna, Nov 18 2007

			Triangle begins:
1;
1, 1;
2, 2, 1;
8, 7, 3, 1;
50, 40, 15, 4, 1;
436, 326, 112, 26, 5, 1;
4912, 3492, 1128, 240, 40, 6, 1;
68098, 46558, 14373, 2881, 440, 57, 7, 1;
1122952, 744320, 221952, 42604, 6135, 728, 77, 8, 1;
21488640, 13889080, 4029915, 748548, 103326, 11565, 1120, 100, 9, 1; ...
Coefficients in iterations of (x+x^2) form table A122888:
1;
1, 1;
1, 2, 2, 1;
1, 3, 6, 9, 10, 8, 4, 1;
1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...; ...
This triangle T transforms one diagonal in the above table into another;
start with the main diagonal of A122888, A112319, which begins:
[1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, ...];
then the transform T*A112319 equals A112317, which begins:
[1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028, ...];
and the transform T*A112317 equals A112320, which begins:
[1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, ...].

Paul D. Hanna, Table of n, a(n) for n=0..495 (rows 0..30)

Cf. columns: A135081, A135082, A135083.
Cf. related tables: A122888, A166900, A187005, A187115, A187120.
Cf. related sequences: A112319, A112317, A112320, A187009.

{T(n,k)=local(F=x,M,N,P,m=max(n,k)); M=matrix(m+2,m+2,r,c,F=x;for(i=1,r+c-2,F=subst(F,x,x+x^2+x*O(x^(m+2))));polcoeff(F,c)); N=matrix(m+1,m+1,r,c,M[r,c]);P=matrix(m+1,m+1,r,c,M[r+1,c]);(P~*N~^-1)[n+1,k+1]}

/* Generate by method given in A187005, A187115, A187120 (faster): */
{T(n,k)=local(Ck=x);for(m=1,n-k+1,Ck=(1/x^k)*subst(truncate(x^k*Ck),x,x+x^2 +x*O(x^m)));polcoeff(Ck,n-k+1,x)}

Columns may be generated by a method illustrated by triangles A187005, A187115, and A187120. The main diagonal of triangles A187005, A187115, and A187120, equals columns 0, 1, and 2, respectively.

Added cross-reference; example corrected and name changed by Paul D. Hanna, Feb 04 2011

A166900 Triangle, read by rows, that transforms rows into diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 9, 21, 9, 1, 64, 156, 84, 16, 1, 630, 1540, 935, 230, 25, 1, 7916, 19160, 12480, 3564, 510, 36, 1, 121023, 288813, 196623, 61845, 10465, 987, 49, 1, 2179556, 5123608, 3591560, 1207696, 228800, 25864, 1736, 64, 1, 45179508, 104657520

Offset: 0

Paul D. Hanna, Nov 27 2009

Compare to the triangle A071207 that transforms rows into diagonals in the table of iterations of x/(1-x), where A071207(n,k) gives the number of labeled free trees with n vertices and k children of the root that have a label smaller than the label of the root. Does this triangle have a similar interpretation?

			Triangle begins:
1;
1, 1;
2, 4, 1;
9, 21, 9, 1;
64, 156, 84, 16, 1;
630, 1540, 935, 230, 25, 1;
7916, 19160, 12480, 3564, 510, 36, 1;
121023, 288813, 196623, 61845, 10465, 987, 49, 1;
2179556, 5123608, 3591560, 1207696, 228800, 25864, 1736, 64, 1;
45179508, 104657520, 74847168, 26415840, 5426949, 695079, 56511, 2844, 81, 1;
1059312264, 2420186616, 1755406674, 642448632, 140247810, 19683060, 1830080, 112520, 4410, 100, 1; ...
Coefficients in self-compositions of (x+x^2) form table A122888:
1;
1, 1;
1, 2, 2, 1;
1, 3, 6, 9, 10, 8, 4, 1;
1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...; ...
This triangle T transforms rows of A122888 into diagonals of A122888;
the initial diagonals begin:
A112319: [1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, ...];
A112317: [1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028,..];
A112320: [1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, ...].
For example:
T * [1, 0, 0, 0, 0, 0, 0,...]~ = A112319;
T * [1, 1, 0, 0, 0, 0, 0,...]~ = A112317;
T * [1, 2, 2, 1, 0, 0, 0,...]~ = A112320.

Paul D. Hanna, Table of n, a(n) for n=0..495 (rows 0..30)

Cf. A112319, A166901, A166902, A166903, A122888, A135080.

{T(n, k)=local(F=x, M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, x+x^2+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, F=x;for(i=1, r, F=subst(F, x, x+x^2+x*O(x^(m+2)))); polcoeff(F, c)); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A122890 Triangle, read by rows, where the g.f. of row n divided by (1-x)^n yields the g.f. of column n in the triangle A122888, for n>=1.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 5, 0, 0, 0, 10, 14, 0, 0, 0, 8, 70, 42, 0, 0, 0, 4, 160, 424, 132, 0, 0, 0, 1, 250, 1978, 2382, 429, 0, 0, 0, 0, 302, 6276, 19508, 12804, 1430, 0, 0, 0, 0, 298, 15674, 106492, 168608, 66946, 4862, 0, 0, 0, 0, 244, 33148, 451948, 1445208, 1337684, 343772, 16796

Offset: 0

Paul D. Hanna, Sep 18 2006

Main diagonal forms the Catalan numbers (A000108). Row sums gives the factorials. In table A122888, row n lists the coefficients of x^k, k = 1..2^n, in the n-th self-composition of (x + x^2) for n >= 0.

Parker gave the following combinatorial interpretation of the numbers: For n > 0, T(n, j) is the number of sequences c_1c_2...c_n of positive integers such that 1 <= c_i <= i for each i in {1, 2, .., n} with exactly j - 1 values of i such that c_i <= c_{i+1}. - Peter Luschny, May 05 2013

			Triangle begins:
1;
0,1;
0,0,2;
0,0,1,5;
0,0,0,10,14;
0,0,0,8,70,42;
0,0,0,4,160,424,132;
0,0,0,1,250,1978,2382,429;
0,0,0,0,302,6276,19508,12804,1430;
0,0,0,0,298,15674,106492,168608,66946,4862;
0,0,0,0,244,33148,451948,1445208,1337684,343772,16796;
0,0,0,0,162,61806,1614906,9459090,16974314,10003422,1744314,58786;
0,0,0,0,84,103932,5090124,51436848,161380816,180308420,71692452,8780912,208012; ...
Table A122888 starts:
1;
1, 1;
1, 2, 2, 1;
1, 3, 6, 9, 10, 8, 4, 1;
1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;
where row n gives the g.f. of the n-th self-composition of (x+x^2).
From _Paul D. Hanna_, Apr 11 2009: (Start)
ROW-REVERSAL yields triangle A158830:
1;
1, 0;
2, 0, 0;
5, 1, 0, 0;
14, 10, 0, 0, 0;
42, 70, 8, 0, 0, 0;
132, 424, 160, 4, 0, 0, 0;
429, 2382, 1978, 250, 1, 0, 0, 0; ...
where
g.f. of row n of A158830 = (1-x)^n*[g.f. of column n of A158825];
g.f. of row n of A158825 = n-th iteration of x*Catalan(x).
RELATED ARRAY A158825 begins:
1,1,2,5,14,42,132,429,1430,4862,16796,58786,...;
1,2,6,21,80,322,1348,5814,25674,115566,528528,...;
1,3,12,54,260,1310,6824,36478,199094,1105478,...;
1,4,20,110,640,3870,24084,153306,993978,...;
1,5,30,195,1330,9380,67844,500619,3755156,...;
1,6,42,315,2464,19852,163576,1372196,11682348,...;
1,7,56,476,4200,38052,351792,3305484,31478628,...;
1,8,72,684,6720,67620,693048,7209036,75915708,...; ...
which consists of successive iterations of x*Catalan(x).
(End)

Toufik Mansour, Mark Shattuck, Statistics on bargraphs of inversion sequences of permutations, Discrete Math. Lett. (2020) Vol. 4, 42-49.
Toufik Mansour, Howard Skogman, Rebecca Smith, Passing through a stack k times, arXiv:1704.04288 [math.CO], 2017.
Susan Field Parker, The Combinatorics of Functional Composition and Inversion, Ph.D. Dissertation, Brandeis Univ. (1993) (Section 2.3.4, p. 27,28.)

Cf. A122888; A122891 (column sums); diagonals: A122892, A000108.

Cf. related tables: A158830, A158825. [Paul D. Hanna, Apr 11 2009]

nmax = 11;
f[0][x_] := x; f[n_][x_] := f[n][x] = f[n - 1][x + x^2] // Expand; T = Table[ SeriesCoefficient[f[n][x], {x, 0, k}], {n, 0, nmax}, {k, 1, nmax}];
row[n_] := CoefficientList[(1-x)^n*(T[[All, n]].x^Range[0, nmax])+O[x]^nmax, x];
Table[row[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Jul 13 2018 *)

From Paul D. Hanna, Apr 11 2009: (Start)
G.f. of row n: (1-x)^n*[g.f. of column n of A122888] where the g.f. of row n of A122888 is the n-th iteration of x+x^2.
Row-reversal forms triangle A158830 where g.f. of row n of A158830 = (1-x)^n*[g.f. of column n of A158825], and the g.f. of row n of array A158825 is the n-th iteration of x*C(x) and C(x) is the g.f. of the Catalan sequence A000108. (End)

A122889 Antidiagonal sums of table A122888.

Original entry on oeis.org

1, 1, 2, 3, 6, 12, 27, 67, 180, 528, 1676, 5721, 20924, 81634, 338322, 1484209, 6870492, 33460830, 170997842, 914764494, 5111445917, 29773148141, 180449216699, 1136060712131, 7418026238304, 50163775035664

Offset: 0

Paul D. Hanna, Sep 18 2006

nonn

In table A122888, row n lists the coefficients of x^k, k=1..2^n, in the n-th self-composition of (x + x^2) for n>=0.

Paul D. Hanna, Table of n, a(n), n=0..100.

Cf. A122888.

{a(n)=local(F=x+x^2, G, RS); RS=0; for(k=0, n, G=x+x*O(x^n); for(i=1, n-k, G=subst(F, x, G)); RS+=polcoeff(G, k, x)); RS} \\ Paul D. Hanna, Nov 28 2009

A122887 An upper diagonal of table A122888.

Original entry on oeis.org

1, 10, 118, 1656, 27326, 520626, 11271436, 273613032, 7365210840, 217823497200, 7022714928604, 245186433880584, 9217032342988575, 371216529334562246, 15948164256621353088, 728062466187371268960, 35197679178007576031784

Offset: 4

Paul D. Hanna, Sep 18 2006

nonn

Cf. A122888; A122892.

{a(n)=local(F=x+x^2, G=x+x*O(x^(n+1))); if(n<3, 0, for(i=1, n-2, G=subst(F, x, G)); return(polcoeff(G, n+0, x)))}

A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1.

Original entry on oeis.org

1, 2, 6, 42, 1806, 3263442, 10650056950806, 113423713055421844361000442, 12864938683278671740537145998360961546653259485195806

Offset: 0

N. J. A. Sloane

Number of ordered trees having nodes of outdegree 0,1,2 and such that all leaves are at level n. Example: a(2)=6 because, denoting by I a path of length 2 and by Y a Y-shaped tree with 3 edges, we have I, Y, I*I, I*Y, Y*I, Y*Y, where * denotes identification of the roots. - Emeric Deutsch, Oct 31 2002
Equivalently, the number of acyclic digraphs (dags) that unravel to a perfect binary tree of height n. - Nachum Dershowitz, Jul 03 2022
a(n) has at least n different prime factors. [Saidak]
Subsequence of squarefree numbers (A005117). - Reinhard Zumkeller, Nov 15 2004 [This has been questioned, see MathOverflow link. - Charles R Greathouse IV, Mar 30 2015]
For prime factors see A007996.
Curtiss shows that if the reciprocal sum of the multiset S = {x_1, x_2, ..., x_n} is 1, then max(S) <= a(n). - Charles R Greathouse IV, Feb 28 2007
The number of reduced ZBDDs for Boolean functions of n variables in which there is no zero sink. (ZBDDs are "zero-suppressed binary decision diagrams.") For example, a(2)=6 because of the 2-variable functions whose truth tables are 1000, 1010, 1011, 1100, 1110, 1111. - Don Knuth, Jun 04 2007
Using the methods of Aho and Sloane, Fibonacci Quarterly 11 (1973), 429-437, it is easy to show that a(n) is the integer just a tiny bit below the real number theta^{2^n}-1/2, where theta =~ 1.597910218 is the exponential of the rapidly convergent series Sum_{n>=0} log(1+1/a_n)/2^{n+1}. For example, theta^32 - 1/2 =~ 3263442.0000000383. - Don Knuth, Jun 04 2007 [Corrected by Darryl K. Nester, Jun 19 2017]
The next term has 209 digits. - Harvey P. Dale, Sep 07 2011
Urquhart shows that a(n) is the minimum size of a tableau refutation of the clauses of the complete binary tree of depth n, see pp. 432-434. - Charles R Greathouse IV, Jan 04 2013
For any positive a(0), the sequence a(n) = a(n-1) * (a(n-1) + 1) gives a constructive proof that there exists integers with at least n distinct prime factors, e.g. a(n). As a corollary, this gives a constructive proof of Euclid's theorem stating that there are an infinity of primes. - Daniel Forgues, Mar 03 2017
Lower bound for A100016 (with equality for the first 5 terms), where a(n)+1 is replaced by nextprime(a(n)). - M. F. Hasler, May 20 2019

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 94.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..12
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
David Adjiashvili, Sandro Bosio and Robert Weismantel, Dynamic Combinatorial Optimization: a complexity and approximability study, 2012.
Gilles Audemard, Steve Bellart, Louenas Bounia, Frédéric Koriche, Jean-Marie Lagniez, and Pierre Marquis, On the Explanatory Power of Decision Trees, arXiv:2108.05266 [cs.AI], 2021.
Arvind Ayyer, Anne Schilling, Benjamin Steinberg and Nicolas M. Thiéry, Markov chains, R-trivial monoids and representation theory, Int. J. Algebra Comput., Vol. 25 (2015), pp. 169-231, arXiv preprint, arXiv:1401.4250 [math.CO], 2014.
Umberto Cerruti, Percorsi tra i numeri (in Italian), page 5.
A. Yu. Chirkov, D. V. Gribanov and N. Yu. Zolotykh, On the Proximity of the Optimal Values of the Multi-Dimensional Knapsack Problem with and without the Cardinality Constraint, arXiv:2004.08589 [math.OC], 2020.
D. R. Curtiss, On Kellogg's Diophantine problem, Amer. Math. Monthly, Vol. 29, No. 10 (1922), pp. 380-387.
Christian Elsholtz and Stefan Planitzer, Sums of four and more unit fractions and approximate parametrizations, arXiv:2012.05984 [math.NT], 2020.
Steven Finch, Exercises in Iterational Asymptotics, arXiv:2411.16062 [math.NT], 2024. See p. 9.
Samuele Giraudo, The combinator M and the Mockingbird lattice, arXiv:2204.03586 [math.CO], 2022.
Murray S. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see p. 577.
Diana Maimuţ and George Teşeleanu, Inferring Bivariate Polynomials for Homomorphic Encryption Application, Cryptology ePrint Archive (2023) Art. 844. See p. 16.
MathOverflow, Is OEIS A007018 really a subsequence of squarefree numbers?.
Marko R. Riedel, Two-colorings of unordered full binary trees on n levels.
Matthew Roughan, Surreal Birthdays and Their Arithmetic, arXiv:1810.10373 [math.HO], 2018.
Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Vol. 113, No. 10 (Dec., 2006), pp. 937-938.
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]
Bertrand Teguia Tabuguia, Computing with D-Algebraic Sequences, arXiv:2412.20630 [math.AG], 2024. See p. 9.
Alasdair Urquhart, The complexity of propositional proofs, Bull. Symbolic Logic, Vol. 1, No. 4 (1995) pp. 425-467, esp. p. 434.
Zalman Usiskin, Letter to N. J. A. Sloane, Oct. 1991.
Index entries for sequences of form a(n+1)=a(n)^2 + ....

Cf. A000058, A003687, A004168, A011782, A077125, A117805, A118227, A371321.
Lower bound for A100016.
Row sums of A122888.

a007018 n = a007018_list !! n
a007018_list = iterate a002378 1  -- Reinhard Zumkeller, Dec 18 2013

[n eq 1 select 1 else Self(n-1)^2 + Self(n-1): n in [1..10]]; // Vincenzo Librandi, May 19 2015

A007018 := proc(n)
    option remember;
    local aprev;
    if n = 0 then
        1;
    else
        aprev := procname(n-1) ;
        aprev*(aprev+1) ;
    end if;
end proc: # R. J. Mathar, May 06 2016

FoldList[#^2 + #1 &, 1, Range@ 8] (* Robert G. Wilson v, Jun 16 2011 *)
NestList[#^2 + #&, 1, 10] (* Harvey P. Dale, Sep 07 2011 *)

a[1]:1$
a[n]:=(a[n-1] + (a[n-1]^2))$
A007018(n):=a[n]$
makelist(A007018(n),n,1,10); /* Martin Ettl, Nov 08 2012 */

a(n)=if(n>0,my(x=a(n-1));x^2+x,1) \\ Edited by M. F. Hasler, May 20 2019 and Jason Yuen, Mar 01 2025

from itertools import islice
def A007018_gen(): # generator of terms
    a = 1
    while True:
        yield a
        a *= a+1
A007018_list = list(islice(A007018_gen(),9)) # Chai Wah Wu, Mar 19 2024

a(n) = A000058(n)-1 = A000058(n-1)^2 - A000058(n-1) = 1/(1-Sum_{jA000058(j)) where A000058 is Sylvester's sequence. - Henry Bottomley, Jul 23 2001
a(n) = floor(c^(2^n)) where c = A077125 = 1.597910218031873178338070118157... - Benoit Cloitre, Nov 06 2002
a(1)=1, a(n) = Product_{k=1..n-1} (a(k)+1). - Benoit Cloitre, Sep 13 2003
a(n) = A139145(2^(n+1) - 1). - Reinhard Zumkeller, Apr 10 2008
If an (additional) initial 1 is inserted, a(n) = Sum_{kFranklin T. Adams-Watters, Jun 11 2009
a(n+1) = a(n)-th oblong (or promic, pronic, or heteromecic) numbers (A002378). a(n+1) = A002378(a(n)) = A002378(a(n-1)) * (A002378(a(n-1)) + 1). - Jaroslav Krizek, Sep 13 2009
a(n) = A053631(n)/2. - Martin Ettl, Nov 08 2012
Sum_{n>=0} (-1)^n/a(n) = A118227. - Amiram Eldar, Oct 29 2020
Sum_{n>=0} 1/a(n) = A371321. - Amiram Eldar, Mar 19 2024

A158825 Square array of coefficients in the successive iterations of xC(x) = (1-sqrt(1-4x))/2 where C(x) is the g.f. of the Catalan numbers (A000108); read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 12, 21, 14, 1, 5, 20, 54, 80, 42, 1, 6, 30, 110, 260, 322, 132, 1, 7, 42, 195, 640, 1310, 1348, 429, 1, 8, 56, 315, 1330, 3870, 6824, 5814, 1430, 1, 9, 72, 476, 2464, 9380, 24084, 36478, 25674, 4862, 1, 10, 90, 684, 4200, 19852, 67844, 153306, 199094, 115566, 16796

Offset: 1

Paul D. Hanna, Mar 28 2009, Mar 29 2009

			Square array of coefficients in iterations of x*C(x) begins:
  1,  1,   2,    5,    14,      42,      132,       429,       1430, ... A000108;
  1,  2,   6,   21,    80,     322,     1348,      5814,      25674, ... A121988;
  1,  3,  12,   54,   260,    1310,     6824,     36478,     199094, ... A158826;
  1,  4,  20,  110,   640,    3870,    24084,    153306,     993978, ... A158827;
  1,  5,  30,  195,  1330,    9380,    67844,    500619,    3755156, ... A158828;
  1,  6,  42,  315,  2464,   19852,   163576,   1372196,   11682348, ...;
  1,  7,  56,  476,  4200,   38052,   351792,   3305484,   31478628, ...;
  1,  8,  72,  684,  6720,   67620,   693048,   7209036,   75915708, ...;
  1,  9,  90,  945, 10230,  113190,  1273668,  14528217,  167607066, ...;
  1, 10, 110, 1265, 14960,  180510,  2212188,  27454218,  344320262, ...;
  1, 11, 132, 1650, 21164,  276562,  3666520,  49181418,  666200106, ...;
  1, 12, 156, 2106, 29120,  409682,  5841836,  84218134, 1225314662, ...;
  1, 13, 182, 2639, 39130,  589680,  8999172, 138755799, 2157976392, ...;
  1, 14, 210, 3255, 51520,  827960, 13464752, 221101608, 3660331064, ...;
  1, 15, 240, 3960, 66640, 1137640, 19640032, 342179672, 6007747368, ...;
  1, 16, 272, 4760, 84864, 1533672, 28012464, 516105720, 9578580504, ...;
ILLUSTRATE ITERATIONS.
       Let G(x) = x*C(x), then the first few iterations of G(x) are:
           G(x) = x +   x^2 +  2*x^3 +   5*x^4 +  14*x^5 + ...;
        G(G(x)) = x + 2*x^2 +  6*x^3 +  21*x^4 +  80*x^5 + ...;
     G(G(G(x))) = x + 3*x^2 + 12*x^3 +  54*x^4 + 260*x^5 + ...;
  G(G(G(G(x)))) = x + 4*x^2 + 20*x^3 + 110*x^4 + 640*x^5 + ...;
...
RELATED TRIANGLES.
The g.f. of column n is (g.f. of row n of A158830)/(1-x)^n
where triangle A158830 begins: 1;
      1,       0;
      2,       0,        0;
      5,       1,        0,        0;
     14,      10,        0,        0,       0;
     42,      70,        8,        0,       0,       0;
    132,     424,      160,        4,       0,       0,     0;
    429,    2382,     1978,      250,       1,       0,     0,   0;
   1430,   12804,    19508,     6276,     302,       0,     0,   0, 0;
   4862,   66946,   168608,   106492,   15674,     298,     0,   0, 0, 0;
  16796,  343772,  1337684,  1445208,  451948,   33148,   244,   0, 0, 0, 0;
  58786, 1744314, 10003422, 16974314, 9459090, 1614906, 61806, 162, 0, 0, 0, 0;
  ...
Triangle A158835 transforms one diagonal into the next:
       1;
       1,      1;
       4,      2,     1;
      27,     11,     3,    1;
     254,     94,    21,    4,   1;
    3062,   1072,   217,   34,   5,  1;
   45052,  15212,  2904,  412,  50,  6, 1;
  783151, 257777, 47337, 6325, 695, 69, 7, 1; ...
so that:
  A158835 * A158831 = A158832;
  A158835 * A158832 = A158833;
  A158835 * A158833 = A158834;
where the diagonals start:
  A158831 = [1, 1,  6,  54,  640,  9380,  163576,  3305484, ...];
  A158832 = [1, 2, 12, 110, 1330, 19852,  351792,  7209036, ...];
  A158833 = [1, 3, 20, 195, 2464, 38052,  693048, 14528217, ...];
  A158834 = [1, 4, 30, 315, 4200, 67620, 1273668, 27454218, ...].

Paul D. Hanna, Table of n, a(n), n = 1..1275 (rows 1..50)
Frédéric Chapoton and Vincent Pilaud, Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra, arXiv:2201.06896 [math.CO], 2022. See p. 26.

Rows: A000108, A121988, A158826, A158827, A158828.
Columns: A000012, A000027, A002378, A160378.
Antidiagonal sums: A158829.
Diagonals: A158831, A158832, A158833, A158834.
Related triangles: A158830, A158835.
Variant: A122888.

Clear[row]; nmax = 12;
row[n_]:= row[n]= CoefficientList[Nest[(1-Sqrt[1-4#])/2&, x, n] + O[x]^(nmax+1), x] //Rest;
T[n_, k_]:= row[n][[k]];
Table[T[n-k+1, k], {n, nmax}, {k, n}]//Flatten (* Jean-François Alcover, Jul 13 2018, updated Aug 09 2018 *)

{T(n,k)= local(F=serreverse(x-x^2+O(x^(k+2))), G=x);
for(i=1, n, G=subst(F,x,G)); polcoeff(G,k)}

G.f. of column n = (g.f. of row n of A158830)/(1-x)^n.
Row k equals the first column of the k-th matrix power of Catalan triangle A033184; thus triangle A033184 transforms row n into row n+1 of this array (A158825). - Paul D. Hanna, Mar 30 2009
From G. C. Greubel, Apr 01 2021: (Start)
T(n, 1) = A000012(n),   T(n, 2) = A000027(n).
T(n, 3) = A002378(n),   T(n, 4) = A160378(n+1). (End)

A112317 Coefficients of x^n in the n-th iteration of (x + x^2) for n>=1.

Original entry on oeis.org

1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028, 3479498880, 90230486346, 2584679465160, 81056989408928, 2762187020749144, 101633218030586364, 4015771398425994048, 169588657820702174728

Offset: 1

Paul D. Hanna, Sep 03 2005

nonn

Forms a diagonal of the tables A122888 and A185755.

			The initial iterations of x + x^2 begin:
F(x) = (1)*x + x^2;
F(F(x)) = x + (2)*x^2 + 2*x^3 + x^4;
F(F(F(x))) = x + 3*x^2 + (6)*x^3 + 9*x^4+ 10*x^5+ 8*x^6+ 4*x^7+ x^8;
F(F(F(F(x)))) = x + 4*x^2 + 12*x^3 + (30)*x^4 + 64*x^5 +...;
F(F(F(F(F(x))))) = x + 5*x^2 + 20*x^3 + 70*x^4 + (220)*x^5 +...;
F(F(F(F(F(F(x)))))) = x + 6*x^2 + 30*x^3 + 135*x^4 + 560*x^5 + (2170)*x^6 +...;
where the terms in parenthesis illustrate how to form this sequence.

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 1..300 (first 100 terms from Paul D. Hanna)

Cf. A112319, A112320, A122888, A185755, A135080, A166900.

{a(n)=local(F=x+x^2, G=x+x*O(x^n));if(n<1,0, for(i=1,n,G=subst(F,x,G));return(polcoeff(G,n,x)))}
for(n=1, 30, print1(a(n), ", "))

a(n) = [x^n] F_n(x) where F_n(x) = F_{n-1}(x+x^2) with F_1(x) = x+x^2.

Added cross-references and comments; name and example changed by Paul D. Hanna, Feb 04 2011

A166880 Triangle T(n,k), read by rows n>=0 with terms k=1..3^n, where row n lists the coefficients in the n-th iteration of (x+x^2+x^3).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 6, 8, 8, 6, 3, 1, 1, 3, 9, 24, 60, 138, 294, 579, 1053, 1767, 2739, 3924, 5196, 6352, 7152, 7389, 6969, 5961, 4587, 3144, 1896, 990, 438, 159, 45, 9, 1, 1, 4, 16, 60, 216, 744, 2460, 7818, 23910, 70446, 200160, 549006, 1455132, 3730846, 9262712

Offset: 0

Paul D. Hanna, Nov 21 2009

			Triangle begins:
1;
1,1,1;
1,2,4,6,8,8,6,3,1;
1,3,9,24,60,138,294,579,1053,1767,2739,3924,5196,6352,7152,7389,6969,5961,4587,3144,1896,990,438,159,45,9,1;
1,4,16,60,216,744,2460,7818,23910,70446,200160,549006,1455132,...;
1,5,25,120,560,2540,11220,48330,203230,835080,3355950,13200648,...;
1,6,36,210,1200,6720,36930,199365,1058175,5526330,28417200,...;
1,7,49,336,2268,15078,98826,639093,4080531,25738755,160474545,...;
1,8,64,504,3920,30128,228984,1722084,12821788,94556532,...;
1,9,81,720,6336,55224,477000,4085028,34700940,292495896,...;
1,10,100,990,9720,94680,915390,8787735,83795085,793894860,...;
1,11,121,1320,14300,153890,1645710,17494455,184915225,...;
1,12,144,1716,20328,239448,2805396,32700558,379309986,...;
1,13,169,2184,28080,359268,4575324,58009614,732380298,...;
1,14,196,2730,37856,522704,7188090,98465913,1343828395,...;
1,15,225,3360,49980,740670,10937010,160947465,2360704815,...;
1,16,256,4080,64800,1025760,16185840,254624520,3993857400,...;
1,17,289,4896,82688,1392368,23379216,391488648,6538326616,...;
1,18,324,5814,104040,1856808,33053814,586957419,10398271833,...;
...
The initial diagonals in this triangle begin:
A166881: [1,1,4,24,216,2540,36930,639093,12821788,292495896,...];
A166882: [1,2,9,60,560,6720,98826,1722084,34700940,793894860,...];
A166883: [1,3,16,120,1200,15078,228984,4085028,83795085,1943920935,...]; ...
The diagonals are transformed one into the other by
triangle A166884, which begins:
1;
1,1;
3,2,1;
15,9,3,1;
114,62,18,4,1;
1159,593,157,30,5,1;
14838,7266,1812,316,45,6,1;
229401,108720,25989,4271,555,63,7,1;
4159662,1922166,445255,70180,8595,890,84,8,1; ...

Cf. diagonals: A166881, A166882, A166883, related triangle: A166884.

Cf. row sums: A166999, variant: A122888.

{T(n, k)=local(F=x+x^2+x^3, G=x+x*O(x^k)); if(n<0, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, k, x)))}

A135081 Column 0 of triangle A135080.

Original entry on oeis.org

1, 1, 2, 8, 50, 436, 4912, 68098, 1122952, 21488640, 468331252, 11456367820, 310888085872, 9269621420284, 301268634277760, 10601062978739338, 401550210033474420, 16291237867482727084, 704847239600911931248

Offset: 0

Paul D. Hanna, Nov 18 2007

nonn

Triangle A135080 transforms diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).

Triangle A187005 is defined by: A187005(n,k) = [y^k] R_{n-1}(y+y^2) for k=1..n where R_n(y) is the n-th row polynomial for n>1 with R_1(y)=y.

Paul D. Hanna, Table of n, a(n) for n=0..50

Cf. A135080 (triangle); other columns: A135082, A135083.

Cf. A187005, A187006, A187007, A187008, A187009.

/* As column 0 of triangle A135080 (slower): */
{a(n)=local(F=x,M,N,P); M=matrix(n+2,n+2,r,c,F=x;for(i=1,r+c-2,F=subst(F,x,x+x^2+x*O(x^(n+2))));polcoeff(F,c)); N=matrix(n+1,n+1,r,c,M[r,c]);P=matrix(n+1,n+1,r,c,M[r+1,c]);(P~*N~^-1)[n+1,1]}

/* As the main diagonal of triangle A187005 (faster): */
{a(n)=local(Rn=y); for(m=1, n+1, Rn=subst(truncate(Rn), y, y+y^2+y*O(y^m))); polcoeff(Rn/y, n, y)}

Equals the main diagonal of triangle A187005.

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A135080 Triangle, read by rows, that transforms diagonals in the table of coefficients in the successive iterations of x+x^2 (cf. A122888).

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A166900 Triangle, read by rows, that transforms rows into diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).

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A122890 Triangle, read by rows, where the g.f. of row n divided by (1-x)^n yields the g.f. of column n in the triangle A122888, for n>=1.

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A122889 Antidiagonal sums of table A122888.

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A122887 An upper diagonal of table A122888.

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A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1.

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A158825 Square array of coefficients in the successive iterations of x*C(x) = (1-sqrt(1-4*x))/2 where C(x) is the g.f. of the Catalan numbers (A000108); read by antidiagonals.

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A112317 Coefficients of x^n in the n-th iteration of (x + x^2) for n>=1.

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A158825 Square array of coefficients in the successive iterations of xC(x) = (1-sqrt(1-4x))/2 where C(x) is the g.f. of the Catalan numbers (A000108); read by antidiagonals.