A091980 -id:A091980 - cp's OEIS frontend

A003095 a(n) = a(n-1)^2 + 1 for n >= 1, with a(0) = 0.

0, 1, 2, 5, 26, 677, 458330, 210066388901, 44127887745906175987802, 1947270476915296449559703445493848930452791205, 3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352026

Offset: 0

N. J. A. Sloane and Richard Stanley

Number of binary trees of height less than or equal to n. [Corrected by Orson R. L. Peters, Jan 03 2020]
The rightmost digits cycle (0,1,2,5,6,7,0,1,2,5,6,7,...). - Jonathan Vos Post, Jul 21 2005
Apart from the initial term, a subsequence of A008318. - Reinhard Zumkeller, Jan 17 2008
Partial sums of A001699. - Jonathan Vos Post, Feb 17 2010
Corresponds to the second and second last diagonals of A119687. - John M. Campbell, Jul 25 2011
This is a divisibility sequence. - Michael Somos, Jan 01 2013
Sum_{n>=1} 1/a(n) = 1.739940825174794649210636285335916041018367182486941... . - Vaclav Kotesovec, Jan 30 2015
From Vladimir Vesic, Oct 03 2015: (Start)
Forming Herbrand's domains of formula: (∃x)(∀y)(∀z)(∃k)(P(x)∨Q(y)∧R(k))
where: x->a
       k->f(y,z)
we get:
  H0 = {a}
  H1 = {a, f(a,a)}
  H2 = {a, f(a,a), f(a,f(a,a)), f(f(a,a),a), f(f(a,a),f(a,a))}
  ...
The number of elements in each domain follows this sequence.
(End)
It is an open question whether or not this sequence satisfies Benford's law [Berger-Hill, 2017] - N. J. A. Sloane, Feb 07 2017
This is a strong divisibility sequence; see A329429. - Clark Kimberling, Nov 13 2019
From Peter Bala, Oct 31 2022: (Start)
Let k be a positive integer. Clearly, the sequence obtained by reducing a(n) modulo k is eventually periodic. Conjectures:
1) The sequence obtained by reducing a(n) modulo 2^k is eventually periodic with period 2.
2) The sequence obtained by reducing a(n) modulo 10^k is eventually periodic with period 6 (the case k = 1 is noted above).
3) The sequence obtained by reducing a(n) modulo 20^k is eventually periodic with period 6.
4) For n >= floor(k/2) and for 1 <= i <= 6, the value of a(6*n+i) mod 10^k is a constant independent of n. The digits of these 6 constant integers, when read from right to left, are the first k digits of the 10-adic numbers A318135 (i = 1), A318136 (i = 2), A318137 (i = 3), A318138 (i = 4), A318139 (i = 5) and A318140 (i = 6), respectively. An example is given below.
  n      a(6*n+1) mod 10^11
1           10066388901
2           72084948901
3           67988948901
4           61588948901
5           01588948901
6           01588948901
7           01588948901
...             ...
A318135 begins 1, 0, 9, 8, 4, 9, 8, 8, 5, 1, 0, 2, .... (End)

Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 443-448.
R. K. Guy, How to factor a number, Proc. 5th Manitoba Conf. Numerical Math., Congress. Num. 16 (1975), 49-89.
R. Penrose, The Emperor's New Mind, Oxford, 1989, p. 122.
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 = 0..13
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!)
A. Berger and T. P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.
Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, Some Facts and Conjectures about Mandelbrot Polynomials, Maple Transactions (2021) Vol. 1, No. 1, Article 1.
P. Flajolet and A. M. Odlyzko, Limit distributions of coefficients of iterates of polynomials with applications to combinatorial enumerations, Math. Proc. Camb. Phil. Soc., 96 (1984), 237-253.
Claudio Gentile, Fabio Vitale, and Anand Rajagopalan, Flattening a Hierarchical Clustering through Active Learning, arXiv:1906.09458 [cs.LG], 2019.
Spencer Hamblen, Rafe Jones, and Kalyani Madhu, The density of primes in orbits of z^d + c, arXiv:1303.6513 [math.NT], 2013; to appear, Int. Math. Res. Not., c. 2015.
Dimitur Krustev, Simple Programs on Binary Trees Testing and Decidable Equivalence, 2016.
Robin Lamarche-Perrin, An Information-theoretic Framework for the Lossy Compression of Link Streams, arXiv:1807.06874 [cs.DS], 2018.
R. Lamarche-Perrin, Y. Demazeau, and J.-M. Vincent, A Generic Algorithmic Framework to Solve Special Versions of the Set Partitioning Problem, Preprint 105, Max-Planck-Institut fur Mathematik in den Naturwissenschaften, Leipzig, 2014.
C. Lenormand, Arbres et permutations II, see p. 6.
Saad Mneimneh, Simple Variations on the Tower of Hanoi to Guide the Study of Recurrences and Proofs by Induction, Department of Computer Science, Hunter College, CUNY, 2019.
Michael Penn, a stylish proof that..., YouTube video, 2021.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
M. Tainiter, Algebraic approach to stopping variable problems: Representation theory and applications, J. Combinatorial Theory 9 1970 148-161.
P. Tarau, A Logic Programming Playground for Lambda Terms, Combinators, Types and Tree-based Arithmetic Computations, arXiv preprint arXiv:1507.06944 [cs.LO], 2015.
Stephan Wagner and Volker Ziegler, Irrationality of growth constants associated with polynomial recursions, arXiv:2004.09353 [math.NT], 2020.
Wikipedia, Herbrand Structure
Damiano Zanardini, Computational Logic, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid.
Index entries for sequences of form a(n+1)=a(n)^2 + ...
Index to divisibility sequences
Index entries for sequences related to Benford's law

Cf. A001699, A004019, A038044, A056207, A076949, A077496, A091980, A143848.
Cf. A143849, A213437, A247981, A248218, A248219, A318135, A318136, A318137.
Cf. A318138, A318139, A318140, A355108.
Cf. A137560, which enumerates binary trees of height less than n and exactly j leaf nodes. - Robert Munafo, Nov 03 2009

function A003095(n)
  if n eq 0 then return 0;
  else return 1 + A003095(n-1)^2;
  end if; return A003095;
end function;
[A003095(n): n in [0..12]]; // G. C. Greubel, Nov 29 2022

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)^2+1) end:
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 11 2019

NestList[#^2+1&,0,10] (* Harvey P. Dale, Dec 17 2010 *)

{a(n) = if( n<1, 0, 1 + a(n-1)^2)}; /* Michael Somos, Jan 01 2013 */

def A003095(n): return 0 if (n==0) else 1 + A003095(n-1)^2
[A003095(n) for n in range(13)] # G. C. Greubel, Nov 29 2022

a(n) = B_{n-1}(1) where B_n(x) = 1 + x*B_{n-1}(x)^2 is the generating function of trees of height <= n.
a(n) is asymptotic to c^(2^n) where c=1.2259024435287485386279474959130085213... (see A076949). - Benoit Cloitre, Nov 27 2002
c = b^(1/4) where b is the constant in Bottomley's formula in A004019. a(n) appears very asymptotic to c^(2^n) - Sum_{k>=1} A088674(k)/(2*c^(2^n))^(2*k-1). - Gerald McGarvey, Nov 17 2007
a(n) = Sum_{i=1..n} A001699(i). - Jonathan Vos Post, Feb 17 2010
G.f. = x + 2*x^2 + 5*x^3 + 26*x^4 + 677*x^5 + 458330*x^6 + 210066388901*x^7 + ... . - Michael Somos, Jan 01 2013
a(2n) mod 2 = 0 ; a(2n+1) mod 2 = 1. - Altug Alkan, Oct 04 2015
a(n) + a(n-1) = A213437(n). - Peter Bala, Feb 03 2017
0 = a(n)^2*(+a(n+1) + a(n+2)) + a(n+1)^2*(-a(n+1) - a(n+2) - a(n+3)) + a(n+2)^3 for all n>=0. - Michael Somos, Feb 10 2017
a(n) = A091980(2^(n-1)) for n > 0. - Alois P. Heinz, Jul 11 2019

Additional comments from Cyril Banderier, Jun 05 2000
Minor edits by Vaclav Kotesovec, Oct 04 2014
Initial term clarified by Clark Kimberling, Nov 13 2019

A309049 Number T(n,k) of (binary) max-heaps on n elements from the set {0,1} containing exactly k 0's; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 4, 2, 1, 1, 1, 4, 6, 6, 5, 2, 1, 1, 1, 4, 7, 8, 7, 5, 2, 1, 1, 1, 5, 10, 12, 11, 8, 5, 2, 1, 1, 1, 5, 11, 16, 17, 13, 9, 5, 2, 1, 1, 1, 6, 15, 23, 27, 24, 16, 10, 5, 2, 1, 1, 1, 6, 16, 27, 34, 34, 27, 18, 11, 5, 2, 1, 1

Offset: 0

Alois P. Heinz, Jul 09 2019

Also the number T(n,k) of (binary) min-heaps on n elements from the set {0,1} containing exactly k 1's.

The sequence of column k satisfies a linear recurrence with constant coefficients of order A063915(k).

			T(6,0) = 1: 111111.
T(6,1) = 3: 111011, 111101, 111110.
T(6,2) = 4: 110110, 111001, 111010, 111100.
T(6,3) = 4: 101001, 110010, 110100, 111000.
T(6,4) = 2: 101000, 110000.
T(6,5) = 1: 100000.
T(6,6) = 1: 000000.
T(7,4) = T(7,7-3) = A000108(3) = 5: 1010001, 1010010, 1100100, 1101000, 1110000.
Triangle T(n,k) begins:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  1,  1;
  1, 2,  2,  1,  1;
  1, 3,  3,  2,  1,  1;
  1, 3,  4,  4,  2,  1,  1;
  1, 4,  6,  6,  5,  2,  1,  1;
  1, 4,  7,  8,  7,  5,  2,  1,  1;
  1, 5, 10, 12, 11,  8,  5,  2,  1, 1;
  1, 5, 11, 16, 17, 13,  9,  5,  2, 1, 1;
  1, 6, 15, 23, 27, 24, 16, 10,  5, 2, 1, 1;
  1, 6, 16, 27, 34, 34, 27, 18, 11, 5, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Columns k=0-10 give: A000012, A110654, A114220 (for n>0), A326504, A326505, A326506, A326507, A326508, A326509, A326510, A326511.
Row sums give A091980(n+1).
T(2n,n) gives A309050.
Rows reversed converge to A000108.
Cf. A000225, A000295, A003095, A024358, A056971, A063915, A137560, A168542, A202019, A309051, A309052.

b:= proc(n) option remember; `if`(n=0, 1, (g-> (f-> expand(
      x^n+b(f)*b(n-1-f)))(min(g-1, n-g/2)))(2^ilog2(n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..14);

b[n_] := b[n] = If[n == 0, 1, Function[g, Function[f, Expand[x^n + b[f]*b[n - 1 - f]]][Min[g - 1, n - g/2]]][2^Floor[Log[2, n]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Oct 04 2019, after Alois P. Heinz *)

Sum_{k=0..n}    k  * T(n,k) = A309051(n).
Sum_{k=0..n} (n-k) * T(n,k) = A309052(n).
Sum_{k=0..2^n-1} T(2^n-1,k) = A003095(n+1).
Sum_{k=0..2^n-1} (2^n-1-k) * T(2^n-1,k) = A024358(n).
Sum_{k=0..n} (T(n,k) - T(n-1,k)) = A168542(n).
T(m,m-n) = A000108(n) for m >= 2^n-1 = A000225(n).
T(2^n-1,k) = A202019(n+1,k+1).

A004019 a(0) = 0; for n > 0, a(n) = (a(n-1) + 1)^2.

Original entry on oeis.org

0, 1, 4, 25, 676, 458329, 210066388900, 44127887745906175987801, 1947270476915296449559703445493848930452791204, 3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352025

Offset: 0

N. J. A. Sloane

Take the standard rooted binary tree of depth n, with 2^(n+1) - 1 labeled nodes. Here is a picture of the tree of depth 3:
             R
            / \
           /   \
          /     \
         /       \
        /         \
       o           o
      / \         / \
     /   \       /   \
    o     o     o     o
   / \   / \   / \   / \
  o   o o   o o   o o   o
Let the number of rooted subtrees be s(n). For example, for n = 1 the s(2) = 4 subtrees are:
  R   R   R      R
     /     \    / \
    o       o  o   o
Then s(n+1) = 1 + 2*s(n) + s(n)^2 = (1+s(n))^2 and so s(n) = a(n+1).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.

Reinhard Zumkeller, Table of n, a(n) for n = 0..11 (shortened by _N. J. A. Sloane_, Jan 13 2019)
Geir Agnarsson, Elie Alhajjar, and Aleyah Dawkins, On locally finite ordered rooted trees and their rooted subtrees, arXiv:2312.11379 [math.CO], 2023.
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!)
Andressa Paola Cordeiro, Alexandre Tavares Baraviera, and Alex Jenaro Becker, Entropy for k-trees defined by k transition matrices, arXiv:2307.05850 [math.DS], 2023. See p. 15.
F. Disanto and N. A. Rosenberg, Enumeration of ancestral configurations for matching gene trees and species trees, J. Comput. Biol. 24 (2017), 831-850.
E. Lappo and N. A. Rosenberg, A lattice structure for ancestral configurations arising from the relationship between gene trees and species trees, Adv. Appl. Math. 343 (2024), 65-81.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
Elmar Teufl and Stephan Wagner, Enumeration problems for classes of self-similar graphs, Journal of Combinatorial Theory, Series A, Volume 114, Issue 7, October 2007, Pages 1254-1277.
Wikipedia, Herbrand Structure
Damiano Zanardini, Computational Logic, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid.
Index entries for sequences of form a(n+1)=a(n)^2 + ...
Index entries for sequences related to rooted trees

Cf. A001699, A056207.
Cf. A000290.
Cf. A003095, A091980, A355108.

a004019 n = a004019_list !! n
a004019_list = iterate (a000290 . (+ 1)) 0
-- Reinhard Zumkeller, Feb 01 2013

[n le 1 select 0 else  (Self(n-1)+1)^2: n in [1..15]]; // Vincenzo Librandi, Oct 05 2015

Table[Nest[(1 + #)^2 &, 0, n], {n, 0, 12}] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)
NestList[(#+1)^2&,0,10] (* Harvey P. Dale, Oct 08 2011 *)

a(n) = if(n==0, 0, (a(n-1) + 1)^2);
vector(20, n, a(n-1)) \\ Altug Alkan, Oct 06 2015

a(n) = A003095(n)^2 = A003095(n+1) - 1 = A056207(n+1) + 1.
It follows from Aho and Sloane that there is a constant c such that a(n) is the nearest integer to c^(2^n). In fact a(n+1) = nearest integer to b^(2^n) - 1 where b = 2.25851845058946539883779624006373187243427469718511465966.... - Henry Bottomley, Aug 30 2005
a(n) is the number of root ancestral configurations for fully symmetric matching gene trees and species trees with 2^n leaves, a(n) = A355108(2^n). - Noah A Rosenberg, Jun 22 2022

One more term from Henry Bottomley, Jul 24 2000

Additional comments from Max Alekseyev, Aug 30 2005

A373451 Number T(n,k) of (binary) heaps of length n whose element set equals [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 7, 3, 0, 1, 9, 23, 23, 8, 0, 1, 14, 55, 92, 70, 20, 0, 1, 24, 147, 386, 502, 320, 80, 0, 1, 34, 281, 1004, 1861, 1880, 985, 210, 0, 1, 54, 633, 3128, 8113, 12008, 10237, 4690, 896, 0, 1, 79, 1241, 8039, 27456, 54900, 66730, 48650, 19600, 3360

Offset: 0

Alois P. Heinz, Jun 05 2024

These heaps may contain repeated elements. Their element sets are gap-free and contain 1 (if nonempty).

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.

			T(4,1) = 1: 1111.
T(4,2) = 5: 2111, 2121, 2211, 2212, 2221.
T(4,3) = 7: 3121, 3211, 3212, 3221, 3231, 3312, 3321.
T(4,4) = 3: 4231, 4312, 4321.
(The examples use max-heaps.)
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  3,   2;
  0, 1,  5,   7,    3;
  0, 1,  9,  23,   23,    8;
  0, 1, 14,  55,   92,   70,    20;
  0, 1, 24, 147,  386,  502,   320,    80;
  0, 1, 34, 281, 1004, 1861,  1880,   985,  210;
  0, 1, 54, 633, 3128, 8113, 12008, 10237, 4690, 896;
  ...

Alois P. Heinz, Rows n = 0..150, flattened
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Columns k=0-4 give A000007, A057427, A091980(n+1)-2, A376962, A376963.
Row sums give A373452.
Row maxima give A373608.
Main diagonal gives A056971.
First lower diagonal gives A373496.
T(2n,n) gives A373500.
Antidiagonal sums give A373632.
Antidiagonal maxima give A373897.
Cf. A019590, A373449.

b:= proc(n, k) option remember; `if`(n=0, 1,
     (g-> (f-> add(b(f, j)*b(n-1-f, j), j=1..k)
             )(min(g-1, n-g/2)))(2^ilog2(n)))
    end:
T:= (n, k)-> add(binomial(k, j)*(-1)^j*b(n, k-j), j=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, k_] := b[n, k] = If[n == 0, 1,
   Function[g, Function[f, Sum[b[f, j]*b[n-1-f, j], {j, 1, k}]][
      Min[g-1, n-g/2]]][2^(Length@IntegerDigits[n, 2]-1)]];
T[n_, k_] := Sum[Binomial[k, j]*(-1)^j*b[n, k-j], {j, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Sep 20 2024, after Alois P. Heinz *)

T(n,k) = Sum_{j=0..k} binomial(k,j) * (-1)^j * A373449(n,k-j).

Sum_{k=0..n} (-1)^k * T(n,n-k) = A019590(n+1).

A372640 Number T(n,k) of defective (binary) heaps on n elements from the set {0,1} where k ancestor-successor pairs do not have the correct order; triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 1, 7, 4, 3, 2, 11, 6, 7, 5, 2, 1, 16, 13, 12, 8, 10, 3, 2, 26, 22, 23, 14, 21, 10, 9, 2, 1, 36, 36, 39, 33, 33, 28, 26, 13, 9, 2, 1, 56, 54, 67, 61, 60, 59, 56, 37, 34, 11, 13, 2, 2, 81, 99, 111, 96, 117, 112, 107, 96, 76, 53, 36, 20, 14, 4, 2

Offset: 0

Alois P. Heinz, May 08 2024

T(n,k) is the number of bit vectors v of length n having exactly k pairs (i,j) in {1,...,n} X {1,...,floor(log_2(i))} such that v[i] > v[floor(i/2^j)].
T(n,0) counts perfect (binary) heaps on n elements from the set {0,1}.
T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

			T(4,0) = 7: 0000, 1000, 1010, 1100, 1101, 1110, 1111.
T(4,1) = 4: 0010, 0100, 1001, 1011.
T(4,2) = 3: 0001, 0101, 0110.
T(4,3) = 2: 0011, 0111.
(The examples use max-heaps.)
Triangle T(n,k) begins:
   1;
   2;
   3,  1;
   5,  2,   1;
   7,  4,   3,  2;
  11,  6,   7,  5,   2,   1;
  16, 13,  12,  8,  10,   3,   2;
  26, 22,  23, 14,  21,  10,   9,  2,  1;
  36, 36,  39, 33,  33,  28,  26, 13,  9,  2,  1;
  56, 54,  67, 61,  60,  59,  56, 37, 34, 11, 13,  2,  2;
  81, 99, 111, 96, 117, 112, 107, 96, 76, 53, 36, 20, 14, 4, 2;
  ...

Alois P. Heinz, Rows n = 0..112, flattened
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Columns k=0-1 give: A091980(n+1), A372643.
Row sums give A000079.
Main diagonal gives A372641.
T(2,n) gives A372642.
Cf. A306393, A370484.

b:= proc(n, t) option remember; `if`(n=0, 1, (g-> (f->
      expand(b(f, t)*b(n-1-f, t)*x^t+b(f, t+1)*b(n-1-f, t+1)
          ))(min(g-1, n-g/2)))(2^ilog2(n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..14);

b[n_, t_] := b[n, t] = If[n == 0, 1, Function[g, Function [f,
   Expand[b[f, t]*b[n-1-f, t]*x^t + b[f, t+1]*b[n-1 - f, t+1]]][
   Min[g-1, n-g/2]]][2^(Length@IntegerDigits[n, 2]-1)]];
T[n_] := CoefficientList[b[n, 0], x];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, May 09 2024, after Alois P. Heinz *)

A370484 Number T(n,k) of defective (binary) heaps on n elements from the set {0,1} with k defects; triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 1, 7, 6, 3, 11, 11, 9, 1, 16, 20, 24, 4, 26, 32, 52, 16, 2, 36, 60, 100, 52, 8, 56, 100, 192, 120, 40, 4, 81, 162, 351, 300, 111, 18, 1, 131, 255, 631, 627, 313, 77, 13, 1, 183, 427, 1067, 1311, 821, 241, 41, 5, 287, 692, 1856, 2484, 1894, 764, 184, 28, 3

Offset: 0

Alois P. Heinz, May 06 2024

A defect in a defective heap is a parent-child pair not having the correct order.
T(n,k) is the number of bit vectors v of length n having exactly k indices i in [n] such that v[i] > v[floor(i/2)].
T(n,0) counts perfect (binary) heaps on n elements from the set {0,1}.
T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

			T(4,0) = 7: 0000, 1000, 1010, 1100, 1101, 1110, 1111.
T(4,1) = 6: 0001, 0010, 0100, 0101, 1001, 1011.
T(4,2) = 3: 0011, 0110, 0111.
(The examples use max-heaps.)
Triangle T(n,k) begins:
    1;
    2;
    3,   1;
    5,   2,    1;
    7,   6,    3;
   11,  11,    9,    1;
   16,  20,   24,    4;
   26,  32,   52,   16,   2;
   36,  60,  100,   52,   8;
   56, 100,  192,  120,  40,   4;
   81, 162,  351,  300, 111,  18,  1;
  131, 255,  631,  627, 313,  77, 13, 1;
  183, 427, 1067, 1311, 821, 241, 41, 5;
  ...

Alois P. Heinz, Rows n = 0..250, flattened
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Columns k=0-1 give: A091980(n+1), A372628.
Row sums give A000079.
T(2n,n) gives A372489.
Cf. A045623, A056971, A139756, A306343, A372640.

b:= proc(n, t) option remember; `if`(n=0, 1, (g-> (f->
      expand(b(f, 1)*b(n-1-f, 1)*t+b(f, x)*b(n-1-f, x)))(
      min(g-1, n-g/2)))(2^ilog2(n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):
seq(T(n), n=0..15);

b[n_, t_] := b[n, t] = If[n == 0, 1, Function[g, Function [f,
   Expand[b[f, 1]*b[n - 1 - f, 1]*t + b[f, x]*b[n - 1 - f, x]]][
   Min[g - 1, n - g/2]]][2^(Length[IntegerDigits[n, 2]] - 1)]];
T[n_] := CoefficientList[b[n, 1], x];
Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, May 09 2024, after Alois P. Heinz *)

Sum_{k>=0} k * T(n,k) = A139756(n) = ceiling((n-1)*2^n/4).

Sum_{k>=0} (k+1) * T(n,k) = A045623(n) = ceiling((n+3)*2^n/4).

A168542 Number of trees that have a maximum 'n'.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 10, 10, 20, 25, 50, 52, 104, 130, 260, 260, 520, 650, 1300, 1352, 2704, 3380, 6760, 6770, 13540, 16925, 33850, 35204, 70408, 88010, 176020, 176020, 352040, 440050, 880100, 915304, 1830608, 2288260, 4576520, 4583290, 9166580, 11458225

Offset: 0

Endi Begeja (andy.bege(AT)libero.it), Nov 29 2009

nonn

a(2^n) = Product_{k=1..n} A003095(k). - Michael Somos, Dec 20 2018

Alois P. Heinz, Table of n, a(n) for n = 0..5654

Cf. A003095, A070939, A309049.

Partial differences of A091980. - Alois P. Heinz, Jul 12 2019

b:= proc(n) option remember; `if`(n=0, 1, (g-> (f->
      1+b(f)*b(n-1-f))(min(g-1, n-g/2)))(2^ilog2(n)))
    end:
a:= n-> b(n)-`if`(n=0, 0, b(n-1)):
seq(a(n), n=0..45);  # Alois P. Heinz, Jul 12 2019

a[ n_] := If[ n < 3, Boole[n > 0], With[{m = BitLength[Quotient[n, 3]] - 1}, Nest[#^2 + 1 &, 2, m] a[n - 2 2^m]]]; (* Michael Somos, Dec 20 2018 *)

{a(n) = if( n<3, n>0, my(m = #binary(n\3)-1, t = 2); for(i=1, m, t = t^2+1); t*a(n - 2*2^m))}; /* Michael Somos, Dec 20 2018 */

a(1) = a(2) = 1, a(3*2^m + k) = A003095(m+2) * a(n - 2*2^m) where 0 <= k < 3*2^m. - Michael Somos, Dec 20 2018

a(n) = Sum_{k=0..n} (A309049(n,k)-A309049(n-1,k)) for n > 0, a(0) = 1. - Alois P. Heinz, Jul 12 2019

a(0)=1 prepended by Alois P. Heinz, Jul 12 2019

A309052 Total number of 1's in all (binary) max-heaps on n elements from the set {0,1}.

Original entry on oeis.org

0, 1, 3, 8, 15, 31, 54, 105, 166, 298, 478, 863, 1307, 2247, 3500, 6136, 9032, 15084, 23039, 39599, 57955, 96019, 145627, 248223, 357650, 583274, 875459, 1476754, 2131618, 3476550, 5210521, 8766473, 12498445, 20138409, 29952394, 50020414, 71658602, 115850282

Offset: 0

Alois P. Heinz, Jul 09 2019

nonn

Also the total number of 0's in all (binary) min-heaps on n elements from the set {0,1}.

			a(4) = 15 = 0+1+2+2+3+3+4, the total number of 1's in 0000, 1000, 1010, 1100, 1101, 1110, 1111.

Alois P. Heinz, Table of n, a(n) for n = 0..5631
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Cf. A024358, A056971, A091980, A309049, A309051.

b:= proc(n) option remember; `if`(n=0, 1, (g-> (f-> expand(
      1+x*b(f)*b(n-1-f)))(min(g-1, n-g/2)))(2^ilog2(n)))
    end:
a:= n-> subs(x=1, diff(b(n), x)):
seq(a(n), n=0..40);

b[n_][x_] := b[n][x] = If[n == 0, 1, Function[g, Function[f, Expand[1 + x b[f][x] b[n - 1 - f][x]]][Min[g - 1, n - g/2]]][2^(Length[IntegerDigits[ n, 2]] - 1)]];
a[n_] := b[n]'[1];
a /@ Range[0, 40] (* Jean-François Alcover, Apr 22 2021, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} (n-k) * A309049(n,k).
a(n) = n * A091980(n+1) - A309051(n).
a(2^n-1) = A024358(n).

A309051 Total number of 0's in all (binary) max-heaps on n elements from the set {0,1}.

Original entry on oeis.org

0, 1, 3, 7, 13, 24, 42, 77, 122, 206, 332, 578, 889, 1484, 2338, 4019, 5960, 9685, 14887, 25134, 37225, 60704, 92919, 156646, 227302, 364551, 550329, 917822, 1337358, 2158150, 3258779, 5441757, 7800755, 12412461, 18546566, 30708486, 44327782, 71090442

Offset: 0

Alois P. Heinz, Jul 09 2019

nonn

Also the total number of 1's in all (binary) min-heaps on n elements from the set {0,1}.

			a(4) = 13 = 4+3+2+2+1+1+0, the total number of 0's in 0000, 1000, 1010, 1100, 1101, 1110, 1111.

Alois P. Heinz, Table of n, a(n) for n = 0..5632
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Cf. A056971, A091980, A309049, A309052.

b:= proc(n) option remember; `if`(n=0, 1, (g-> (f-> expand(
      x^n+b(f)*b(n-1-f)))(min(g-1, n-g/2)))(2^ilog2(n)))
    end:
a:= n-> subs(x=1, diff(b(n), x)):
seq(a(n), n=0..40);

b[n_][x_] := b[n][x] = If[n == 0, 1, Function[g, Function[f, Expand[x^n + b[f][x] b[n - 1 - f][x]]][Min[g - 1, n - g/2]]][2^(Length[IntegerDigits[ n, 2]] - 1)]];
a[n_] := b[n]'[1];
a /@ Range[0, 40] (* Jean-François Alcover, Apr 22 2021, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} k * A309049(n,k).

a(n) = n * A091980(n+1) - A309052(n).

A355108 Maximal number of root ancestral configurations among matching gene trees and species trees with n leaves.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 15, 25, 35, 55, 80, 130, 182, 286, 416, 676, 936, 1456, 2106, 3406, 4758, 7462, 10842, 17602, 24372, 37912, 54837, 88687, 123891, 194299, 282309, 458329, 634349, 986389, 1426439, 2306539, 3221843, 5052451, 7340711, 11917231, 16500521

Offset: 1

Noah A Rosenberg, Jun 19 2022

nonn

An ancestral configuration is a set of gene lineages present immediately before a node of a species tree is reached, looking backward in time, and a root ancestral configuration is an ancestral configuration at the root node. The term a(n) gives the largest number of root ancestral configurations among pairs (G,S) where G is a labeled gene tree topology, S is a bijectively labeled species tree topology, G and S have n leaves, and G=S.

F. Disanto and N. A. Rosenberg, Enumeration of ancestral configurations for matching gene trees and species trees, J. Comput. Biol. 24 (2017), 831-850.

Cf. A003095, A004019, A091980, A306390.

a:= proc(n) option remember; `if`(n=1, 0, (g-> (f->
     (1+a(f))*(1+a(n-f)))(min(g, n-g/2)))(2^ilog2(n)))
    end:
seq(a(n), n=1..42);  # Alois P. Heinz, Jun 19 2022

b[n_] := b[n] = If[n == 1, 1, 1+Max[Table[b[i] b[n-i], {i, n-1}]]];
a[n_] := b[n]-1;
Array[a, 42] (* Jean-François Alcover, Jun 25 2022 *)

a(n) = max_{i=1..floor(n/2)} (a(i)+1)*(a(n-i)+1), with a(1)=0.
a(n) = A091980(n) - 1.
a(2^n) = A004019(n) = A003095(n)^2.

cp's OEIS Frontend

A003095 a(n) = a(n-1)^2 + 1 for n >= 1, with a(0) = 0.

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A309049 Number T(n,k) of (binary) max-heaps on n elements from the set {0,1} containing exactly k 0's; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A004019 a(0) = 0; for n > 0, a(n) = (a(n-1) + 1)^2.

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A373451 Number T(n,k) of (binary) heaps of length n whose element set equals [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A372640 Number T(n,k) of defective (binary) heaps on n elements from the set {0,1} where k ancestor-successor pairs do not have the correct order; triangle T(n,k), n>=0, read by rows.

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A370484 Number T(n,k) of defective (binary) heaps on n elements from the set {0,1} with k defects; triangle T(n,k), n>=0, read by rows.

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A168542 Number of trees that have a maximum 'n'.