A309049 -id:A309049 - cp's OEIS frontend

A091980 Recursive sequence; one more than maximum of products of pairs of previous terms with indices summing to current index.

1, 2, 3, 5, 7, 11, 16, 26, 36, 56, 81, 131, 183, 287, 417, 677, 937, 1457, 2107, 3407, 4759, 7463, 10843, 17603, 24373, 37913, 54838, 88688, 123892, 194300, 282310, 458330, 634350, 986390, 1426440, 2306540, 3221844, 5052452, 7340712, 11917232, 16500522

Offset: 1

Franklin T. Adams-Watters, Mar 15 2004

The maximum is always obtained by taking i as the power of 2 nearest to n/2. - Anna de Mier, Mar 12 2012

a(n) is the number of (binary) max-heaps on n-1 elements from the set {0,1}. a(7) = 16: 000000, 100000, 101000, 101001, 110000, 110010, 110100, 110110, 111000, 111001, 111010, 111011, 111100, 111101, 111110, 111111. - Alois P. Heinz, Jul 09 2019

A. de Mier and M. Noy, On the maximum number of cycles in outerplanar and series-parallel graphs, Graphs Combin., 28 (2012), 265-275.

Alois P. Heinz, Table of n, a(n) for n = 1..5652
F. Disanto and N. A. Rosenberg, Enumeration of ancestral configurations for matching gene trees and species trees, J. Comput. Biol. 24 (2017), 831-850. See Section 4.2.
A. de Mier and M. Noy, On the maximum number of cycles in outerplanar and series-parallel graphs, Elect. Notes Discr. Math 34 (2009) 489-493
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Cf. A003095, A056971, A309049.
Partial differences give A168542.
a(n) = A355108(n)+1.
Column k=0 of A370484 and of A372640.

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-1):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 09 2019

a[n_] := a[n] = 1 + Max[Table[a[i] a[n-i], {i, n-1}]]; a[1] = 1;
Array[a, 50] (* Jean-François Alcover, Apr 30 2020 *)

a(n) = 1 + max_{i=1..n-1} a(i)*a(n-i) for n > 1, a(1) = 1.
From Alois P. Heinz, Jul 09 2019: (Start)
a(n) = Sum_{k=0..n-1} A309049(n-1,k).
a(2^(n-1)) = A003095(n). (End)

A114220 a(n) = Sum_{k=0..floor(n/2)} (k - (k-1)0^(n-2k)).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 4, 6, 7, 10, 11, 15, 16, 21, 22, 28, 29, 36, 37, 45, 46, 55, 56, 66, 67, 78, 79, 91, 92, 105, 106, 120, 121, 136, 137, 153, 154, 171, 172, 190, 191, 210, 211, 231, 232, 253, 254, 276, 277, 300, 301, 325, 326, 351, 352, 378, 379, 406, 407, 435, 436

Offset: 0

Paul Barry, Nov 18 2005

Diagonal sums of A114219.

It appears that the sequence terms from a(4) = 2 onwards are the exponents in the expansion of Sum_{n >= 0} q^(3*n-3) * Product_{k >= n} 1 - q^(2*k) = 1 - q^2 + q^3 - q^4 + q^6 - q^7 + q^10 - q^11 + - .... - Peter Bala, Jan 17 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1)

Cf. A055802, A114219.

Column k=2 of A309049 (for n>0).

[(2*n^2-2*n+7 + (9-2*n)*(-1)^n)/16: n in [0..80]]; // G. C. Greubel, Oct 21 2024

CoefficientList[Series[(1-x-x^2+2x^3)/((1-x)(1-x^2)^2), {x,0,80}],x] (* Harvey P. Dale, Mar 24 2011 *)

def A114220(n): return (2*n^2-2*n+7 + (9-2*n)*(-1)^n)//16
[A114220(n) for n in range(81)] # G. C. Greubel, Oct 21 2024

G.f.: (1-x-x^2+2x^3)/((1-x)*(1-x^2)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (2*n^2-2*n+7 + (9-2*n)*(-1)^n)/16.
a(n) = A055802(n+1), n > 1. - R. J. Mathar, Aug 11 2008
E.g.f.: (1/16)*((9 + 2*x)*exp(-x) + (7 + 2*x^2)*exp(x)). - G. C. Greubel, Oct 21 2024

A137560 Let f(z) = z^2 + c, then row k lists the expansion of the n-fold composition f(f(...f(0)...)) in rising powers of c.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 5, 6, 6, 4, 1, 0, 1, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1, 0, 1, 1, 2, 5, 14, 42, 100, 221, 470, 958, 1860, 3434, 6036, 10068, 15864, 23461, 32398, 41658, 49700, 54746, 55308, 50788, 41944, 30782, 19788

Offset: 0

Roger L. Bagula, Apr 25 2008

The root of one of these polynomials gives Julia Douady's rabbit.
These polynomials are basic to the theory of "cycles" in complex dynamics.
These polynomials are also described in a comment by Donald D. Cross in the entry for the Catalan numbers, A000108.
Except for the first row, row sums are A003095 (a(n) = a(n-1)^2 + 1). - Gerald McGarvey, Sep 26 2008
The coefficients also enumerate the ways to divide a line segment into at most j pieces, with 0 <= j <= 2^n, in which every piece is a power of two in size (for example, 1/4 is allowed but 3/8 is not), no piece is less than 1/2^n of the whole, and every piece is aligned on a power of 2 boundary (so 1/4+1/2+1/4=1 is not allowed). See the everything2 web link (which treats the segment as a musical measure). - Robert Munafo, Oct 29 2009
Also the number of binary trees with exactly J leaf nodes and a height no greater than N. See the Munafo web page and note the connection to A003095. - Robert Munafo, Nov 03 2009
The sequence of polynomials is conjectured to tend to the Catalan numbers (A000108). - Jon Perry, Oct 31 2010
It can be shown that the initial n nonzero terms of row n are the first Catalan numbers. - Joerg Arndt, Jun 04 2016
From Jianing Song, Mar 23 2021: (Start)
Let P_0(z) = 0, P_{n+1}(z) = P_n(z)^2 + z for n >= 0. For n > 0, the n-th row gives the coefficients of P_n(z) (a polynomial with degree 2^(n-1) for n > 0) in rising powers of z. Note that the famous Mandelbrot set is Intersect_{n>=0} {z: |P_n(z)| <= 2}. In particular, the Mandelbrot set is compact since it is closed and bounded.
Let P(z) = (1 - sqrt(1-4*z))/2. For every 0 < r < 1/4, P_n(z) converges uniformly to P(z) on the disk {z: |z| <= r}, because |P_n(z) - P(z)| <= (1/2)*(1 - sqrt(1-4*r))^(n+1) for every |z| <= r. Note that P(z)/z is the generating function for Catalan numbers, which explains the comment from Joerg Arndt above. Is the convergence uniform on the disk {z: |z| <= 1/4}? (End)

			Triangle starts:
  {1},
  {0, 1},
  {0, 1, 1},
  {0, 1, 1, 2, 1},
  {0, 1, 1, 2, 5, 6, 6, 4, 1},
  {0, 1, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},
  {0, 1, 1, 2, 5, 14, 42, 100, 221, 470, 958, 1860, 3434, 6036, 10068, 15864, 23461, 32398, 41658, 49700, 54746, 55308, 50788, 41944, 30782, 19788, 10948, 5096, 1932, 568, 120, 16, 1},
  ...

Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer, New York, 1993, pp 128-129

Alois P. Heinz, Rows n = 0..13, flattened (rows n=0..8 from Roger L. Bagula)
Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, Some Facts and Conjectures about Mandelbrot Polynomials, Maple Trans., Vol. 1, No. 1, Article 1 (July 2021).
Robert Munafo, Lemniscates [From _Robert Munafo_, Oct 29 2009]
Everything2 user ferrouslepidoptera, How many melodies are there in the universe? [From _Robert Munafo_, Oct 29 2009]
Wikipedia, Mandelbrot set

A052154 gives the same array read by antidiagonals.
A137867 gives the related Misiurewicz polynomials. [From Robert Munafo, Dec 12 2009]
Cf. A202019 (reversed rows).
Cf. A309049.

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-> `if`(n=0, 1, (m-> (p-> seq(coeff(p, x, m-i),
                  i=-1..m))(b(m)))(2^(n-1)-1)):
seq(T(n), n=0..7);  # Alois P. Heinz, Jul 11 2019

f[z_] = z^2 + x; g = Join[{1}, ExpandAll[NestList[f, x, 7]]]; a = Table[CoefficientList[g[[n]], x], {n, 1, Length[g]}]; Flatten[a] Table[Apply[Plus, CoefficientList[g[[n]], x]], {n, 1, Length[g]}];

p = vector(6); p[1] = x; for(n=2,6, p[n] = p[n-1]^2 + x); print1("1"); for(n=1,6, for(m=0,poldegree(p[n]), print1(", ",polcoeff(p[n],m)))) \\ Gerald McGarvey, Sep 26 2008

Edited by N. J. A. Sloane, Apr 26 2008

Offset set to 0 and new name from Joerg Arndt, Jun 04 2016

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).

A024358 Sum of the sizes of binary subtrees of the perfect binary tree of height n.

Original entry on oeis.org

0, 1, 8, 105, 6136, 8766473, 8245941529080, 3508518207951157937469961, 311594265746788494170062926869662848646207622648, 1217308491239906829392988008143949647398943617188660186130545502913055217344025410733271773705

Offset: 0

Cyril Banderier, Jun 09 2000

Size of binary tree = number of internal nodes.

Alois P. Heinz, Table of n, a(n) for n = 0..12
Cyril Banderier, On the sum of the sizes of binary subtrees of a perfect binary tree, personal note, 2000.

Cf. A003095, A309049, A309052.

B:= proc(n) B(n):= `if`(n<0, 0, expand(1+x*B(n-1)^2)) end:
a:= n-> subs(x=1, diff(B(n), x)):
seq(a(n), n=0..9);  # Alois P. Heinz, Jul 12 2019

B[n_] := If[n<0, 0, Expand[1+x*B[n-1]^2]];
a[n_] := D[B[n], x] /. x -> 1;
Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Oct 13 2022, after Alois P. Heinz *)

a(n) = B'n(1) where B{n+1}(x) = 1 + x*B_n(x)^2.
From Alois P. Heinz, Jul 12 2019: (Start)
a(n) = Sum_{k=0..2^n-1} (2^n-1-k) * A309049(2^n-1,k).
a(n) = A309052(2^n-1). (End)

a(0) changed to 0 by Alois P. Heinz, Jul 12 2019

A202019 Triangle by rows, related to the numbers of binary trees of height less than n, derived from the Mandelbrot set.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 4, 6, 6, 5, 2, 1, 1, 1, 8, 28, 60, 94, 116, 114, 94, 69, 44, 26, 14, 5, 2, 1, 1, 1, 16, 120, 568, 1932, 5096, 10948, 19788, 30782, 41944, 50788, 55308, 54746, 49700, 41658, 32398, 23461, 15864, 10068, 6036, 3434, 1860, 958, 470, 221, 100, 42, 14, 5, 2, 1, 1

Offset: 1

Gary W. Adamson, Dec 08 2011

As shown on p. 74 [Diaconis & Graham], n-th row polynomials are cyclic with period n, given real roots, if the polynomials are divided through by n. For example, taking x^3 + 2x^2 + x + 1 = 0, the real root = -1.75487766... = c. Then using x^2 + c, we obtain the period three trajectory: -1.75487... -> 1.32471...-> 0.
The shuffling connection [p.75], resulting in a permutation that is the Gilbreath shuffle: "To make the connection with shuffling cards, write down a periodic sequence starting at zero. Write a one above the smallest point, a two above the next smallest point and so on. For example, if c = -1.75486...(a period three point), we have:
2.............1.............3......
0........-1.75487........1.32471... For a fixed value of c, the numbers written on top code up a permutation that is a Gilbreath shuffle".
Row sums = A003095: (1, 2, 5, 26, 677,...) relating to the number of binary trees of height less than n.
Let f(z) = z^2 + c, then row k lists the expansion of the n-fold composition f(f(...f(0)...)) in falling powers of c (with the coefficients for c^0 omitted). The n initial terms of the reversed n-th row are the Catalan numbers (cf. A137560). - Joerg Arndt, Jun 04 2016

			Row 4 = (1, 4, 6, 6, 5, 2, 1, 1) since (x^4 + 2x^3 + x^2 + x)^2 + x = x^8 + 4x^7 + 6x^6 + 6x^5 + 5x^4 + 2x^3 + x^2 + x.
Triangle begins:
  1;
  1, 1;
  1, 2,  1,  1;
  1, 4,  6,  6,  5,   2,   1,  1;
  1, 8, 28, 60, 94, 116, 114, 94, 69, 44, 26, 14, 5, 2, 1, 1;
  ...

Persi Diaconis & R. L. Graham, "Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks", Princeton University Press, 2012; pp. 73-83.

Alois P. Heinz, Rows n = 1..13, flattened
Juan Carlos Nuño and Francisco J. Muñoz, Entropy-Variance curves of binary sequences generated by random substitutions of constant length, arXiv:2112.15563 [math.PR], 2021.

Row sums are A003095.
Cf. A137560 (reversed rows).
Cf. A309049.

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..degree(p)))(b(2^(n-1)-1)):
seq(T(n), n=1..7);  # Alois P. Heinz, Jul 11 2019

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^(Length[IntegerDigits[n, 2]]-1)]];
T[n_] := CoefficientList[b[2^(n-1)-1], x];
Array[T, 7] // Flatten (* Jean-François Alcover, Feb 19 2021, after Alois P. Heinz *)

Coefficients of x by rows such that given n-th row p(x), the next row is (p(x))^2 + x; starting x -> (x^2 + x) -> (x^4 + 2*x^3 + x^2 + x)...

T(n,k) = A309049(2^(n-1)-1,k-1). - Alois P. Heinz, Jul 11 2019

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).

A309050 Number of (binary) max-heaps on 2n elements from the set {0,1} containing n 0's and n 1's.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 27, 54, 109, 219, 460, 962, 1986, 4044, 8516, 18058, 37801, 77701, 164300, 350336, 738945, 1530521, 3250659, 6962248, 14735660, 30625898, 65206770, 140040538, 297712980, 622136512, 1328716192, 2861101350, 6086238317, 12716525621, 27172910036

Offset: 0

Alois P. Heinz, Jul 09 2019

nonn

Also the number of (binary) min-heaps on 2n elements from the set {0,1} containing n 0's and n 1's.

			a(0) = 1: ().
a(1) = 1: 10.
a(2) = 2: 1010, 1100.
a(3) = 4: 101001, 110010, 110100, 111000.
a(4) = 7: 10100110, 11010001, 11011000, 11100010, 11100100, 11101000, 11110000.
a(5) = 13: 1101000110, 1101100001, 1101100010, 1101100100, 1110011000, 1110100001, 1110101000, 1110110000, 1111000010, 1111000100, 1111001000, 1111010000, 1111100000.

Alois P. Heinz, Table of n, a(n) for n = 0..3000
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Cf. A056971, A309049.

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-> coeff(b(2*n), x, n):
seq(a(n), n=0..40);

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^(Length@IntegerDigits[n, 2] - 1)]];
a[n_] := Coefficient[b[2n], x, n];
a /@ Range[0, 40] (* Jean-François Alcover, Apr 19 2021, after Alois P. Heinz *)

a(n) = A309049(2n,n).

A326504 Number of (binary) max-heaps on n elements from the set {0,1} containing exactly three 0's.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 12, 16, 23, 27, 38, 44, 60, 66, 88, 96, 125, 133, 170, 180, 226, 236, 292, 304, 371, 383, 462, 476, 568, 582, 688, 704, 825, 841, 978, 996, 1150, 1168, 1340, 1360, 1551, 1571, 1782, 1804, 2036, 2058, 2312, 2336, 2613, 2637, 2938, 2964, 3290

Offset: 3

Alois P. Heinz, Jul 09 2019

Alois P. Heinz, Table of n, a(n) for n = 3..10000
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,0,0,-2,2,1,-1).

Column k=3 of A309049.

LinearRecurrence[{1,2,-2,0,0,-2,2,1,-1},{1,1,2,4,6,8,12,16,23},60] (* Harvey P. Dale, Mar 11 2023 *)

G.f.: x^3*(2*x^6-2*x^5+2*x^3-x^2+1)/((x^2+1)*(x+1)^3*(x-1)^4).

cp's OEIS Frontend

A091980 Recursive sequence; one more than maximum of products of pairs of previous terms with indices summing to current index.

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A114220 a(n) = Sum_{k=0..floor(n/2)} (k - (k-1)*0^(n-2*k)).

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A137560 Let f(z) = z^2 + c, then row k lists the expansion of the n-fold composition f(f(...f(0)...)) in rising powers of c.

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Maple

Mathematica

PARI

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

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Maple

Mathematica

PARI

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

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Maple

Mathematica

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A024358 Sum of the sizes of binary subtrees of the perfect binary tree of height n.

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Maple

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A202019 Triangle by rows, related to the numbers of binary trees of height less than n, derived from the Mandelbrot set.

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A114220 a(n) = Sum_{k=0..floor(n/2)} (k - (k-1)0^(n-2k)).