A000992 -id:A000992 - cp's OEIS frontend

A178833 Partial sums of "Half-Catalan numbers" A000992.

1, 2, 3, 5, 8, 14, 25, 49, 96, 199, 413, 894, 1924, 4261, 9392, 21205, 47534, 108492, 246313, 568003, 1302431, 3024429, 6990985, 16343338, 38026783, 89322813, 208986625, 493184761, 1159317065, 2745547588, 6480141829, 15399987104, 36475269692, 86916706534, 206503331542

Offset: 1

Jonathan Vos Post, Jan 01 2011

The subsequence of primes begins: 2, 3, 5, 199, 4261, 493184761.

The subsequence of perfect powers begins: 1, 8, 25, 49.

			A000992 starts with 1, 1, 1, 2, 3, ... giving partial sums 1, 2, 3, 5, 8 ...

Cf. A000992, A001190, A093637, A124973.

b:= proc(n) option remember; `if`(n=1, 1,
      add(b(j)*b(n-j), j=1..n/2))
    end:
a:= proc(n) option remember; `if`(n<1, 0, b(n)+a(n-1)) end:
seq(a(n), n=1..42);  # Alois P. Heinz, Nov 04 2024

lista(nn) = for (k=1, nn, print1(vecsum(A000992_list(k)), ", ")); \\ Michel Marcus, Feb 16 2015

a(n) = Sum_{i=1..n} A000992(i).

A196545 Number of weakly ordered plane trees with n leaves.

Original entry on oeis.org

1, 1, 2, 5, 12, 34, 92, 277, 806, 2500, 7578, 24198, 75370, 243800, 776494, 2545777, 8223352, 27221690, 88984144, 296856400, 979829772, 3287985078, 10934749788, 36912408342, 123519937044, 418650924886, 1408867195252, 4794243983204, 16205061000480

Offset: 1

Gus Wiseman, Oct 03 2011

nonn

A weakly ordered plane tree (p-tree) of weight n > 1 is a sequence (t_1, t_2, ..., t_k) where k > 1 and for some integer partition y of length k and sum n, the term t_i is a p-tree of weight y_i, for 1 <= i <= k. For n = 1, the only p-tree is a single node.

This definition precludes nodes with only one branch, and non-leaf nodes have weight 0. If the above is changed so that k >= 1 and y is partition of n-1, we get the trees counted by A093637. Binary p-trees are counted by A000992.

			Let o denote a single node. The 12 p-trees of weight 5 are: ((((oo)o)o)o), (((ooo)o)o), (((oo)(oo))o), (((oo)oo)o), ((oooo)o), (((oo)o)(oo)), ((ooo)(oo)), (((oo)o)oo), ((ooo)oo), ((oo)(oo)o), ((oo)ooo), (ooooo).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Callan, The weakly ordered plane trees of size up to 4
Gus Wiseman, All 277 weakly ordered plane trees of weight 8

Cf. A000041, A063834, A093637, A000992.

b:= proc(n, i) option remember;
      `if`(i>n, 0, a(i)*`if`(i=n, 1, b(n-i, i)))+`if`(i>1, b(n, i-1), 0)
    end:
a:= proc(n) option remember;
      `if`(n=1, 1, add(a(k)*b(n-k, min(n-k, k)), k=1..n-1))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 06 2012

PTNum[n_] := PTNum[n] =
    If[n === 1, 1, Plus @@ Function[y,
    Times @@ PTNum /@ y] /@ Rest[Partitions[n]]]; Array[PTNum, 20]
(* Second program: *)
b[n_, i_] := b[n, i] = If[i>n, 0, a[i]*If[i == n, 1, b[n-i, i]]] + If[i>1, b[n, i-1], 0];
a[n_] := a[n] = If[n == 1, 1, Sum[a[k]*b[n-k, Min[n-k, k] ], {k, 1, n-1}]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *)

seq(n)={my(v=vector(n)); v[1] = 1; for(n=2, n, v[n] = polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); v} \\ Andrew Howroyd, Aug 26 2018

@cached_function
def A196545(n):
    if n == 1: return 1
    return sum(prod(A196545(t) for t in p) for p in Partitions(n,min_length=2))
# D. S. McNeil, Oct 03 2011

a(1) = 1; for n > 1, a(n) = Sum_{y} Product_{i} a(y_i) where the sum is over all partitions of n with at least two parts.

The generating function is characterized by the formal equation 1 + 2*S(x) = x + 1/P(x) where S(x) = Sum_{n>0} a(n)*x^n and P(x) = Product_{n>0} (1 - a(n)*x^n) are the formal infinite series and formal infinite product with a(n) as coefficients and roots respectively.

A201204 Half-convolution of Catalan sequence A000108 with itself.

Original entry on oeis.org

1, 1, 3, 7, 23, 66, 227, 715, 2529, 8398, 30275, 104006, 380162, 1337220, 4939443, 17678835, 65844845, 238819350, 895451117, 3282060210, 12374186318, 45741281820, 173257703723, 644952073662, 2452607696798, 9183676536076, 35042725663002, 131873975875180, 504697422982484, 1907493251046152

Offset: 0

Wolfdieter Lang, Jan 02 2012

In general the half-convolution of a sequence {b(n)}_0^infty with itself is defined by chat(n):=sum(b(k)*b(n-k), k=0..floor(n/2)), n>=0. The o.g.f. of the sequence {chat(n)} is obtained from the bisection 2*chat(2*k) - b(k)^2 = c(2*k), k>=0, with the ordinary convolution c(n):=sum(b(k)*b(n-k),k=0..n), n>=0, and 2*chat(2*k+1) = c(2*k+1), k>=0. This leads to the o.g.f.s  for the corresponding even (e) and odd (o) parts:
  2*Chate(x) - B2(x) = Ce(x) and 2*Chato(x) = Co(x), where Chate(x):= sum(chat(2*k)*x^k,k=0..infty), Chato(x):= sum(chat(2*k+1)*x^k,k=0..infty), B2(x) := sum(b(k)^2*x^k, k=0..infty), Ce(x) := sum(c(2*k)*x^k, k=0..infty) and Co(x) := sum(c(2*k+1)*x^k, k=0..infty). Thus Chate(x)=(Ce(x) + B2(x))/2 and Chato(x)=Co(x)/2. Expressing this in terms of C(x), the o.g.f. of {c(n)}, and B2(x) leads to the result: Chat(x)= (C(x) + B2(x^2))/2.
In the Catalan case b(n)=A000108(n), c(n)=b(n+1), C(x)= (cata(x)+1)/x, with the o.g.f. of A000108 cata(x)=(1-sqrt(1-4*x))/(2*x), and B2(x) is found under A001246 to be (-1 + hypergeom([-1/2,-1/2],[1],16*x))/(4*x). This produces the o.g.f. given in the formula section.
This computation was motivated by a question about the o.g.f. of A000992 ("half-Catalan numbers"). Note, however, that this sequence is not the half-convolution of the Catalan numbers presented here.
Apparently the number of hills to the left of or at the midpoint in all Dyck paths of semilength n+1. [David Scambler, Apr 30 2013]

Robert Israel, Table of n, a(n) for n = 0..170

A000108, bisection: A201205 and A065097.

C:= n -> binomial(2*n,n)/(n+1):
A:= n -> add(C(k)*C(n-k),k=0..floor(n/2));
seq(A(i),i=1..100); # Robert Israel, Jun 06 2014

Table[Sum[CatalanNumber[k]CatalanNumber[n-k],{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, Jun 12 2012 *)
Table[CatalanNumber[n + 1]/2 + 2^(2 n + 1) Binomial[1/2, n/2 + 1]^2, {n, 0, 30}] (* Vladimir Reshetnikov, Oct 03 2016 *)

a(n) = sum(Catalan(k)*Catalan(n-k),k=0..floor(n/2)), n>=0, with Catalan(n)=A000108(n).
O.g.f.: G(x)=(catalan(x)-1)/(2*x)+(-1+hypergeom([-1/2,-1/2],[1],16*x^2))/(8*x^2), with the o.g.f. catalan(x) of the Catalan numbers (see also the comment section).
a(n) ~ 2^(2*n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 15 2014
a(n) = A000108(n+1)/2 + 2^(2*n+1) * binomial(1/2, n/2+1)^2. - Vladimir Reshetnikov, Oct 03 2016
D-finite with recurrence: (n+1)*(n+2)^2*a(n) +6*(n-2)*(n+1)^2*a(n-1) +4*(-16*n^3+25*n^2+4*n-4)*a(n-2) +16*(-4*n^3+25*n^2-56*n+41)*a(n-3) +192*(4*n-7)*(n-3)^2*a(n-4) -256*(2*n-7)*(n-4)^2*a(n-5)=0. - R. J. Mathar, Feb 21 2020

A300443 Number of binary enriched p-trees of weight n.

Original entry on oeis.org

1, 1, 2, 3, 8, 15, 41, 96, 288, 724, 2142, 5838, 17720, 49871, 151846, 440915, 1363821, 4019460, 12460721, 37374098, 116809752, 353904962, 1109745666, 3396806188, 10712261952, 33006706419, 104357272687, 323794643722, 1027723460639, 3204413808420, 10193485256501

Offset: 0

Gus Wiseman, Mar 05 2018

nonn

A binary enriched p-tree of weight n is either a single node of weight n, or an ordered pair of binary enriched p-trees with weakly decreasing weights summing to n.

			The a(4) = 8 binary enriched p-trees: 4, (31), (22), ((21)1), ((11)2), (2(11)), (((11)1)1), ((11)(11)).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000992, A001190, A063834, A196545, A273873, A289501, A300354, A300439, A300442.

a:= proc(n) option remember;
      1+add(a(j)*a(n-j), j=1..n/2)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 06 2018

j[n_]:=j[n]=1+Sum[Times@@j/@y,{y,Select[IntegerPartitions[n],Length[#]===2&]}];
Array[j,40]
(* Second program: *)
a[n_] := a[n] = 1 + Sum[a[j]*a[n-j], {j, 1, n/2}];
a /@ Range[0, 40] (* Jean-François Alcover, May 12 2021, after Alois P. Heinz *)

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(k=1, n\2, v[k]*v[n-k])); concat([1], v)} \\ Andrew Howroyd, Aug 26 2018

a(n) = 1 + Sum_{x + y = n, 0 < x <= y < n} a(x) * a(y).

A300442 Number of binary strict trees of weight n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 23, 46, 108, 231, 561, 1285, 3139, 7348, 18265, 43907, 109887, 267582, 675866, 1669909, 4238462, 10555192, 26955062, 67706032, 173591181, 438555624, 1129088048, 2869732770, 7410059898, 18911818801, 48986728672, 125562853003, 326011708368

Offset: 0

Gus Wiseman, Mar 05 2018

nonn

A binary strict tree of weight n > 0 is either a single node of weight n, or an ordered pair of binary strict trees with strictly decreasing weights summing to n.

			The a(5) = 6 binary strict trees: 5, (41), (32), ((31)1), ((21)2), (((21)1)1).
The a(6) = 10 binary strict trees:
  6,
  (51), (42),
  ((41)1), ((32)1), ((31)2),
  (((31)1)1), (((21)2)1), (((21)1)2),
  ((((21)1)1)1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000992, A001190, A063834, A196545, A273873, A289501, A292432, A293511, A300352, A300440, A300443.

a:= proc(n) option remember;
      1+add(a(j)*a(n-j), j=1..(n-1)/2)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 06 2018

k[n_]:=k[n]=1+Sum[Times@@k/@y,{y,Select[IntegerPartitions[n],Length[#]===2&&UnsameQ@@#&]}];
Array[k,40]
(* Second program: *)
a[n_] := a[n] = 1 + Sum[a[j]*a[n - j], {j, 1, (n - 1)/2}];
a /@ Range[0, 40] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(k=1, (n-1)\2, v[k]*v[n-k])); concat([1], v)} \\ Andrew Howroyd, Aug 25 2018

a(n) = 1 + Sum_{x + y = n, 0 < x < y < n} a(x) * a(y).

A301368 Regular triangle where T(n,k) is the number of binary enriched p-trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 2, 4, 5, 3, 1, 3, 7, 12, 12, 6, 1, 3, 9, 19, 28, 25, 11, 1, 4, 14, 36, 65, 81, 63, 24, 1, 4, 16, 48, 107, 172, 193, 136, 47, 1, 5, 22, 75, 192, 369, 522, 522, 331, 103, 1, 5, 25, 96, 284, 643, 1108, 1420, 1292, 750, 214, 1, 6

Offset: 1

Gus Wiseman, Mar 19 2018

A binary enriched p-tree of weight n is either a single node of weight n, or an ordered pair of binary enriched p-trees with weakly decreasing weights summing to n.

			Triangle begins:
  1
  1   1
  1   1   1
  1   2   3   2
  1   2   4   5   3
  1   3   7  12  12   6
  1   3   9  19  28  25  11
  1   4  14  36  65  81  63  24
  1   4  16  48 107 172 193 136  47
  1   5  22  75 192 369 522 522 331 103
  ...
The T(6,3) = 7 binary enriched p-trees: ((41)1), ((32)1), (4(11)), ((31)2), ((22)2), (3(21)), ((21)3).

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Last entries of each row give A000992. Row sums are A300443.

Cf. A001190, A008284, A055277, A063834, A196545, A273873, A289501, A292050, A298422, A298426, A300354, A300439, A300442, A301344, A301364-A301367.

bintrees[n_]:=Prepend[Join@@Table[Tuples[bintrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]===2&]}],n];
Table[Length[Select[bintrees[n],Count[#,_Integer,{-1}]===k&]],{n,13},{k,n}]

A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + sum(k=1, n\2, v[k]*v[n-k])); apply(p->Vecrev(p/y), v)}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 26 2018

A327643 Number of refinement sequences n -> ... -> {1}^n, where in each step one part is replaced by a partition of itself into two smaller parts (in weakly decreasing order).

Original entry on oeis.org

1, 1, 1, 3, 6, 24, 84, 498, 2220, 15108, 92328, 773580, 5636460, 53563476, 471562512, 5270698716, 52117937052, 637276396764, 7317811499736, 100453675122444, 1276319138168796, 19048874583061716, 270233458572751440, 4442429353548965628, 68384217440167826412

Offset: 1

Alois P. Heinz, Sep 20 2019

nonn

Number of proper (n-1)-times partitions of n, cf. A327639.
Might be called "Half-Factorial numbers" analog to the "Half-Catalan numbers" (A000992).
The recursion formula is a special case of the formula given in A327729.
a(n+1)/(n*a(n)) tends to 0.67617164... - Vaclav Kotesovec, Apr 28 2020

			a(1) = 1:
  1
a(2) = 1:
  2 -> 11
a(3) = 1:
  3 -> 21 -> 111
a(4) = 3:
  4 -> 31 -> 211 -> 1111
  4 -> 22 -> 112 -> 1111
  4 -> 22 -> 211 -> 1111
a(5) = 6:
  5 -> 41 -> 311 -> 2111 -> 11111
  5 -> 41 -> 221 -> 1121 -> 11111
  5 -> 41 -> 221 -> 2111 -> 11111
  5 -> 32 -> 212 -> 1112 -> 11111
  5 -> 32 -> 212 -> 2111 -> 11111
  5 -> 32 -> 311 -> 2111 -> 11111

Alois P. Heinz, Table of n, a(n) for n = 1..481
Vaclav Kotesovec, Plot of a(n+1)/(n*a(n)) for n = 1..10000
Wikipedia, Partition (number theory)

Cf. A000142, A000992, A002846 (only one part of each size is replaceable), A327631, A327639, A327697, A327698, A327699, A327702, A327729.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
      b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> add(b(n$2, i)*(-1)^(n-1-i)*binomial(n-1, i), i=0..n-1):
seq(a(n), n=1..29);
# second Maple program:
a:= proc(n) option remember; `if`(n=1, 1,
      add(a(j)*a(n-j)*binomial(n-2, j-1), j=1..n/2))
    end:
seq(a(n), n=1..29);

a[n_] := a[n] = Sum[Binomial[n-2, j-1] a[j] a[n-j], {j, n/2}]; a[1] = 1;
Array[a, 25] (* Jean-François Alcover, Apr 28 2020 *)

a(n) = Sum_{j=1..floor(n/2)} C(n-2,j-1) a(j)*a(n-j) for n > 1, a(1) = 1.

a(n) = A327639(n,n-1) = A327631(n,n-1)/n.

A300866 Signed recurrence over binary strict trees: a(n) = 1 - Sum_{x + y = n, 0 < x < y < n} a(x) * a(y).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, -1, 1, 1, -2, 3, -1, -3, 8, -8, 1, 14, -26, 22, 10, -59, 90, -52, -74, 238, -291, 80, 417, -930, 915, 124, -1991, 3483, -2533, -2148, 9011, -12596, 5754, 14350, -37975, 42735, -4046, -77924, 154374, -133903, -56529, 376844, -591197, 355941, 522978, -1706239

Offset: 0

Gus Wiseman, Mar 13 2018

sign

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000992, A001190, A007317, A063834, A099323, A196545, A220418, A273866, A273873, A289501, A290261, A300442, A300443, A300862, A300863, A300864, A300865.

a[n_]:=a[n]=1-Sum[a[k]*a[n-k],{k,1,(n-1)/2}];
Array[a,40]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 - sum(k=1, (n-1)\2, v[k]*v[n-k])); concat([1], v)} \\ Andrew Howroyd, Aug 27 2018

A030036 a(n+1) = Sum_{k=0..floor(n/5)} a(k) * a(n-k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 5, 8, 13, 26, 47, 86, 159, 292, 584, 1121, 2156, 4153, 8014, 16028, 31472, 61823, 121490, 238827, 477654, 947294, 1878560, 3725648, 7389473, 14900436, 29796719, 59593438, 119178862, 238341696, 482287600, 970054641, 1951222247, 3924973230

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000992, A030032, A030034, A030037, A030038, A030039.

lista(nn) = {my(v=vector(nn+2, i, 1)); for(n=3, nn, v[n+2]=sum(k=1, 1+n\5, v[k]*v[n+2-k])); v; } \\ Jinyuan Wang, Mar 18 2020

More terms from Jinyuan Wang, Mar 18 2020

A300863 Signed recurrence over enriched p-trees: a(n) = (-1)^(n - 1) + Sum_{y1 + ... + yk = n, y1 >= ... >= yk > 0, k > 1} a(y1) * ... * a(yk).

Original entry on oeis.org

1, 0, 2, 2, 6, 14, 34, 82, 214, 566, 1482, 4058, 10950, 30406, 83786, 235714, 658286, 1874254, 5293674, 15189810, 43312542, 125075238, 359185586, 1043712922, 3015569582, 8800146182, 25565402802, 74918274562, 218572345718, 642783954238, 1882606578002

Offset: 1

Gus Wiseman, Mar 13 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..500

Cf. A000992, A063834, A099323, A196545, A220418, A273866, A273873, A289501, A290261, A300862, A300864, A300865, A300866.

a[n_]:=a[n]=(-1)^(n-1)+Sum[Times@@a/@y,{y,Select[IntegerPartitions[n],Length[#]>1&]}];
Array[a,40]

O.g.f.: (-1/(1+x) + Product 1/(1-a(n)x^n))/2.

cp's OEIS Frontend

A178833 Partial sums of "Half-Catalan numbers" A000992.

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A196545 Number of weakly ordered plane trees with n leaves.

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A201204 Half-convolution of Catalan sequence A000108 with itself.

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A300443 Number of binary enriched p-trees of weight n.

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A300442 Number of binary strict trees of weight n.

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A301368 Regular triangle where T(n,k) is the number of binary enriched p-trees of weight n with k leaves.

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Mathematica

PARI

A327643 Number of refinement sequences n -> ... -> {1}^n, where in each step one part is replaced by a partition of itself into two smaller parts (in weakly decreasing order).

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