A256116 -id:A256116 - cp's OEIS frontend

A183135 Square array A(n,k) by antidiagonals. A(n,k) is the number of length 2n k-ary words (n,k>=0) that can be built by repeatedly inserting doublets into the initially empty word.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 20, 1, 0, 1, 5, 28, 87, 70, 1, 0, 1, 6, 45, 232, 543, 252, 1, 0, 1, 7, 66, 485, 2092, 3543, 924, 1, 0, 1, 8, 91, 876, 5725, 19864, 23823, 3432, 1, 0, 1, 9, 120, 1435, 12786, 71445, 195352, 163719, 12870, 1, 0

Offset: 0

Alois P. Heinz, Dec 26 2010

A(n,k) is also the number of rooted closed walks of length 2n on the infinite rooted k-ary tree. - Danny Rorabaugh, Oct 31 2017

A(n,2k) is the number of unreduced words of length 2n that reduce to the empty word in the free group with k generators. - Danny Rorabaugh, Nov 09 2017

			A(2,2) = 6, because 6 words of length 4 can be built over 2-letter alphabet {a, b} by repeatedly inserting doublets (words with two equal letters) into the initially empty word: aaaa, aabb, abba, baab, bbaa, bbbb.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,     1,  ...
  0,  1,   2,    3,     4,     5,  ...
  0,  1,   6,   15,    28,    45,  ...
  0,  1,  20,   87,   232,   485,  ...
  0,  1,  70,  543,  2092,  5725,  ...
  0,  1, 252, 3543, 19864, 71445,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Jason Bell, Marni Mishna, On the Complexity of the Cogrowth Sequence, arXiv:1805.08118 [math.CO], 2018.
Beth Bjorkman et al., k-foldability of words, arXiv preprint arXiv:1710.10616 [math.CO], 2017.

Columns k=0-10 give: A000007, A000012, A000984, A089022, A035610, A130976, A130977, A130978, A130979, A130980, A131521.
Rows n=0-3 give: A000012, A001477, A000384, A027849(k-1) for k>0.
Main diagonal gives A294491.
Coefficients of row polynomials in k, (k-1) are given by A157491, A039599.
Cf. A007318, A183134, A256116, A256117.

A:= proc(n, k) local j;
      if n=0 then 1
             else k/n *add(binomial(2*n,j) *(n-j) *(k-1)^j, j=0..n-1)
      fi
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[, 1] = 1; A[n, k_] := If[n == 0, 1, k/n*Sum[Binomial[2*n, j]*(n - j)*(k - 1)^j, {j, 0, n - 1}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

A(n,k) = k/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*(k-1)^j if n>0, A(0,k) = 1.
A(n,k) = A183134(n,k) if n=0 or k<2, A(n,k) = A183134(n,k)*k otherwise.
G.f. of column k: 1/(1-k*x) if k<2, 2*(k-1)/(k-2+k*sqrt(1-(4*k-4)*x)) otherwise.

A256117 Number T(n,k) of length 2n words such that all letters of the k-ary alphabet occur at least once and are introduced in ascending order and which can be built by repeatedly inserting doublets into the initially empty word; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 9, 5, 0, 1, 34, 56, 14, 0, 1, 125, 465, 300, 42, 0, 1, 461, 3509, 4400, 1485, 132, 0, 1, 1715, 25571, 55692, 34034, 7007, 429, 0, 1, 6434, 184232, 657370, 647920, 231868, 32032, 1430, 0, 1, 24309, 1325609, 7488228, 11187462, 6191808, 1447992, 143208, 4862

Offset: 0

Alois P. Heinz, Mar 15 2015

In general, column k>2 is asymptotic to (4*(k-1))^n / ((k-2)^2 * (k-2)! * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

			T(0,0) = 1: (the empty word).
T(1,1) = 1: aa.
T(2,1) = 1: aaaa.
T(2,2) = 2: aabb, abba.
T(3,1) = 1: aaaaaa.
T(3,2) = 9: aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba.
T(3,3) = 5: aabbcc, aabccb, abbacc, abbcca, abccba.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,    2;
  0, 1,    9,      5;
  0, 1,   34,     56,     14;
  0, 1,  125,    465,    300,     42;
  0, 1,  461,   3509,   4400,   1485,    132;
  0, 1, 1715,  25571,  55692,  34034,   7007,   429;
  0, 1, 6434, 184232, 657370, 647920, 231868, 32032, 1430;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A057427, A010763(n-1) (for n>1), A258490, A258491, A258492, A258493, A258494, A258495, A258496, A258497.
Main diagonal gives A000108.
T(n+2,n+1) gives A002055(n+5).
Row sums give A258498.
T(2n,n) gives A258499.
Cf. A183135, A256116, A256311.

A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
      add(binomial(2*n, j)*(n-j)*(k-1)^j, j=0..n-1))
    end:
T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

A[n_, k_] := A[n, k] = If[n == 0, 1, k/n*Sum[Binomial[2*n, j]*(n - j)*If[j == 0, 1, (k - 1)^j], {j, 0, n - 1}]];
T[n_, k_] := Sum[(-1)^i*A[n, k - i]/(i!*(k - i)!), {i, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz, updated Jan 01 2021 *)

T(n,k) = Sum_{i=0..k} (-1)^i * A183135(n,k-i) / (i!*(k-i)!).

T(n,k) = A256116(n,k) / (k-1)! for k > 0.

A065866 a(n) = n! * Catalan(n+1).

Original entry on oeis.org

1, 2, 10, 84, 1008, 15840, 308880, 7207200, 196035840, 6094932480, 213322636800, 8303173401600, 355850288640000, 16653793508352000, 845180020548864000, 46236318771202560000, 2712530701243883520000, 169890080762116915200000, 11314679378756986552320000

Offset: 0

Len Smiley, Dec 06 2001

nonn

From Noam Zeilberger, Mar 19 2019: (Start)
a(n) is the number of flags in the associahedron of dimension n. For example, there are a(2) = 10 flags in the associahedron of dimension 2, a pentagon. (In this case a flag corresponds to a triple v:e:f of a mutually incident vertex v, edge e, and face f, with f necessarily the unique face of the pentagon.)
Equivalently, a(n) is the number of maximal sequences of consistent parenthesizations of a string of n + 2 letters, starting with n + 1 pairs of parentheses, then removing one pair, and so on, ending with the trivial (outermost) parenthesization. For example, (a(b(cd))):(ab(cd)):(abcd) and (a(b(cd))):(a(bcd)):(abcd) are two of the a(2) = 10 maximal sequences of consistent parenthesizations of the letters abcd. (End)

			G.f. = 1 + 2*x + 10*x^2 + 84*x^3 + 1008*x^4 + 15840*x^5 + 308880*x^6 + ...

R. L. Graham, D. E. Knuth, and O. Patashnik, "Concrete Mathematics", Addison-Wesley, 1994, pp. 200-204.

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A000108, A001761.
Cf. A370055, A370058.
Equals 2 * A102693(n+1), n > 0.
Main diagonal of A256116.

List([0..20], n-> 2*Factorial(2*n+1)/Factorial(n+2)); # G. C. Greubel, Mar 19 2019

[Factorial(n)*Catalan(n+1): n in [0..20]]; // G. C. Greubel, Mar 19 2019

with(combstruct): ZL:=[T, {T=Union(Z, Prod(Epsilon, Z, T), Prod(T, Z, Epsilon), Prod(T, T, Z))}, labeled]: seq(count(ZL, size=i+1)/(i+1), i=0..18); # Zerinvary Lajos, Dec 16 2007
a := n -> (2^(2*n+2)*GAMMA(n+3/2))/(sqrt(Pi)*(n+1)*(n+2)):
seq(simplify(a(n)), n=0..17); # Peter Luschny, Mar 20 2019

Table[2*(2n+1)!/(n+2)!, {n,0,20}] (* G. C. Greubel, Mar 19 2019 *)
Table[n! CatalanNumber[n+1],{n,0,20}] (* Harvey P. Dale, Feb 02 2023 *)

{ for (n = 0, 100, a = 2 * (2*n + 1)!/(n + 2)!; write("b065866.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 02 2009

[factorial(n)*catalan_number(n+1) for n in (0..20)] # G. C. Greubel, Mar 19 2019

a(n) = 2 * (2n+1)!/(n+2)!.
E.g.f.: (1-2*x-sqrt(1-4*x))/(2*x^2) = (O.g.f. for A000108)^2 = B_2(x)^2 (cf. GKP reference).
0 = a(n)*(-7200*a(n+2) + 2700*a(n+3) + 246*a(n+4) - 33*a(n+5)) + a(n+1)*(+36*a(n+2) + 372*a(n+3) + 36*a(n+4) - a(n+5)) + a(n+2)*(-18*a(n+2) + 9*a(n+3) + a(n+4)) for n >= 0. - Michael Somos, Apr 14 2015
The e.g.f. A(x) satisfies 0 = -2 + A(x) * (6*x - 2) + A'(x) * (4*x^2 - x). - Michael Somos, Apr 14 2015
(n+2)*a(n) - 2*n*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Oct 31 2015
a(n) ~ 4^n*exp(-n)*n^(n - 2)*sqrt(2)*(24*n - 61)/6. - Peter Luschny, Mar 20 2019
Sum_{n>=0} 1/a(n) = (25*exp(1/4)*sqrt(Pi)*erf(1/2) + 22)/32, where erf is the error function. - Amiram Eldar, Dec 04 2022
a(n) = 2 * Sum_{k=0..n} (n+2)^(k-1) * |Stirling1(n,k)|. - Seiichi Manyama, Aug 31 2024
E.g.f.: (1/x) * Series_Reversion( x/(1 + x)^2 ). - Seiichi Manyama, Feb 06 2025

A294603 Number of words of semilength n over n-ary alphabet, either empty or beginning with the first letter of the alphabet, such that the index set of occurring letters is an integer interval [1, k], that can be built by repeatedly inserting doublets into the initially empty word.

Original entry on oeis.org

1, 1, 3, 20, 231, 3864, 85360, 2353546, 77963599, 3019479344, 133966276692, 6702399275538, 373406941221160, 22930441709648290, 1539004344848618466, 112089683771614695478, 8805334896381292460191, 742162775145283382779168, 66809386370870410069346476

Offset: 0

Alois P. Heinz, Nov 03 2017

nonn

			a(0) = 1: the empty word.
a(1) = 1: aa.
a(2) = 3: aaaa, aabb, abba.
a(3) = 20: aaaaaa, aaaabb, aaabba, aabaab, aabbaa, aabbbb, aabbcc, aabccb, aacbbc, aaccbb, abaaba, abbaaa, abbabb, abbacc, abbbba, abbcca, abccba, acbbca, accabb, accbba.

Alois P. Heinz, Table of n, a(n) for n = 0..350

Row sums of A256116.

Cf. A258498.

A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
      add(binomial(2*n, j) *(n-j) *(k-1)^j, j=0..n-1))
    end:
T:= proc(n, k) option remember;
      add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)/`if`(k=0, 1, k)
    end:
a:= n-> add(T(n, k), k=0..n):
seq(a(n), n=0..20);

A[n_, k_] := A[n, k] = If[n == 0, 1, k/n*
     Sum[Binomial[2*n, j]*(n-j) *If[j == 0, 1, (k - 1)^j], {j, 0, n - 1}]];
T[n_, k_] := T[n, k] =
     Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}]/If[k == 0, 1, k];
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} A256116(n,k).

cp's OEIS Frontend

A183135 Square array A(n,k) by antidiagonals. A(n,k) is the number of length 2n k-ary words (n,k>=0) that can be built by repeatedly inserting doublets into the initially empty word.

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A256117 Number T(n,k) of length 2n words such that all letters of the k-ary alphabet occur at least once and are introduced in ascending order and which can be built by repeatedly inserting doublets into the initially empty word; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A065866 a(n) = n! * Catalan(n+1).

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A294603 Number of words of semilength n over n-ary alphabet, either empty or beginning with the first letter of the alphabet, such that the index set of occurring letters is an integer interval [1, k], that can be built by repeatedly inserting doublets into the initially empty word.

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