A213027 -id:A213027 - cp's OEIS frontend

A047099 a(n) = A047098(n)/2.

1, 4, 19, 98, 531, 2974, 17060, 99658, 590563, 3540464, 21430267, 130771376, 803538100, 4967127736, 30866224824, 192696614730, 1207967820099, 7600482116932, 47981452358201, 303820299643138, 1929099000980219, 12279621792772822, 78346444891033856

Offset: 1

Clark Kimberling, Dec 11 1999

nonn

T(2*n,n)/2, with array T as in A047110.
Also given by a recurrence that features row 3 of the Pascal triangle (Mathematica code): u[0,0]=1; u[n_,k_]/;k<0 || k>n := 0; u[n_,k_]/;0<=k<=n := u[n,k] = u[n-1,k-1] + 3u[n-1,k] + 3u[n-1,k+1] + u[n-1,k+2]; u[n_]:=Sum[u[n,k],{k,0,n}]; Table[u[n],{n,0,10}]. - David Callan, Jul 22 2008
INVERT transform of (1,3,12,55,273,...), the ternary numbers A001764. - David Callan, Nov 21 2011

Alois P. Heinz, Table of n, a(n) for n = 1..400
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
J.-P. Bultel and S. Giraudo, Combinatorial Hopf algebras from PROs, arXiv preprint arXiv:1406.6903 [math.CO], 2014-2016.
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Noga Alon and Noah Kravitz, Counting Dope Matrices, arXiv:2205.09302 [math.CO], 2022.

Cf. A066380, A005809.
Column k=2 of A213027.
Cf. A001764.

f := n -> binomial(3*n, n) - (1/2)*add(binomial(3*n, k), k=0..n):
seq(f(n), n=1..20);

Table[Binomial[3 n, n] - Sum[Binomial[3 n, k], {k, 0, n}]/2, {n, 20}] (* Wesley Ivan Hurt, Jun 13 2014 *)

{a(n)=local(A=1+x); A=x*exp(sum(m=1, n+1, sum(j=0, m, binomial(3*m, j))*x^m/m +x*O(x^n))); polcoeff(A, n)} \\ Paul D. Hanna, Sep 04 2012

a(n) = binomial(3*n, n) - (1/2)*Sum_{k=0..n} binomial(3*n, k). - Vladeta Jovovic, Mar 22 2003
a(n) = A047098(n)/2. - Benoit Cloitre, Jan 28 2004
From Gary W. Adamson, Jul 28 2011: (Start)
a(n) is the upper left term in M^n, where M is the infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  3, 3, 1, 0, 0, 0, ...
  3, 3, 3, 1, 0, 0, ...
  1, 1, 3, 3, 1, 0, ...
  0, 0, 1, 3, 3, 0, ...
  0, 0, 0, 1, 3, 0, ...
  ... (End)
G.f.: x*exp( Sum_{n>=1} A066380*x^n/n ) where A066380(n) = Sum_{k=0..n} binomial(3*n,k). - Paul D. Hanna, Sep 04 2012
G.f.: (F(x)-1)/(2-F(x)), where F(x) is g.f. of A001764. - Vladimir Kruchinin, Jun 13 2014.
a(n) = (1/n)*Sum_{k=1..n} k*C(3*n,n-k). - Vladimir Kruchinin, Oct 03 2022
From Paul D. Hanna, Jun 06 2025: (Start)
G.f. A(x) = Series_Reversion( x*(1 + x)^2 / (1 + 2*x)^3 ).
G.f. satisfies A(x) = x*(1 + 2*A(x))^3 / (1 + A(x))^2.
G.f. satisfies A'(x) = A(x) * (1 + A(x)) * (1 + 2*A(x)) / (x*(1 - A(x))).
(End)

Comment revised by Clark Kimberling, Dec 08 2006

Edited by N. J. A. Sloane, Dec 21 2006

A213028 Number A(n,k) of 3n-length k-ary words that can be built by repeatedly inserting triples of identical letters into the initially empty word; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 8, 1, 0, 1, 4, 21, 38, 1, 0, 1, 5, 40, 183, 196, 1, 0, 1, 6, 65, 508, 1773, 1062, 1, 0, 1, 7, 96, 1085, 7240, 18303, 5948, 1, 0, 1, 8, 133, 1986, 20425, 110524, 197157, 34120, 1, 0, 1, 9, 176, 3283, 46476, 412965, 1766416, 2189799, 199316, 1, 0

Offset: 0

Alois P. Heinz, Jun 03 2012

			A(0,k) = 1: the empty word.
A(n,1) = 1: (aaa)^n.
A(2,2) = 8: there are 8 words of length 6 over alphabet {a,b} that can be built by repeatedly inserting triples of identical letters into the initially empty word: aaaaaa, aaabbb, aabbba, abbbaa, baaabb, bbaaab, bbbaaa, bbbbbb.
A(1,3) = 3: aaa, bbb, ccc.
A(2,3) = 21: aaaaaa, aaabbb, aaaccc, aabbba, aaccca, abbbaa, acccaa, baaabb, bbaaab, bbbaaa, bbbbbb, bbbccc, bbcccb, bcccbb, caaacc, cbbbcc, ccaaac, ccbbbc, cccaaa, cccbbb, cccccc.
Square array A(n,k) begins:
  1, 1,    1,      1,       1,       1,        1, ...
  0, 1,    2,      3,       4,       5,        6, ...
  0, 1,    8,     21,      40,      65,       96, ...
  0, 1,   38,    183,     508,    1085,     1986, ...
  0, 1,  196,   1773,    7240,   20425,    46476, ...
  0, 1, 1062,  18303,  110524,  412965,  1170066, ...
  0, 1, 5948, 197157, 1766416, 8755985, 30921756, ...

Alois P. Heinz, Antidiagonals n = 1..140, flattened

Rows n=0-2 give: A000012, A001477, A000567.
Columns k=0-2 give: A000007, A000012, A047098.
Cf. A183134, A183135, A213027, A256311.

A:= (n, k)-> `if`(n=0, 1,
    k/n *add(binomial(3*n, j) *(n-j) *(k-1)^j, j=0..n-1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Unprotect[Power]; 0^0 = 1; A[n_, k_] := If[n==0, 1, k/n*Sum[Binomial[3*n, j]*(n-j)*(k-1)^j, {j, 0, n-1}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

A(n,k) = k/n * Sum_{j=0..n-1} C(3*n,j) * (n-j) * (k-1)^j if n>0, k>1; A(0,k) = 1; A(n,k) = k if n>0, k<2.

A(n,k) = k * A213027(n,k) if n>0, k>1; else A(n,k) = A213027(n,k).

A218473 Number of 3n-length 3-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

Original entry on oeis.org

1, 1, 7, 61, 591, 6101, 65719, 729933, 8297247, 96044101, 1128138567, 13411861629, 161066465583, 1950996039669, 23808159962839, 292413627476141, 3611870017079871, 44838216520062117, 559127724970143079, 7000374603097246173, 87964883375131331151

Offset: 0

Alois P. Heinz, Oct 29 2012

Alois P. Heinz, Table of n, a(n) for n = 0..300

Column k=3 of A213027. Cf. A000139, A000309, A001764.

a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*2^j, j=0..n-1)/n):
seq(a(n), n=0..20);
# second Maple program
a:= proc(n) a(n):= `if`(n<3, [1, 1, 7][n+1], (-81*(3*n-1)*(3*n-5)*a(n-2)
       +(81*n^2-81*n+15)*a(n-1))/ ((2*n-1)*n))
    end:
seq(a(n), n=0..20);

Flatten[{1,Table[1/n*Sum[Binomial[3*n,j]*(n-j)*2^j,{j,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2013 *)
Flatten[{1,Table[FullSimplify[SeriesCoefficient[(1/(81*x-3)+2/((3-81*x)*(1-27*x-3*Sqrt[3*x*(27*x-2)])^(2/3))),{x,0,n}]],{n,1,10}]}] (* Vaclav Kotesovec, Jul 06 2013 *)

a(n) = (1/n) * Sum_{j=0..n-1} binomial(3*n,j)*(n-j)*2^j for n>0, a(0) = 1.
a(n) ~ 3^(3*n-3/2)/(sqrt(Pi)*2^(n-1)*n^(3/2)). - Vaclav Kotesovec, May 22 2013
G.f. (for n>0): (1/(81*x-3)+2/((3-81*x)*(1-27*x-3*sqrt(3*x*(27*x-2)))^(2/3))). - Vaclav Kotesovec, Jul 06 2013
From Peter Bala, Feb 06 2022: (Start)
The o.g.f. A(x) satisfies the algebraic equation 8*x - 36*x*A(x) + (54*x - 1)*A(x)^2 + (-27*x + 1)*A(x)^3 = 0.
A(x) = (6 - 4*T(2*x))/(2*T(2*x)^2 - 9*T(2*x) + 9), where T(x) = 1 + x*T(x)^3 is the o.g.f. of A001764.
A(x) = 1 + 2*x*B'(2*x)/B(2*x), where B(x) = 2 + x + 2*x^2 + 6*x^3 + 22*x^4 + 91*x^5 + ... is the o.g.f. of A000139.
exp(Sum_{n >= 1} a(n)*x*n/n) = 1 + x + 4*x^2 + 24*x^3 + 176*x^4 + 1456*x^5 + ... is the o.g.f. of A000309, a power series with integral coefficients. It follows that the Gauss congruences a(n*p^k) == a(n*p*(k-1)) (mod p^k) hold for all prime p and positive integers n and k. (End)

A218472 Number of 3n-length n-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

Original entry on oeis.org

1, 1, 4, 61, 1810, 82593, 5153626, 410380885, 39868799482, 4579454148865, 607729841261560, 91553310170011501, 15441283593044256696, 2883254656878648757729, 590578881927993264483880, 131681888589427990097216549, 31753512197914767223878851626

Offset: 0

Alois P. Heinz, Oct 29 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Diagonal of A213027.

a:= n-> `if`(n<2, 1, add(binomial(3*n, j)*(n-j)*(n-1)^j, j=0..n-1)/n):
seq(a(n), n=0..20);

a(n) = 1/n * Sum_{j=0..n-1} C(3*n,j)*(n-j)*(n-1)^j for n>1, a(n) = 1 else.

a(n) ~ 3^(3*n + 1/2) * n^(n - 5/2) / (sqrt(Pi) * exp(1) * 2^(2*n+2)). - Vaclav Kotesovec, Mar 25 2016

A218474 Number of 3n-length 4-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

Original entry on oeis.org

1, 1, 10, 127, 1810, 27631, 441604, 7293700, 123485914, 2131511455, 37368531010, 663539143015, 11908626395320, 215670579863428, 3936425910379840, 72335601620713432, 1337149262553687658, 24847762997547701695, 463900901255209923310, 8697278488612398979645

Offset: 0

Alois P. Heinz, Oct 29 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Column k=4 of A213027.

a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*3^j, j=0..n-1)/n):
seq(a(n), n=0..20);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 1, 10][n+1],
      ((2359*n^3 -5063*n^2 +2898*n -360)*a(n-1)
       -576*(3*n-5)*(7*n-2)*(3*n-4)*a(n-2))/
       (2*(2*n-1)*(7*n-9)*n))
    end:
seq(a(n), n=0..30);

a[n_] := If[n == 0, 1, Sum[Binomial[3n, j] (n - j) 3^j, {j, 0, n - 1}]/n];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 18 2020, after Maple *)

a(n) = 1/n * Sum_{j=0..n-1} C(3*n,j)*(n-j)*3^j for n>0, a(0) = 1.

a(n) ~ 3^(4*n+3/2) / (25*sqrt(Pi)*n^(3/2)*4^n). - Vaclav Kotesovec, Jul 16 2014

A218475 Number of 3n-length 5-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

Original entry on oeis.org

1, 1, 13, 217, 4085, 82593, 1751197, 38413481, 864413317, 19842830065, 462825376685, 10937407206265, 261311076852245, 6301225556698177, 153160687795008445, 3748598210810053449, 92303640047399410341, 2285025852515378528913, 56836898766186234593485

Offset: 0

Alois P. Heinz, Oct 29 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Column k=5 of A213027.

a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*4^j, j=0..n-1)/n):
seq(a(n), n=0..20);

a(n) = 1/n * Sum_{j=0..n-1} C(3*n,j)*(n-j)*4^j for n>0, a(0) = 1.
Recurrence: n*(2*n-1)*(4*n-5)*a(n) = (1216*n^3 - 2452*n^2 + 1267*n - 120)*a(n-1) - 750*(3*n-5)*(3*n-4)*(4*n-1)*a(n-2). - Vaclav Kotesovec, Aug 31 2014
a(n) ~ 4 * 3^(3*n+1/2) / (49 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 31 2014

A218476 Number of 3n-length 6-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

Original entry on oeis.org

1, 1, 16, 331, 7746, 195011, 5153626, 140995716, 3958980906, 113434797571, 3303283462836, 97478710394451, 2908594804576416, 87605427983818356, 2659959016770389896, 81330226479826092536, 2501989790308939894026, 77386492111973937031491, 2405093253522796180052056

Offset: 0

Alois P. Heinz, Oct 29 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=6 of A213027.

a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*5^j, j=0..n-1)/n):
seq(a(n), n=0..20);

a(n) = 1/n * Sum_{j=0..n-1} C(3*n,j)*(n-j)*5^j for n>0, a(0) = 1.
Recurrence: 2*n*(2*n-1)*(9*n-11)*a(n) = 3*(2997*n^3 - 5769*n^2 + 2754*n - 200)*a(n-1) - 3240*(3*n-5)*(3*n-4)*(9*n-2)*a(n-2). - Vaclav Kotesovec, Aug 31 2014
a(n) ~ 3^(3*n-7/2) * 5^(n+1) / (sqrt(Pi) * n^(3/2) * 4^n). - Vaclav Kotesovec, Aug 31 2014

A218477 Number of 3n-length 7-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

Original entry on oeis.org

1, 1, 19, 469, 13123, 395461, 12517939, 410380885, 13811907043, 474457464613, 16567069507219, 586287339402997, 20980966876537411, 757961579781924805, 27605221102084999411, 1012488016842242735509, 37364825362229946450595, 1386427393386051832383589

Offset: 0

Alois P. Heinz, Oct 29 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=7 of A213027.

a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*6^j, j=0..n-1)/n):
seq(a(n), n=0..20);

a(n) = 1/n * Sum_{j=0..n-1} C(3*n,j)*(n-j)*6^j for n>0, a(0) = 1.
Recurrence: n*(2*n-1)*(5*n-6)*a(n) = (3835*n^3 - 7127*n^2 + 3201*n - 180)*a(n-1) - 3087*(3*n-5)*(3*n-4)*(5*n-1)*a(n-2). - Vaclav Kotesovec, Aug 31 2014
a(n) ~ 3^(4*n+3/2) / (121 * 2^(n-1) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 31 2014

A218478 Number of 3n-length 8-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

Original entry on oeis.org

1, 1, 22, 631, 20546, 721071, 26594464, 1016157668, 39868799482, 1596785816431, 65014851904262, 2683064838415039, 111976833827421368, 4717961282984709124, 200410768873037098384, 8573481927644738115096, 369045717586929668129706, 15972561730958196240953007

Offset: 0

Alois P. Heinz, Oct 29 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=8 of A213027.

a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*7^j, j=0..n-1)/n):
seq(a(n), n=0..20);

a(n) = 1/n * Sum_{j=0..n-1} C(3*n,j)*(n-j)*7^j for n>0, a(0) = 1.
Recurrence: 2*n*(2*n-1)*(11*n-13)*a(n) = (24607*n^3 - 44503*n^2 + 19066*n - 840)*a(n-1) - 10752*(3*n-5)*(3*n-4)*(11*n-2)*a(n-2). - Vaclav Kotesovec, Aug 31 2014
a(n) ~ 3^(3*n+1/2) * 7^(n+1) / (169 * sqrt(Pi) * 4^n * n^(3/2)). - Vaclav Kotesovec, Aug 31 2014

A218479 Number of 3n-length 9-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

Original entry on oeis.org

1, 1, 25, 817, 30345, 1215521, 51186169, 2233558545, 100094682729, 4579454148865, 213011954187225, 10043235050384625, 478897402022246985, 23054568307757054433, 1118987674598361862585, 54698321862693296502737, 2690410083349269351772713

Offset: 0

Alois P. Heinz, Oct 29 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=9 of A213027.

a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*8^j, j=0..n-1)/n):
seq(a(n), n=0..20);

a(n) = 1/n * Sum_{j=0..n-1} C(3*n,j)*(n-j)*8^j for n>0, a(0) = 1.
Recurrence: n*(2*n-1)*(6*n-7)*a(n) = 3*(3132*n^3 - 5544*n^2 + 2289*n - 80)*a(n-1) - 8748*(3*n-5)*(3*n-4)*(6*n-1)*a(n-2). - Vaclav Kotesovec, Aug 31 2014
a(n) ~ 2^(n+3) * 3^(3*n-3/2) / (25 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 31 2014

cp's OEIS Frontend

A047099 a(n) = A047098(n)/2.

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A213028 Number A(n,k) of 3n-length k-ary words that can be built by repeatedly inserting triples of identical letters into the initially empty word; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A218473 Number of 3n-length 3-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

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A218472 Number of 3n-length n-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

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A218474 Number of 3n-length 4-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

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A218475 Number of 3n-length 5-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

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A218476 Number of 3n-length 6-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

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A218477 Number of 3n-length 7-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

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