A047099 -id:A047099 - cp's OEIS frontend

A213027 Number A(n,k) of 3n-length k-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; square array A(n,k), n>=0, k>=0, by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 4, 1, 0, 1, 1, 7, 19, 1, 0, 1, 1, 10, 61, 98, 1, 0, 1, 1, 13, 127, 591, 531, 1, 0, 1, 1, 16, 217, 1810, 6101, 2974, 1, 0, 1, 1, 19, 331, 4085, 27631, 65719, 17060, 1, 0, 1, 1, 22, 469, 7746, 82593, 441604, 729933, 99658, 1, 0

Offset: 0

Alois P. Heinz, Jun 03 2012

In general, column k > 1 is asymptotic to a(n) ~ 3^(3*n+1/2) * (k-1)^(n+1) / (sqrt(Pi) * (2*k-3)^2 * 4^n * n^(3/2)). - Vaclav Kotesovec, Aug 31 2014

			A(0,k) = 1: the empty word.
A(n,1) = 1: (aaa)^n.
A(2,2) = 4: there are 4 words of length 6 over alphabet {a,b}, 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: aaaaaa, aaabbb, aabbba, abbbaa.
A(2,3) = 7: aaaaaa, aaabbb, aaaccc, aabbba, aaccca, abbbaa, acccaa.
A(3,2) = 19: aaaaaaaaa, aaaaaabbb, aaaaabbba, aaaabbbaa, aaabaaabb, aaabbaaab, aaabbbaaa, aaabbbbbb, aabaaabba, aabbaaaba, aabbbaaaa, aabbbabbb, aabbbbbba, abaaabbaa, abbaaabaa, abbbaaaaa, abbbaabbb, abbbabbba, abbbbbbaa.
Square array A(n,k) begins:
  1, 1,    1,     1,      1,       1,       1, ...
  0, 1,    1,     1,      1,       1,       1, ...
  0, 1,    4,     7,     10,      13,      16, ...
  0, 1,   19,    61,    127,     217,     331, ...
  0, 1,   98,   591,   1810,    4085,    7746, ...
  0, 1,  531,  6101,  27631,   82593,  195011, ...
  0, 1, 2974, 65719, 441604, 1751197, 5153626, ...

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

Columns k=0-10 give: A000007, A000012, A047099, A218473, A218474, A218475, A218476, A218477, A218478, A218479, A218480.
Rows n=0-3 give: A000012, A057427, A016777(k-1), A127854(k-1).
Main diagonal gives: A218472.
Cf. A183134, A183135, A213028.

A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k,
    1/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);

a[0, ] = 1; a[, k_ /; k < 2] := k; a[n_, k_] := 1/n*Sum[Binomial[3*n, j]*(n-j)*(k-1)^j, {j, 0, n-1}]; Table[a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013 *)

A(n,k) = 1/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) = 1/k * A213028(n,k) if n>0, k>1; else A(n,k) = A213028(n,k).

A047098 a(n) = 2binomial(3n, n) - Sum_{k=0..n} binomial(3*n, k).

Original entry on oeis.org

1, 2, 8, 38, 196, 1062, 5948, 34120, 199316, 1181126, 7080928, 42860534, 261542752, 1607076200, 9934255472, 61732449648, 385393229460, 2415935640198, 15200964233864, 95962904716402, 607640599286276, 3858198001960438, 24559243585545644, 156692889782067712

Offset: 0

Clark Kimberling, Aug 15 1998

nonn

T(2n,n), array T as in A047089. [Corrected Dec 08 2006]
Let B_3^+ denote the semigroup with presentation . Let D=aba be the 'fundamental word'. Then this sequence is also equal to the number of words in B_3^+ equal in B_3^+ to D^n, n >= 0. - Stephen P. Humphries, Jan 20 2004
In the language of Riordan arrays, row sums of (1/(1+x), x/(1+x)^3)^-1, where (1/(1+x), x/(1+x)^3) has general term (-1)^(n-k)*binomial(n+2k, 3k). - Paul Barry, May 09 2005
Hankel transform is 2^n*A051255(n) where A051255 is the Hankel transform of C(3n,n)/(2n+1). - Paul Barry, Jan 21 2007

Michael De Vlieger, Table of n, a(n) for n = 0..1211
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Christopher R. Cornwell and Stephen P. Humphries, Counting fundamental paths in certain Garside semigroups, Journal of Knot Theory and Its Ramifications, Vol. 17 (2008), No. 02, pp. 191-211.

Column k=2 of A213028.

Cf. A047089, A047099, A107026, A321957.

A047098 := n -> 2*binomial(3*n, n)-add(binomial(3*n, k), k=0..n);

Table[2Binomial[3n,n]-Sum[Binomial[3n,k],{k,0,n}],{n,0,35}] (* Harvey P. Dale, Jul 27 2011 *)

a(n)=if(n<0,0,polcoeff((((1+10*x-2*x^2)+(1-4*x)*sqrt(1-4*x+x*O(x^n)))/2)^n,n))

a(n)=if(n<0,0, 2*binomial(3*n,n)-sum(k=0,n,binomial(3*n,k)))

G.f. A(x)=y satisfies (8x-1)y^3-y^2+y+1=0. - Michael Somos, Jan 28 2004
Coefficient of x^n in ((1+10x-2x^2+(1-4x)^(3/2))/2)^n. - Michael Somos, Sep 25 2003
a(n) = Sum_{k = 0..n} A109971(k)*2^k; a(0) = 1, a(n) = Sum_{k = 0..n} 2^k*C(3n-k,n-k)*2*k/(3*n-k), n > 0. - Paul Barry, Jan 21 2007
Conjecture: 2*n*(2*n-1)*a(n) +(-71*n^2+112*n-48)*a(n-1) +3*(131*n^2-391*n+296)*a(n-2) -72*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Nov 30 2012
a(n) = A321957(n) + 2*binomial(3*n, n) - 8^n. - Peter Luschny, Nov 22 2018
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. - Peter Bala, Mar 05 2022

Clark Kimberling, Dec 08 2006, changed "T(3n,2n)" to "T(2n,n)" in the comment line, but observes that some of the other comments seem to apply to the sequence T(3n,2n) rather than to the sequence T(2n,n).

Edited by N. J. A. Sloane, Dec 21 2006, replacing the old definition in terms of A047089 by an explicit formula supplied by Benoit Cloitre, Oct 25 2003.

A387033 a(n) = Sum_{k=0..n} binomial(3*n-1,k).

Original entry on oeis.org

1, 3, 16, 93, 562, 3473, 21778, 137980, 880970, 5658537, 36519556, 236618693, 1538132224, 10026362492, 65513177704, 428957009288, 2813768603466, 18486790962201, 121634649321208, 801330506737399, 5285305708097522, 34896814868837161, 230631268849574378

Offset: 0

Seiichi Manyama, Aug 13 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A066380, A160906, A385823, A386006, A387007, A387008.
Cf. A114121, A387037.
Cf. A001764, A047099, A165817.

[&+[Binomial(3*n-1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Table[Sum[Binomial[3*n-1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)

a(n) = sum(k=0, n, binomial(3*n-1, k));

a(n) = [x^n] (1+x)^(3*n-1)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n-1) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-1,k) * binomial(3*n-k-2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-2,n-k).
G.f.: 1/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
D-finite with recurrence: 24*(5*n+6)*(3*n-4)*(3*n-5)*a(n-2)-(295*n^3-451*n^2-234*n+360)*a(n-1)+2*n*(5*n+1)*(2*n-3)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 3^(3*n - 1/2) / (sqrt(Pi*n) * 2^(2*n-1)). - Vaclav Kotesovec, Aug 27 2025

A385823 a(n) = Sum_{k=0..n} binomial(3*n-3,k).

Original entry on oeis.org

1, 1, 7, 42, 256, 1586, 9949, 63004, 401930, 2579130, 16628809, 107636402, 699030226, 4552602248, 29722279084, 194458630304, 1274628824490, 8368726082346, 55027110808177, 362301656545966, 2388274575638228, 15760514137668514, 104108685843640517, 688331413734386356

Offset: 0

Seiichi Manyama, Aug 13 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A066380, A160906, A386006, A387007, A387008, A387033.

Cf. A001764, A047099, A165817.

[&+[Binomial(3*n-3,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Table[Sum[Binomial[3*n-3,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)

a(n) = sum(k=0, n, binomial(3*n-3, k));

a(n) = [x^n] (1+x)^(3*n-3)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n-3) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-3,k) * binomial(3*n-k-4,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-4,n-k).
G.f.: 1/(g^2 * (2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.

A386006 a(n) = Sum_{k=0..n} binomial(3*n-2,k).

Original entry on oeis.org

1, 2, 11, 64, 386, 2380, 14893, 94184, 600370, 3850756, 24821333, 160645504, 1043243132, 6794414896, 44360053772, 290244832992, 1902631226010, 12493030680180, 82153313341429, 540953389469312, 3566279609565226, 23536562549993228, 155489358646406149

Offset: 0

Seiichi Manyama, Aug 13 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A066380, A160906, A385823, A387007, A387008, A387033.

Cf. A001764, A047099, A165817.

[&+[Binomial(3*n-2,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Table[Sum[Binomial[3*n-2,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)

a(n) = sum(k=0, n, binomial(3*n-2, k));

a(n) = [x^n] (1+x)^(3*n-2)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n-2) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-2,k) * binomial(3*n-k-3,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-3,n-k).
G.f.: 1/(g * (2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.

A127898 Inverse of Riordan array (1/(1+x)^3, x/(1+x)^3).

Original entry on oeis.org

1, 3, 1, 12, 6, 1, 55, 33, 9, 1, 273, 182, 63, 12, 1, 1428, 1020, 408, 102, 15, 1, 7752, 5814, 2565, 760, 150, 18, 1, 43263, 33649, 15939, 5313, 1265, 207, 21, 1, 246675, 197340, 98670, 35880, 9750, 1950, 273, 24, 1

Offset: 0

Paul Barry, Feb 04 2007

The convolution triangle of A001764 (number of ternary trees). - Peter Luschny, Oct 09 2022

			Triangle begins:
1,
3, 1,
12, 6, 1,
55, 33, 9, 1,
273, 182, 63, 12, 1,
1428, 1020, 408, 102, 15, 1,
7752, 5814, 2565, 760, 150, 18, 1,
43263, 33649, 15939, 5313, 1265, 207, 21, 1,
246675, 197340, 98670, 35880, 9750, 1950, 273, 24, 1,
1430715, 1170585, 610740, 237510, 71253, 16443, 2842, 348, 27, 1,
8414640, 7012200, 3786588, 1553472, 503440, 129456, 26040, 3968, 432, 30, 1

G. C. Greubel, Rows n=0..100 of triangle, flattened
Paul Drube, Generalized Path Pairs and Fuss-Catalan Triangles, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8.

First column is A001764(n+1).
Row sums are A047099.
Inverse of A127895.

Flat(List([0..10],n->List([0..n],k->(k+1)/(n+1)*Binomial(3*n+3,n-k)))); # Muniru A Asiru, Apr 30 2018

/* As triangle: */ [[(k+1)/(n+1)*Binomial(3*n+3,n-k): k in [0..n]]: n in [0..8]];  // Bruno Berselli, Jan 17 2013

# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> binomial(3*n, n)/(2*n+1)); # Peter Luschny, Oct 09 2022

Table[If[k == 0, Binomial[3*n, n-k]/(2*n+1), ((k+1)/n)*Binomial[3*n, n-k -1]], {n,1,10}, {k,0,n-1}]//Flatten (* G. C. Greubel, Apr 29 2018 *)

for(n=1,10, for(k=0,n-1, print1(if(k==0, binomial(3*n, n-k)/( 2*n +1), ((k+1)/n)*binomial(3*n, n-k-1)), ", "))) \\ G. C. Greubel, Apr 29 2018

T(n,k) = (k+1)/(n+1)*binomial(3*n+3,n-k). - Vladimir Kruchinin, Jan 17 2013
G.f.: 1/(-y + 1/(-1 + (2*sin(1/3 *arcsin((3*sqrt(3*x))/2)))/(
  sqrt(3*x))))/x.  - Vladimir Kruchinin, Feb 14 2023

A064282 Triangle read by rows: T(n,k) = binomial(3n+3, k)*(n-k+1)/(n+1).

Original entry on oeis.org

1, 1, 3, 1, 6, 12, 1, 9, 33, 55, 1, 12, 63, 182, 273, 1, 15, 102, 408, 1020, 1428, 1, 18, 150, 760, 2565, 5814, 7752, 1, 21, 207, 1265, 5313, 15939, 33649, 43263, 1, 24, 273, 1950, 9750, 35880, 98670, 197340, 246675, 1, 27, 348, 2842, 16443, 71253, 237510

Offset: 0

Henry Bottomley, Sep 24 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000

Columns include A000012 and A008585. Right hand columns include A001764 and A006630. Row sums are A047099. Triangles A010054 (Triangle Indicator), A007318 (Pascal) and A050166 form a sequence which has this as its next member.

Flatten[Table[Binomial[3n+3,k] (n-k+1)/(n+1),{n,0,10},{k,0,n}]] (* Harvey P. Dale, Dec 26 2014 *)

{ n=-1; for (m=0, 10^9, for (k=0, m, a=binomial(3*m + 3, k)*(m - k + 1)/(m + 1); write("b064282.txt", n++, " ", a); if (n==1000, break)); if (n==1000, break) ) } \\ Harry J. Smith, Sep 11 2009

T(n, k) = T(n-1, k) + 3*T(n-1, k-1) + 3*T(n-1, k-2) + T(n-1, k-3) [starting with T(0,0)=1 and T(n,k)=0 if n < 0 or n < k].

A382100 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of 1/(2 - B_k(x)), where B_k(x) = 1 + x*B_k(x)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 10, 8, 1, 1, 1, 5, 19, 35, 16, 1, 1, 1, 6, 31, 98, 126, 32, 1, 1, 1, 7, 46, 213, 531, 462, 64, 1, 1, 1, 8, 64, 396, 1556, 2974, 1716, 128, 1, 1, 1, 9, 85, 663, 3651, 11843, 17060, 6435, 256, 1

Offset: 0

Seiichi Manyama, Mar 15 2025

			Square array begins:
  1,  1,   1,    1,     1,     1,     1, ...
  1,  1,   1,    1,     1,     1,     1, ...
  1,  2,   3,    4,     5,     6,     7, ...
  1,  4,  10,   19,    31,    46,    64, ...
  1,  8,  35,   98,   213,   396,   663, ...
  1, 16, 126,  531,  1556,  3651,  7391, ...
  1, 32, 462, 2974, 11843, 35232, 86488, ...

Columns k=0..5 give A000012, A011782, A088218, A047099 (for n > 0), A107026(n)/2 (for n > 0), A304979(n)/2 (for n > 0).

Cf. A107027, A107030, A355262, A382101.

a(n, k) = polcoef(1/(2-sum(j=0, n, binomial(k*j+1, j)/(k*j+1)*x^j+x*O(x^n))), n);

a(n, k) = if(n==0, 1, (binomial(k*n, n)-(k-2)*sum(j=0, n-1, binomial(k*n, j)))/2);

A(n,k) = ( binomial(k*n,n) - (k-2) * Sum_{j=0..n-1} binomial(k*n,j) )/2 for n > 0.

G.f. of column k: 1/( 1 - Series_Reversion( x/(1+x)^k ) ).

cp's OEIS Frontend

A213027 Number A(n,k) of 3n-length k-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; square array A(n,k), n>=0, k>=0, by antidiagonals.

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A047098 a(n) = 2*binomial(3*n, n) - Sum_{k=0..n} binomial(3*n, k).

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A387033 a(n) = Sum_{k=0..n} binomial(3*n-1,k).

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A385823 a(n) = Sum_{k=0..n} binomial(3*n-3,k).

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A386006 a(n) = Sum_{k=0..n} binomial(3*n-2,k).

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A127898 Inverse of Riordan array (1/(1+x)^3, x/(1+x)^3).

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A064282 Triangle read by rows: T(n,k) = binomial(3n+3, k)*(n-k+1)/(n+1).

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A047098 a(n) = 2binomial(3n, n) - Sum_{k=0..n} binomial(3*n, k).