A047098 -id:A047098 - 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

A127635 Hankel transform of A047098.

Original entry on oeis.org

1, 4, 44, 1360, 118864, 29454720, 20723316480, 41430374667264, 235483137163985920, 3806579106735674587136, 175045931960590896961989632, 22902901668710944230193460535296, 8527272133354589357030560193109508096

Offset: 0

Paul Barry, Jan 21 2007

Cf. A047098, A051255.

a(n) = 2^n*A051255(n).

A107027 Number triangle associated to the Riordan arrays (1/(1+x),x/(1+x)^k),k>=0.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 6, 8, 2, 1, 2, 8, 20, 16, 2, 1, 2, 10, 38, 70, 32, 2, 1, 2, 12, 62, 196, 252, 64, 2, 1, 2, 14, 92, 426, 1062, 924, 128, 2, 1, 2, 16, 128, 792, 3112, 5948, 3432, 256, 2, 1, 2, 18, 170, 1326, 7302, 23686, 34120, 12870, 512, 2

Offset: 0

Paul Barry, May 09 2005

As a number square read by antidiagonals, the rows represent the row sums of the inverses of the Riordan arrays (1/(1+x),x/(1+x)^k), k>=0. The rows are then given by T(n,k)=(n-1)C(n*k,k)-(n-2)*sum{j=0..k, C(n*k,j)}.

			Triangle begins
  1;
  1, 2;
  1, 2, 2;
  1, 2, 4,  2;
  1, 2, 6,  8,  2;
  1, 2, 8, 20, 16, 2;

T(n,n) is A040000, T(n+1,n) is A000079, T(n+2,n) is A000984, T(n+3,n) is A047098.
The reverse of this triangle is A107030.
Row sums are A107028.
Diagonal sums are A107029.

Number triangle T(n, k)=if(k<=n, (n-k-1)C((n-k)*k, k)-(n-k-2)*sum{j=0..k, C((n-k)*k, j)}, 0).

A107026 Row sums of inverse of Riordan array (1/(1+x),x/(1+x)^4).

Original entry on oeis.org

1, 2, 10, 62, 426, 3112, 23686, 185684, 1488554, 12144248, 100489320, 841268078, 7112138790, 60629940152, 520591221412, 4498091003272, 39079909924522, 341193986978008, 2991881019936760, 26338436818801496, 232688056611178216

Offset: 0

Paul Barry, May 09 2005

The Riordan array (1/(1+x),x/(1+x)^4) has general term (-1)^(n-k)*binomial(n+3k,4k).

Cf. A047098, A107027, A107030.

A107026 := proc(n)
    3*binomial(4*n,n)-2*add(binomial(4*n,k),k=0..n) ;
end proc: # R. J. Mathar, Feb 20 2015

G.f.: A(x)=y satisfies (2y)^4*x-(y+1)^3*(y-1)=0.
a(n) = 3*binomial(4*n, n) - 2*Sum_{k=0..n} binomial(4*n, k).
Conjecture: +189*n*(3*n-1)*(3*n-2)*a(n) +72*(-1034*n^3+3098*n^2-3754*n+1655)*a(n
-1) +384*(2700*n^3-12828*n^2+20426*n-10785)*a(n-2) +4096*(-1066*n^3+6666*n^2-129
50*n+7365)*a(n-3) -65536*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Feb 20 2015
Conjecture: 3*n*(3*n-1)*(3*n-2)*(22*n^2-62*n+43)*a(n) +8*(-1892*n^5+8280*n^4-13330*n^3+9660*n^2-3048*n+315)*a(n-1) +128*(4*n-7)*(2*n-3)*(4*n-5)*(22*n^2-18*n+3)*a(n-2)=0. - R. J. Mathar, Feb 20 2015

A107030 Number triangle associated with the Riordan arrays (1/(1+x),x/(1+x)^k),k>=0.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 8, 6, 2, 1, 2, 16, 20, 8, 2, 1, 2, 32, 70, 38, 10, 2, 1, 2, 64, 252, 196, 62, 12, 2, 1, 2, 128, 924, 1062, 426, 92, 14, 2, 1, 2, 256, 3432, 5948, 3112, 792, 128, 16, 2, 1, 2, 512, 12870, 34120, 23686, 7302, 1326, 170, 18, 2, 1

Offset: 0

Paul Barry, May 09 2005

			Triangle begins
  1;
  2,  1;
  2,  2,  1;
  2,  4,  2, 1;
  2,  8,  6, 2, 1;
  2, 16, 20, 8, 2, 1;

Reversal of A107027.
Row sums are A107028.
Diagonal sums are A107031.
Columns include A040000, A000079, A000984, A047098.

Number triangle T(n, k)=(k-1)*C(k(n-k), n-k)-(k-2)*sum{j=0..n-k, C(k(n-k), j)}

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).

A307468 Cogrowth sequence for the Heisenberg group.

Original entry on oeis.org

1, 4, 28, 232, 2156, 21944, 240280, 2787320, 33820044, 424925872, 5486681368, 72398776344, 972270849512, 13247921422480, 182729003683352, 2546778437385032, 35816909974343308, 507700854900783784, 7246857513425470288, 104083322583897779656

Offset: 0

Igor Pak, Apr 09 2019

This is the number of words of length 2n in the letters x,x^{-1},y,y^{-1} that equal the identity of the Heisenberg group H=.

Also, this is the number of closed walks of length 2n on the square lattice enclosing algebraic area 0 [Béguin et al.]. - Andrey Zabolotskiy, Sep 15 2021

			For n=1 the a(1)=4 words are x^{-1}x, xx^{-1}, y^{-1}y, yy^{-1}.

Jay Pantone, Table of n, a(n) for n = 0..200
Cédric Béguin, Alain Valette and Andrzej Zuk, On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator, Journal of Geometry and Physics, 21 (1997), 337-356.
D. Lind and K, Schmidt, A survey of algebraic actions of the discrete Heisenberg group, arXiv:1502.06243 [math.DS], 2015; Russian Mathematical Surveys, 70:4 (2015), 77-142.

Related cogrowth sequences: Z A000984, Z^2 A002894, Z^3 A002896, (Z/kZ)^*2 for k = 2..5: A126869, A047098, A107026, A304979, Richard Thompson's group F A246877. The cogrowth sequences for BS(N,N) for N = 2..10 are A229644, A229645, A229646, A229647, A229648, A229649, A229650, A229651, A229652.

Cf. A352838, A178106.

Asymptotics: a(n) ~ (1/2) * 16^n * n^(-2).

A304979 The nonzero terms of the cogrowth sequence of (Z/5Z)^2 = with respect to the generating set {(x,1), (1,y)}.

Original entry on oeis.org

1, 2, 12, 92, 792, 7302, 70464, 702536, 7178568, 74771570, 790906012, 8472417384, 91724327928, 1001987961834, 11030476949952, 122247789508992, 1362840516623944, 15272530735735338, 171946029518128956, 1943927810200670820, 22059590401383177792, 251183781609841838444

Offset: 0

Marni Mishna, May 22 2018

nonn

The number of paths on the Cayley graph of ((Z/5Z)^*2, {(x,1), (1,y)} which start and end at the identity.

a(n) is the number of words of length 5n over the alphabet {x,y} that reduce to the empty string upon iteratively removing factors of x^5 and y^5. For n=2, the a(2)=12 words of length 10 that reduce to the empty string are x^10, y^10, x^i y^5 x^(5-i), for i=1..5, and y^i x^5 y^(5-i), for i=1..5.

J. P. Bell and M. J. Mishna, On the Complexity of the Cogrowth Sequence, arXiv:1805.08118 [math.CO], 2018-2019.

Related cogrowth sequences (Z/2Z)^*2: A126869; (Z/3Z)^*2: A047098; (Z/4Z)^*2: A107026.

terms = 22;
A[] = 0; Do[A[x] = (1 + 4 A[x] + 6 A[x]^2 + 4 A[x]^3 + A[x]^4 + 32 x A[x]^5)/(1 + A[x])^4 + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 16 2018 *)

G.f.: A(x) satisfies 32*x*A(x)^5 - (A(x)-1)*(A(x)+1)^4 = 0.
a(n) satisfies the recurrence (2120000*(5*n+1))*(5*n+2)*(5*n+3)*(5*n+4)*a(n) + (250*(160980199*n^4 + 1129209134*n^3 + 2872721885*n^2 + 3155706646*n + 1267579560))*a(n+1) - (50*(109722203*n^4 + 959367613*n^3 + 3144281425*n^2 + 4572924587*n + 2485585548))*a(n+2) + (60*(4290021*n^4 + 51502996*n^3 + 243316306*n^2 + 532456081*n + 451079946))*a(n+3) - (3*(2673299*n^4 + 44756419*n^3 + 283571239*n^2 + 805783469*n + 866093430))*a(n+4) + (4008*(n+5))*(4*n+17)*(2*n+9)*(4*n+19)*a(n+5) = 0.
a(n) ~ 5^(5*n + 1/2) / (9 * sqrt(Pi) * n^(3/2) * 2^(8*n - 3/2)). - Vaclav Kotesovec, Oct 24 2023
a(n) = binomial(5*n,n) - 3 * Sum_{k=0..n-1} binomial(5*n,k). - Seiichi Manyama, Apr 05 2024

cp's OEIS Frontend

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

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A127635 Hankel transform of A047098.

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A107027 Number triangle associated to the Riordan arrays (1/(1+x),x/(1+x)^k),k>=0.

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A107026 Row sums of inverse of Riordan array (1/(1+x),x/(1+x)^4).

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A107030 Number triangle associated with the Riordan arrays (1/(1+x),x/(1+x)^k),k>=0.

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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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A307468 Cogrowth sequence for the Heisenberg group.

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A304979 The nonzero terms of the cogrowth sequence of (Z/5Z)^*2 = * with respect to the generating set {(x,1), (1,y)}.

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A304979 The nonzero terms of the cogrowth sequence of (Z/5Z)^2 = with respect to the generating set {(x,1), (1,y)}.