A183135 -id:A183135 - cp's OEIS frontend

A183134 Square array A(n,k) by antidiagonals. A(n,k) is the number of length 2n k-ary words (n,k>=0), either empty or beginning with the first character of the alphabet, that can be built by repeatedly inserting doublets into the initially empty word.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 5, 10, 1, 0, 1, 1, 7, 29, 35, 1, 0, 1, 1, 9, 58, 181, 126, 1, 0, 1, 1, 11, 97, 523, 1181, 462, 1, 0, 1, 1, 13, 146, 1145, 4966, 7941, 1716, 1, 0, 1, 1, 15, 205, 2131, 14289, 48838, 54573, 6435, 1, 0

Offset: 0

Alois P. Heinz, Dec 26 2010

Column k > 2 is asymptotic to 2^(2*n) * (k-1)^(n+1) / ((k-2)^2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 07 2014

			A(3,2) = 10, because 10 words of length 6 beginning with the first character of the 2-letter alphabet {a, b} can be built by repeatedly inserting doublets (words with two equal letters) into the initially empty word: aaaaaa, aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba.
Square array A(n,k) begins:
  1,  1,   1,    1,    1,     1,  ...
  0,  1,   1,    1,    1,     1,  ...
  0,  1,   3,    5,    7,     9,  ...
  0,  1,  10,   29,   58,    97,  ...
  0,  1,  35,  181,  523,  1145,  ...
  0,  1, 126, 1181, 4966, 14289,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
C. Kassel and C. Reutenauer, Algebraicity of the zeta function associated to a matrix over a free group algebra, arXiv preprint arXiv:1303.3481, 2013
A. Lakshminarayan, Z. Puchala, K. Zyczkowski, Diagonal unitary entangling gates and contradiagonal quantum states, arXiv preprint arXiv:1407.1169, 2014

Rows 0-10 give: A000012, A057427, A004273, A079273(k) for k>0, A194716, A194717, A194718, A194719, A194720, A194721, A194722.
Columns 0-10 give: A000007, A000012, A001700(n-1) for n>0, A194723, A194724, A194725, A194726, A194727, A194728, A194729, A194730.
Main diagonal gives A248828.
Coefficients of row polynomials for k>0 in k, (k+1) are given by A050166, A157491.
Cf. A007318, A183135.

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

a[n_, k_] := If[ n == 0, 1 , If[ k <= 1, k, Sum [Binomial[2*n, j]*(n-j)*(k-1)^j, {j, 0, n-1}] / n ] ]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

A(n,k) = 1 if n=0, A(n,k) = k if n>0 and k<=1, and A(n,k) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*(k-1)^j else.
A(n,k) = A183135(n,k) if n=0 or k<2, A(n,k) = A183135(n,k)/k else.
G.f. of column k: 1/(1-k*x) if k<2, (1-1/k) * (1 + 2 / (k-2 + k * sqrt (1-(4*k-4)*x))) else.

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.

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.

Original entry on oeis.org

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).

A089022 Number of walks of length 2n on the 3-regular tree beginning and ending at some fixed vertex.

Original entry on oeis.org

1, 3, 15, 87, 543, 3543, 23823, 163719, 1143999, 8099511, 57959535, 418441191, 3043608351, 22280372247, 164008329423, 1213166815047, 9012417249663, 67208553680247, 502920171632943, 3775020828459687, 28415858155984863, 214444848602732247, 1622146752543427983

Offset: 0

Paul Boddington, Nov 11 2003

The generating function for the corresponding sequence for the m-regular tree is 2*(m-1)/(m-2+m*sqrt(1-4*(m-1)*x)). When m=2 this reduces to the usual generating function for the central binomial coefficients.

Hankel transform is A133460. - Philippe Deléham, Dec 01 2007

			a(2) = 15 because there are 3*3=9 walks whose second step is to return to the starting vertex and 3*2=6 walks whose second step is to move away from the starting vertex.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Libor Caha and Daniel Nagaj, The pair-flip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quant-ph], 2018.
Pakawut Jiradilok and Supanat Kamtue, Transportation Distance between Probability Measures on the Infinite Regular Tree, arXiv:2107.09876 [math.CO], 2021.
Ed Pegg Jr., K-Cayley Trees

Column k=3 of A183135.

A000602 := x -> 2^x*binomial(2*x, x)-9^x+1/3*2^x*binomial(2*x, x) * hypergeom([1, 2*x+1], [x+1], 2/3); # Tobias Friedrich (tfried(AT)mpi-inf.mpg.de), Jun 12 2007

Table[2^n*Binomial[2*n,n]-3^(n-1)*Sum[(2/3)^k*Binomial[n+k,n],{k,0,n-1}],{n,0,20}] (* or *)
CoefficientList[Series[4/(1+3*Sqrt[1-8*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

my(x='x+O('x^30)); Vec(4/(1+3*sqrt(1-8*x))) \\ Joerg Arndt, May 10 2013

G.f.: 4/(1+3*sqrt(1-8*x)).
a(n) = 2^n * binomial(2*n,n) - 3^(n-1) * Sum_{j=0..n-1} (2/3)^j*binomial(n+j,n). - Tobias Friedrich (tfried(AT)mpi-inf.mpg.de), Jun 12 2007
a(n) = Sum_{k=0..n} A039599(n,k)*2^(n-k). - Philippe Deléham, Aug 25 2007
From Paul Barry, Sep 04 2009: (Start)
G.f.: 1/(1-3x/(1-2x/(1-2x/(1-2x/(1-2x/(1-... (continued fraction);
G.f.: (1-2*x*c(x))/(1-3*x-2*x*c(x)), where c(x) is the g.f. of A000108. (End)
a(n) = A126087(2n). - Philippe Deléham, Nov 02 2011
D-finite with recurrence n*a(n) + (12-17*n)*a(n-1) + 36*(2n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
a(n) ~ 6*8^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
From Karol A. Penson, Sep 06 2014: (Start)
a(n) is the (2*n)-th moment of a positive function W(x) = (3/Pi)*sqrt(8-x^2)/(9-x^2), on the segment x = (0,2*sqrt(2)): a(n) = Integral_{x=0..2*sqrt(2)} x^(2*n)*W(x) dx;
a(n) is the special value of hypergeometric function 2F1, in Maple notation: a(n)=2*8^n*GAMMA(n+1/2)*hypergeom([1,n+1/2],[n+2],8/9)/(3*sqrt(Pi)*(n+1)!). (End)
a(n) = A151374(n)*hypergeom([1,n+1/2],[n+2],8/9)*(2/3). - Peter Luschny, Sep 06 2014

A035610 G.f.: 3/(1 + 2sqrt(1-12x)).

Original entry on oeis.org

1, 4, 28, 232, 2092, 19864, 195352, 1970896, 20275660, 211823800, 2240795848, 23951289520, 258255469816, 2805534253552, 30675477376432, 337306474674592, 3727578443380492, 41376874025687032, 461121792658583272, 5157384457905440752, 57869888433073055272, 651266142688270063312, 7349148747954997832272

Offset: 0

N. J. A. Sloane

nonn

Number of walks of length 2n on the 4-regular tree beginning and ending at some fixed vertex. The generating function for the corresponding sequence for the m-regular tree is 2*(m-1)/(m-2+m*sqrt(1-4*(m-1)*x)). When m=2 this reduces to the usual generating function for the central binomial coefficients. - Paul Boddington, Nov 11 2003
Main diagonal of the array A(0,j)=A(i,0)=1 for i,j>=0 and for i,j>=1 A(i,j)=min{A(i,j-1)+3*A(i-1,j); 3*A(i,j-1)+A(i-1,j)}. - Benoit Cloitre, Aug 05 2004
Hankel transform is A133461. - Philippe Deléham, Dec 01 2007
Also the number of length 2n words over an alphabet of size 4 that can be built by repeatedly inserting doublets into the initially empty word.
The sequence {b(n)}, where b(2n)=a(n), b(2n+1)=0, is the cogrowth function of the free group of rank 2. - Murray Elder, Jun 28 2016

			a(2)=28 because there are 4*4=16 walks whose second step is to return to the starting vertex and 4*3=12 walks whose second step is to move away from the starting vertex.

Robert Israel, Table of n, a(n) for n = 0..837
Murray Elder, Table of n, b(n) for n = 0..200, where b(n) is mentioned in a comment above.
Libor Caha and Daniel Nagaj, The pair-flip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quant-ph], 2018.
M. Elder and A. Rechnitzer, T. Wong, On the cogrowth of Thompson's group F, Groups, Complexity, Cryptology 4(2) (2012), 301-320.
Pakawut Jiradilok and Supanat Kamtue, Transportation Distance between Probability Measures on the Infinite Regular Tree, arXiv:2107.09876 [math.CO], 2021.
J. Novak, Three lectures on free probability, arXiv preprint arXiv:1205.2097, 2012. - _N. J. A. Sloane_, Oct 15 2012
Gregory Quenell, Combinatorics of free product graphs, Contemp. Math (1994) 257-281 (Eq. 19).
Ian M. Wanless, Counting Matchings and Tree-Like Walks in Regular Graphs, Combinatorics, Probability and Computing (2010) 19, 463-480 (Lemma 3.1).

Cf. A089022.

4th column of A183135.

a:= n-> `if`(n=0, 1, 4/n*add(binomial(2*n, j) *(n-j)*3^j, j=0..n-1)):
seq(a(n), n=0..20);
# Alternative:
f:= gfun:-rectoproc({(-192*n-288)*a(n+1)+(28*n+66)*a(n+2)+(-n-3)*a(n+3)=0,a(0)=1,a(1)=4,a(2)=28},a(n),remember):
map(f, [$0..50]); # Robert Israel, Jul 06 2015

CoefficientList[ Series[3/(1 + 2Sqrt[1 - 12x]), {x, 0, 19}], x] (* Robert G. Wilson v, Nov 12 2003 *)

x='x+O('x^66); Vec(3/(1+2*sqrt(1-12*x))) \\ Joerg Arndt, Sep 06 2014

a(n) = Sum_{k=0..n} A039599(n,k)*3^(n-k). - Philippe Deléham, Aug 25 2007
From Paul Barry, Sep 15 2009: (Start)
G.f.: 1/(1-4x*c(3x)), c(x) the g.f. of A000108;
G.f.: 1/(1-4x/(1-3x/(1-3x/(1-3x/(1-3x/(1-.... (continued fraction);
G.f.: 1/(1-4x-12x^2/(1-6x-9x^2/(1-6x-9x^2/(1-6x-9x^2/(1-... (continued fraction).
Integral representation: a(n) = (2/Pi)*Integral_{x=0..12} x^n*sqrt(x*(12-x))/(16-x). (End)
a(0) = 1; a(n) = (4/n) * Sum_{j=0..n-1} C(2*n,j) * (n-j) * 3^j for n > 0.
a(n) = upper left term in M^n, M = an infinite square production matrix as follows:
  4, 4, 0, 0, 0, 0, ...
  3, 3, 3, 0, 0, 0, ...
  3, 3, 3, 3, 0, 0, ...
  3, 3, 3, 3, 3, 0, ...
  3, 3, 3, 3, 3, 3, ...
  ...
- Gary W. Adamson, Jul 15 2011
D-finite with recurrence: n*a(n) + 2*(9-14*n)*a(n-1) + 96*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
P-recurrence confirmed using differential equation (-96*x+10)*g(x) + (-192*x^2+28*x-1)*g'(x) - 6 = 0 satisfied by the generating function. - Robert Israel, Jul 06 2015
a(n) ~ 3*12^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013
a(n) are special values of the hypergeometric function 2F1, in Maple notation: a(n) = 3*12^n*GAMMA(n+1/2)*hypergeom([1,n+1/2],[n+2],3/4)/(4*sqrt(Pi)*(n+1)!), n=0,1,... . - Karol A. Penson, Jul 06 2015

Edited by Alois P. Heinz, Jan 20 2011

A130976 G.f.: 8/(3 + 5sqrt(1-16x)).

Original entry on oeis.org

1, 5, 45, 485, 5725, 71445, 925965, 12335685, 167817405, 2321105525, 32536755565, 461181239205, 6598203881245, 95157851939285, 1381842797170125, 20187779510360325, 296499276685062525, 4375281190871356725, 64836419120040890925

Offset: 0

Philippe Deléham, Aug 23 2007

nonn

Number of walks of length 2n on the 5-regular tree beginning and ending at some fixed vertex. Hankel transform is A135292. - Philippe Deléham, Feb 25 2009

Also the number of length 2n words over an alphabet of size 5 that can be built by repeatedly inserting doublets into the initially empty word.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Libor Caha and Daniel Nagaj, The pair-flip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quant-ph], 2018.
Pakawut Jiradilok and Supanat Kamtue, Transportation Distance between Probability Measures on the Infinite Regular Tree, arXiv:2107.09876 [math.CO], 2021.

Column k=5 of A183135.

Cf. A007318.

a:= n-> `if`(n=0, 1, 5/n*add(binomial(2*n, j) *(n-j)*4^j, j=0..n-1)):
seq(a(n), n=0..20);

CoefficientList[Series[8/(3+5*Sqrt[1-16*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

a(n) = Sum_{k=0..n} A039599(n,k) * 4^(n-k). - Philippe Deléham, Aug 25 2007
a(0) = 1; a(n) = (5/n) * Sum_{j=0..n-1} C(2*n,j) * (n-j) * 4^j for n > 0.
a(n) = upper left term in M^n, M = an infinite square production matrix as follows:
  5, 5, 0, 0, 0, 0, ...
  4, 4, 4, 0, 0, 0, ...
  4, 4, 4, 4, 0, 0, ...
  4, 4, 4, 4, 4, 0, ...
  4, 4, 4, 4, 4, 4, ...
  ...
- Gary W. Adamson, Jul 13 2011
D-finite with recurrence: n*a(n) = (41*n-24)*a(n-1) - 200*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 20*16^n/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012
From Karol A. Penson, Jul 02 2015: (Start)
Special values of the hypergeometric function 2F1, in Maple notation:
a(n) = 4*16^n*GAMMA(n+1/2)*hypergeom([1, n+1/2], [n+2], 16/25)/(5*sqrt(Pi)*(n+1)!), n=0,1,...
Moment representation as the 2n-th moment of the positive function
  W(x) = 5*sqrt(16-x^2)/(Pi*(25-x^2)) on (0,4):
  a(n) = int(x^(2*n)*W(x),x=0..4), n=0,1,... . (End)

More terms from Olivier Gérard, Sep 22 2007

Edited by Alois P. Heinz, Jan 17 2011

A130977 G.f.: 5/(2 + 3sqrt(1-20x)).

Original entry on oeis.org

1, 6, 66, 876, 12786, 197796, 3183156, 52718616, 892401426, 15368638836, 268388185596, 4741271556456, 84573471344916, 1521119577791976, 27554494253636136, 502257203287150896, 9205363627419463506

Offset: 0

Philippe Deléham, Aug 23 2007

nonn

Number of walks of length 2n on the 6-regular tree beginning and ending at some fixed vertex. Hankel transform is A135349. - Philippe Deléham, Feb 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Column k=6 of A183135.

CoefficientList[Series[5/(2+3*Sqrt[1-20*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

a(n) = Sum_{k=0..n} A039599(n,k)*5^(n-k). - Philippe Deléham, Aug 25 2007
From Gary W. Adamson, Jul 22 2011: (Start)
a(n) = upper left term in M^n, M = an infinite square production matrix as follows:
  6, 6, 0, 0, 0, 0, ...
  5, 5, 5, 0, 0, 0, ...
  5, 5, 5, 5, 0, 0, ...
  5, 5, 5, 5, 5, 0, ...
  5, 5, 5, 5, 5, 5, ...
  ... (End)
D-finite with recurrence: n*a(n) = 2*(28*n-15)*a(n-1) - 360*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 3*2^(2*n-3)*5^(n+1)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012

More terms from Olivier Gérard, Sep 22 2007

A130978 G.f.: 12/(5 + 7sqrt(1-24x)).

Original entry on oeis.org

1, 7, 91, 1435, 24955, 460747, 8859739, 175466347, 3553964155, 73266506635, 1532152991131, 32420721097387, 692865048943291, 14932919812627915, 324195908270339035, 7083228794200550635

Offset: 0

Philippe Deléham, Aug 23 2007

nonn

Number of walks of length 2n on the 7-regular tree beginning and ending at some fixed vertex. Hankel transform is A135314. - Philippe Deléham, Feb 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Column k=7 of A183135.

g:=12/(5+7*sqrt(1-24*x));gser:=series(g,x=0,20); seq(coeff(gser,x,n),n=0..15); # Emeric Deutsch, Aug 26 2007

CoefficientList[Series[12/(5+7*Sqrt[1-24*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

a(n) = Sum_{k=0..n} A039599(n,k)*6^(n-k). - Philippe Deléham, Aug 25 2007
D-finite with recurrence: n*a(n) = (73*n-36)*a(n-1) - 588*(2*n-3)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 7*2^(3*n+1)*3^(n+1)/(25*sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012

More terms from Emeric Deutsch, Aug 26 2007

A130979 G.f.: 7/(3 + 4sqrt(1-28x)).

Original entry on oeis.org

1, 8, 120, 2192, 44248, 949488, 21237168, 489517344, 11544312984, 277190766896, 6753051796240, 166505875155936, 4146984734796016, 104174408364697952, 2636346768784492128, 67149645964991840832, 1720072455615130558488

Offset: 0

Philippe Deléham, Aug 23 2007

nonn

Number of walks of length 2n on the 8-regular tree beginning and ending at some fixed vertex. Hankel transform is A135315. - Philippe Deléham, Feb 25 2009

G. C. Greubel, Table of n, a(n) for n = 0..690

Column k=8 of A183135.

CoefficientList[Series[7/(3 + 4*Sqrt[1 - 28*x]), {x,0,50}], x] (* G. C. Greubel, Jan 28 2017 *)

Vec(7/(3 + 4*sqrt(1-28*x)) + O(x^50)) \\ G. C. Greubel, Jan 28 2017

a(n) = Sum_{k=0..n} A039599(n,k)*7^(n-k). - Philippe Deléham, Aug 25 2007
a(n) ~ 14*28^n/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013
D-finite with recurrence: n*a(n) +2*(-46*n+21)*a(n-1) +896*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020

More terms from Olivier Gérard, Sep 22 2007

A130980 G.f.: 16/(7 + 9sqrt(1 - 32x)).

Original entry on oeis.org

1, 9, 153, 3177, 73017, 1785609, 45543897, 1197639081, 32231934585, 883404542025, 24570973169433, 691759954058985, 19674867844155321, 564462038150345097, 16315646312285498457, 474680922491822688297

Offset: 0

Philippe Deléham, Aug 23 2007

nonn

Number of walks of length 2n on the 9-regular tree beginning and ending at some fixed vertex. Hankel transform is A135320. - Philippe Deléham, Feb 25 2009

G. C. Greubel, Table of n, a(n) for n = 0..660

Column k=9 of A183135.

CoefficientList[Series[16/(7+9*Sqrt[1-32*x]),{x,0,30}],x] (* Harvey P. Dale, Feb 21 2013 *)

Vec(16/(7 + 9*sqrt(1-32*x)) + O(x^50)) \\ G. C. Greubel, Jan 28 2017

a(n) = Sum_{k=0..n} A039599(n,k)*8^(n-k). - Philippe Deléham, Aug 25 2007
a(n) ~ 72/49*32^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013
D-finite with recurrence: n*a(n) +(-113*n+48)*a(n-1) +1296*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020

More terms from Olivier Gérard, Sep 22 2007

cp's OEIS Frontend

A183134 Square array A(n,k) by antidiagonals. A(n,k) is the number of length 2n k-ary words (n,k>=0), either empty or beginning with the first character of the alphabet, 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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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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A089022 Number of walks of length 2n on the 3-regular tree beginning and ending at some fixed vertex.

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A035610 G.f.: 3/(1 + 2*sqrt(1-12*x)).

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A130976 G.f.: 8/(3 + 5*sqrt(1-16*x)).

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A130977 G.f.: 5/(2 + 3*sqrt(1-20*x)).

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A035610 G.f.: 3/(1 + 2sqrt(1-12x)).

A130976 G.f.: 8/(3 + 5sqrt(1-16x)).

A130977 G.f.: 5/(2 + 3sqrt(1-20x)).

A130978 G.f.: 12/(5 + 7sqrt(1-24x)).

A130979 G.f.: 7/(3 + 4sqrt(1-28x)).

A130980 G.f.: 16/(7 + 9sqrt(1 - 32x)).