cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A131521 Expansion of 9/(4 + 5*sqrt(1-36*x)).

Original entry on oeis.org

1, 10, 190, 4420, 113950, 3128140, 89608780, 2647358920, 80065458910, 2466432898300, 77115832253380, 2440820453410360, 78053018025315340, 2517915855707814520, 81839894422876183000, 2677554649095487584400
Offset: 0

Views

Author

Philippe Deléham, Aug 23 2007

Keywords

Comments

Number of walks of length 2n on the 10-regular tree beginning and ending at some fixed vertex. Hankel transform is A135321. - Philippe Deléham, Feb 25 2009

Links

Crossrefs

Column k=10 of A183135.
Cf. A039599, A135321.

Programs

  • Mathematica
    CoefficientList[Series[9/(4+5*Sqrt[1-36*x]),{x,0,30}],x] (* Harvey P. Dale, Aug 21 2012 *)
  • PARI
    Vec(9/(4 + 5*sqrt(1-36*x)) + O(x^50)) \\ G. C. Greubel, Jan 28 2017

Formula

G.f.: 9/(4 + 5*sqrt(1-36*x)).
a(n) = Sum_{k=0..n} A039599(n,k)*9^(n-k). - Philippe Deléham, Aug 25 2007
a(n) ~ 45*36^n/(32*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013
D-finite with recurrence: n*a(n) +2*(-68*n+27)*a(n-1) +1800*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020

Extensions

More terms from Olivier Gérard, Sep 22 2007

A256116 Number T(n,k) of length 2n k-ary words, either empty or beginning with the first letter of the alphabet and using each letter at least once, that 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, 10, 0, 1, 34, 112, 84, 0, 1, 125, 930, 1800, 1008, 0, 1, 461, 7018, 26400, 35640, 15840, 0, 1, 1715, 51142, 334152, 816816, 840840, 308880, 0, 1, 6434, 368464, 3944220, 15550080, 27824160, 23063040, 7207200
Offset: 0

Views

Author

Alois P. Heinz, Mar 15 2015

Keywords

Examples

			T(3,2) = 9: aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba.
T(3,3) = 10: aabbcc, aabccb, aacbbc, aaccbb, abbacc, abbcca, abccba, acbbca, accabb, accbba.
T(4,2) = 34: aaaaaabb, aaaaabba, aaaabaab, aaaabbaa, aaaabbbb, aaabaaba, aaabbaaa, aaabbabb, aaabbbba, aabaaaab, aabaabaa, aabaabbb, aababbab, aabbaaaa, aabbaabb, aabbabba, aabbbaab, aabbbbaa, aabbbbbb, abaaaaba, abaabaaa, abaababb, abaabbba, ababbaba, abbaaaaa, abbaaabb, abbaabba, abbabaab, abbabbaa, abbabbbb, abbbaaba, abbbbaaa, abbbbabb, abbbbbba.
T(4,4) = 84: aabbccdd, aabbcddc, aabbdccd, aabbddcc, aabccbdd, aabccddb, aabcddcb, aabdccdb, aabddbcc, aabddccb, aacbbcdd, aacbbddc, aacbddbc, aaccbbdd, aaccbddb, aaccdbbd, aaccddbb, aacdbbdc, aacddbbc, aacddcbb, aadbbccd, aadbbdcc, aadbccbd, aadcbbcd, aadccbbd, aadccdbb, aaddbbcc, aaddbccb, aaddcbbc, aaddccbb, abbaccdd, abbacddc, abbadccd, abbaddcc, abbccadd, abbccdda, abbcddca, abbdccda, abbddacc, abbddcca, abccbadd, abccbdda, abccddba, abcddcba, abdccdba, abddbacc, abddbcca, abddccba, acbbcadd, acbbcdda, acbbddca, acbddbca, accabbdd, accabddb, accadbbd, accaddbb, accbbadd, accbbdda, accbddba, accdbbda, accddabb, accddbba, acdbbdca, acddbbca, acddcabb, acddcbba, adbbccda, adbbdacc, adbbdcca, adbccbda, adcbbcda, adccbbda, adccdabb, adccdbba, addabbcc, addabccb, addacbbc, addaccbb, addbbacc, addbbcca, addbccba, addcbbca, addccabb, addccbba.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,    2;
  0, 1,    9,    10;
  0, 1,   34,   112,     84;
  0, 1,  125,   930,   1800,   1008;
  0, 1,  461,  7018,  26400,  35640,  15840;
  0, 1, 1715, 51142, 334152, 816816, 840840, 308880;

Links

Crossrefs

Columns k=0-2 give: A000007, A057427, A010763(n-1) for n>0.
Main diagonal gives A065866(n-1) (for n>0).
Row sums give A294603.
Cf. A183135, A256117.

Programs

  • Maple
    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(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)/
    `if`(k=0, 1, k):
seq(seq(T(n, k), k=0..n), n=0..12);
  • Mathematica
    Unprotect[Power]; 0^0 = 1; A[n_, k_] := A[n, k] = If[n==0, 1, k/n*Sum[ Binomial[2*n, j]*(n-j)*(k-1)^j, {j, 0, n-1}]];
T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]/If[k==0, 1, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

Formula

T(n,k) = (Sum_{i=0..k} (-1)^i * C(k,i) * A183135(n,k-i)) / A028310(k).
T(n,k) = (k-1)! * A256117(n,k) for k > 0.

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

Views

Author

Alois P. Heinz, Jun 03 2012

Keywords

Examples

			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, ...

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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).

A294491 Number of length 2n n-ary words that can be built by repeatedly inserting doublets into the initially empty word.

Original entry on oeis.org

1, 1, 6, 87, 2092, 71445, 3183156, 175466347, 11544312984, 883404542025, 77115832253380, 7564442149980111, 823833773843404776, 98644885379708947357, 12880909497761085034632, 1821689155897508835803475, 277402856595034529463789616, 45253909471856604392088994065
Offset: 0

Views

Author

Alois P. Heinz, Oct 31 2017

Keywords

Comments

Also the number of rooted closed walks of length 2n on the infinite rooted n-ary tree.

Examples

			a(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.

Links

Crossrefs

Main diagonal of A183135.
Cf. A248828.

Programs

  • Maple
    a:= n-> `if`(n=0, 1, add(binomial(2*n, j)*(n-j)*(n-1)^j, j=0..n-1)):
seq(a(n), n=0..21);

Formula

a(n) = Sum_{j=0..n-1} binomial(2*n,j)*(n-j)*(n-1)^j for n>0, a(0) = 1.
a(n) = [x^n] 2*(n-1)/(n-2+n*sqrt(1-(4*n-4)*x)) for n>1, a(n) = 1 for n<2.
a(n) = A183135(n,n).
a(n) = n * A248828(n) for n>0, a(0) = 1.
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