A183134 -id:A183134 - cp's OEIS frontend

A194730 Number of 10-ary words either empty or beginning with the first character of the alphabet, that can be built by inserting n doublets into the initially empty word.

1, 1, 19, 442, 11395, 312814, 8960878, 264735892, 8006545891, 246643289830, 7711583225338, 244082045341036, 7805301802531534, 251791585570781452, 8183989442287618300, 267755464909548758440, 8810802978165549384355, 291414010749705281701270

Offset: 0

Alois P. Heinz, Sep 02 2011

nonn

			a(2) = 19: aaaa, aabb, aacc, aadd, aaee, aaff, aagg, aahh, aaii, aajj, abba, acca, adda, aeea, affa, agga, ahha, aiia, ajja (with 10-ary alphabet {a,b,c,d,e,f,g,h,i,j}).

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

Column k=10 of A183134.

a:= n-> `if`(n=0, 1, add(binomial (2*n, j) *(n-j) *9^j, j=0..n-1) /n):
seq(a(n), n=0..20);

G.f.: 9/10 + 9/(5*(8+10*sqrt(1-36*x))).
a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*9^j for n>0.
a(n) ~ 9 * 36^n / (64 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 07 2014

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).

A248828 Number of 2n-length words, either empty or beginning with the first character of an n-ary alphabet, that can be built by repeatedly inserting doublets into the initially empty word.

Original entry on oeis.org

1, 1, 3, 29, 523, 14289, 530526, 25066621, 1443039123, 98156060225, 7711583225338, 687676559089101, 68652814486950398, 7588068106131457489, 920064964125791788188, 121445943726500589053565, 17337678537189658091486851, 2661994674815094376005234945

Offset: 0

Alois P. Heinz, Oct 15 2014

nonn

			a(2) = 3: aaaa, aabb, abba (with alphabet {a,b}).

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

Main diagonal of A183134.

Cf. A294491.

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

Flatten[{1,1,Table[Sum[Binomial[2*n, j]*(n-j)*(n-1)^j, {j,0,n-1}]/n,{n,2,20}]}] (* Vaclav Kotesovec, Oct 15 2014 *)

a(n) = A183134(n,n).
a(n) ~ exp(-1) * 4^n * n^(n-5/2) / sqrt(Pi). - Vaclav Kotesovec, Oct 15 2014
a(n) = A294491(n) / n for n>0, a(0) = 1. - Alois P. Heinz, Oct 31 2017

A194727 Number of 7-ary words either empty or beginning with the first character of the alphabet, that can be built by inserting n doublets into the initially empty word.

Original entry on oeis.org

1, 1, 13, 205, 3565, 65821, 1265677, 25066621, 507709165, 10466643805, 218878998733, 4631531585341, 98980721277613, 2133274258946845, 46313701181477005, 1011889827742935805, 22232378278653590125, 490899296804667191005, 10887346288742800406605

Offset: 0

Alois P. Heinz, Sep 02 2011

nonn

			a(2) = 13: aaaa, aabb, aacc, aadd, aaee, aaff, aagg, abba, acca, adda, aeea, affa, agga (with 7-ary alphabet {a,b,c,d,e,f,g}).

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=7 of A183134.

Cf. A194728, A194729, A194723.

a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *6^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, 13][n+1],
      ((73*n-36)*a(n-1) -(1176*n-1764)*a(n-2))/n)
    end:
seq(a(n), n=0..30);

G.f.: 6/7 + 12/(7*(5+7*sqrt(1-24*x))).
a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*6^j for n>0.
a(n) ~ 6 * 24^n / (25 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 07 2014
D-finite with recurrence n*a(n) +(-73*n+36)*a(n-1) +588*(2*n-3)*a(n-2)=0. - R. J. Mathar, Mar 14 2015
From Karol A. Penson, Jul 16 2015: (Start)
Special values of the hypergeometric function 2F1, in Maple notation:
a(n+1) = (12/7)^2*24^n*GAMMA(n+3/2)*hypergeom([1,n+3/2],[n+3],24/49)/(sqrt(Pi)*(n+2)!), n=0,1,... .
Integral representation as the n-th moment of a positive function W(x) = sqrt(x*(24-x))/(2*Pi*(49-x)) on (0,24): a(n+1) = int(x^n*W(x), x=0..24), n=0,1,... . This representation is unique as W(x) is the solution of the Hausdorff moment problem. (End)

A194716 Number of n-ary words beginning with the first character of the alphabet, that can be built by inserting four doublets into the initially empty word.

Original entry on oeis.org

0, 1, 35, 181, 523, 1145, 2131, 3565, 5531, 8113, 11395, 15461, 20395, 26281, 33203, 41245, 50491, 61025, 72931, 86293, 101195, 117721, 135955, 155981, 177883, 201745, 227651, 255685, 285931, 318473, 353395, 390781, 430715, 473281, 518563, 566645, 617611

Offset: 0

Alois P. Heinz, Sep 02 2011

nonn

			a(2) = 35: aaaaaaaa, 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 (with 2-ary alphabet {a,b}).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Row n=4 of A183134.

a:= n-> `if`(n=0, 0, (x-> 1+(6+(14+14*x)*x)*x)(n-1)):
seq(a(n), n=0..40);

G.f.: x*(1+31*x+47*x^2+5*x^3) / (x-1)^4.

a(0) = 0, a(n) = 1+(6+(14+14*(n-1))*(n-1))*(n-1) for n>0.

A194717 Number of n-ary words beginning with the first character of the alphabet, that can be built by inserting five doublets into the initially empty word.

Original entry on oeis.org

0, 1, 126, 1181, 4966, 14289, 32966, 65821, 118686, 198401, 312814, 470781, 682166, 957841, 1309686, 1750589, 2294446, 2956161, 3751646, 4697821, 5812614, 7114961, 8624806, 10363101, 12351806, 14613889, 17173326, 20055101, 23285206, 26890641, 30899414, 35340541

Offset: 0

Alois P. Heinz, Sep 02 2011

nonn

			a(1) = 1: aaaaaaaaaa (with 1-ary alphabet {a}).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Row n=5 of A183134.

a:= n-> `if`(n=0, 0, (x-> 1+(8+(27+(48+42*x)*x)*x)*x)(n-1)):
seq(a(n), n=0..40);

G.f.: -x*(1+121*x+561*x^2+311*x^3+14*x^4) / (x-1)^5.

a(0) = 0, a(n) = 1+(8+(27+(48+42*(n-1))*(n-1))*(n-1))*(n-1) for n>0.

A194718 Number of n-ary words beginning with the first character of the alphabet, that can be built by inserting six doublets into the initially empty word.

Original entry on oeis.org

0, 1, 462, 7941, 48838, 185193, 530526, 1265677, 2654646, 5060433, 8960878, 14964501, 23826342, 36463801, 53972478, 77642013, 108971926, 149687457, 201755406, 267399973, 349118598, 449697801, 572229022, 720124461, 897132918, 1107355633, 1355262126, 1645706037

Offset: 0

Alois P. Heinz, Sep 02 2011

			a(1) = 1: aaaaaaaaaaaa (with 1-ary alphabet {a}).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Row n=6 of A183134.

a:= n-> `if`(n=0, 0, (x-> 1+(10+(44+(110+(165+132*x)*x)*x)*x)*x)(n-1)):
seq(a(n), n=0..30);

LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,462,7941,48838,185193,530526},30] (* Harvey P. Dale, Oct 23 2015 *)

G.f.: x*(1+456*x+5184*x^2+8102*x^3+2055*x^4+42*x^5) / (x-1)^6.

a(0) = 0, a(n) = 1+(10+(44+(110+(165+132*(n-1))*(n-1))*(n-1))*(n-1)) * (n-1) for n>0.

A194719 Number of n-ary words beginning with the first character of the alphabet, that can be built by inserting seven doublets into the initially empty word.

Original entry on oeis.org

0, 1, 1716, 54573, 492724, 2467137, 8786436, 25066621, 61189668, 133071009, 264735892, 490704621, 858686676, 1432583713, 2295801444, 3554870397, 5343375556, 7826194881, 11204046708, 15718346029, 21656369652, 29356730241, 39215159236, 51690598653, 67311601764

Offset: 0

Alois P. Heinz, Sep 02 2011

nonn

			a(1) = 1: a^14 (with 1-ary alphabet {a}).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Row n=7 of A183134.

a:= n-> `if`(n=0, 0, (x-> 1+(12+(65+(208+(429+(572+429*x)*x)*x)
                          *x)*x)*x)(n-1)):
seq(a(n), n=0..30);

G.f.: -x*(1+1709*x+42582*x^2+146714*x^3+104077*x^4+13665*x^5+132*x^6) / (x-1)^7.

a(0) = 0, a(n) = 1+(12+(65+(208+(429+(572+429*(n-1)) * (n-1)) * (n-1)) * (n-1)) * (n-1)) * (n-1) for n>0.

A194720 Number of n-ary words beginning with the first character of the alphabet, that can be built by inserting eight doublets into the initially empty word.

Original entry on oeis.org

0, 1, 6435, 381333, 5068915, 33563481, 148733571, 507709165, 1443039123, 3581326065, 8006545891, 16475259141, 31690921395, 57644499913, 100028603715, 166732334301, 268424064211, 419229350625, 637511191203, 946759829365, 1376599316211, 1963918036281

Offset: 0

Alois P. Heinz, Sep 02 2011

nonn

			a(1) = 1: a^16 (with 1-ary alphabet {a}).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Row n=8 of A183134.

a:= n-> `if`(n=0, 0, (x-> 1+(14+(90+(350+(910+(1638+(2002+1430*x)*
                             x)*x)*x)*x)*x)*x)(n-1)):
seq(a(n), n=0..30);

G.f.: x * (1 +6427*x +329881*x^2 +2198375*x^3 +3329195*x^4 +1251089*x^5 +91803*x^6 +429*x^7) / (x-1)^8.

a(0) = 0, a(n) = 1 +(14 +(90 +(350 +(910 +(1638 +(2002 +1430 * (n-1)) * (n-1)) * (n-1)) * (n-1)) * (n-1)) * (n-1)) * (n-1) for n>0.

A194721 Number of n-ary words beginning with the first character of the alphabet, that can be built by inserting nine doublets into the initially empty word.

Original entry on oeis.org

0, 1, 24310, 2699837, 52955950, 464221105, 2561439806, 10466643805, 34648845862, 98156060225, 246643289830, 563506356061, 1191627482750, 2363434581937, 4441172224750, 7969478316605, 13742556531766, 22888430598145, 36972962559062, 58126513174525, 89196318660430

Offset: 0

Alois P. Heinz, Sep 02 2011

nonn

			a(1) = 1: a^18 (with 1-ary alphabet {a}).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Row n=9 of A183134.

a:= n-> `if`(n=0, 0, (x-> 1+(16+(119+(544+(1700+(3808+(6188+(7072+4862
                          *x)*x)*x)*x)*x)*x)*x)*x)(n-1)):
seq(a(n), n=0..30);

G.f.: -x * (1 +24301*x +2481083*x^2 +29532493*x^3 +82769773*x^4 +66140687*x^5 +14462017*x^6 +624055*x^7 +1430*x^8) / (x-1)^9.

a(0) = 0, a(n) = 1 +(16 +(119 +(544 +(1700 +(3808 +(6188 +(7072 +4862 * (n-1)) *(n-1)) *(n-1)) *(n-1)) *(n-1)) *(n-1)) *(n-1)) *(n-1) for n>0.

cp's OEIS Frontend

A194730 Number of 10-ary words either empty or beginning with the first character of the alphabet, that can be built by inserting n doublets into the initially empty word.

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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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A248828 Number of 2n-length words, either empty or beginning with the first character of an n-ary alphabet, that can be built by repeatedly inserting doublets into the initially empty word.

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A194727 Number of 7-ary words either empty or beginning with the first character of the alphabet, that can be built by inserting n doublets into the initially empty word.

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A194716 Number of n-ary words beginning with the first character of the alphabet, that can be built by inserting four doublets into the initially empty word.

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A194717 Number of n-ary words beginning with the first character of the alphabet, that can be built by inserting five doublets into the initially empty word.

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A194718 Number of n-ary words beginning with the first character of the alphabet, that can be built by inserting six doublets into the initially empty word.

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A194719 Number of n-ary words beginning with the first character of the alphabet, that can be built by inserting seven doublets into the initially empty word.

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