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.

Showing 1-5 of 5 results.

A351638 Number of length n word structures with all distinct run-lengths using an infinite alphabet.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 17, 19, 31, 45, 177, 191, 335, 469, 733, 2679, 3063, 5129, 7445, 11431, 15667, 59025, 65301, 112379, 159827, 248185, 336913, 505683, 1660611, 1909901, 3184601, 4576771, 6994351, 9606093, 14229033, 19085255, 61388207, 69587029, 116257501, 164298495, 252820047
Offset: 0

Views

Author

Andrew Howroyd, Feb 15 2022

Keywords

Comments

Permuting the symbols will not change the structure.
Equivalently, a(n) is the number of restricted growth strings [s(0), s(1), ..., s(n-1)] where s(0)=0 and s(i) <= 1 + max(prefix) for i >= 1 and every run has a different length.

Examples

			The a(3) = 3 words are 111, 112, 122.
The a(4) = 3 words are 1111, 1112, 1222. The word 1122 is not included because both runs have the same length.
The a(6) = 17 words are 111111, 111112, 111122, 111211, 111221, 112111, 112221, 112222, 122111, 122211, 122222, 111223, 111233, 112333, 112223, 122333, 122233.

Links

Crossrefs

Row sums of A351637.
Cf. A351292, A351642.

Programs

  • PARI
    P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
R(u, k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
seq(n)={my(u=P(n)); concat([1], sum(k=1, n, R(u, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)/r!) ))}

A351641 Triangle read by rows: T(n,k) is the number of length n word structures with all distinct runs using exactly k different symbols.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 3, 1, 0, 1, 8, 12, 4, 1, 0, 1, 17, 28, 22, 5, 1, 0, 1, 26, 81, 68, 35, 6, 1, 0, 1, 45, 177, 251, 135, 51, 7, 1, 0, 1, 76, 410, 704, 610, 236, 70, 8, 1, 0, 1, 121, 906, 2068, 2086, 1266, 378, 92, 9, 1
Offset: 0

Views

Author

Andrew Howroyd, Feb 15 2022

Keywords

Comments

Permuting the symbols will not change the structure.
Equivalently, T(n,k) is the number of restricted growth strings [s(0), s(1), ..., s(n-1)] where s(0)=0 and s(i) <= 1 + max(prefix) for i >= 1, the maximum value is k and all runs are distinct.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,   1;
  0, 1,  5,   3,   1;
  0, 1,  8,  12,   4,   1;
  0, 1, 17,  28,  22,   5,  1;
  0, 1, 26,  81,  68,  35,  6, 1;
  0, 1, 45, 177, 251, 135, 51, 7, 1;
  ...
The T(4,1) = 1 word is 1111.
The T(4,2) = 5 words are 1112, 1121, 1122, 1211, 1222.
The T(4,3) = 3 words are 1123, 1223, 1233.
The T(4,4) = 1 word is 1234.

Links

Crossrefs

Row sums are A351642.
Partial row sums include A000007, A000012, A351018, A351644.
Column k=3 is A351643.
Cf. A111596, A351637, A351640.

Programs

  • PARI
    \\ here LahI is A111596 as row polynomials.
LahI(n, y)={sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
S(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); 1 + sum(i=1, (sqrtint(8*n+1)-1)\2, polcoef(p, i, y)*LahI(i, y))}
R(q)={[subst(serlaplace(p), y, 1) | p<-Vec(q)]}
T(n)={my(q=S(n), v=concat([1], sum(k=1, n, R(q^k-1)*sum(r=k, n, y^r*binomial(r, k)*(-1)^(r-k)/r!) ))); [Vecrev(p) | p<-v]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

Formula

T(n,k) = A351640(n,k)/k!.

A350824 Triangle read by rows: T(n,k) is the number of patterns of length n with all distinct run lengths and maximum value k, n >= 0, k = 0..floor(sqrt(8*n+1)-1/2).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 4, 0, 1, 4, 0, 1, 8, 0, 1, 20, 36, 0, 1, 24, 36, 0, 1, 36, 72, 0, 1, 52, 108, 0, 1, 112, 576, 576, 0, 1, 128, 612, 576, 0, 1, 200, 1116, 1152, 0, 1, 264, 1584, 1728, 0, 1, 384, 2520, 2880, 0, 1, 700, 8064, 20736, 14400, 0, 1, 868, 9432, 22464, 14400
Offset: 0

Views

Author

Andrew Howroyd, Feb 12 2022

Keywords

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1;
  0, 1,   4;
  0, 1,   4;
  0, 1,   8;
  0, 1,  20,   36;
  0, 1,  24,   36;
  0, 1,  36,   72;
  0, 1,  52,  108;
  0, 1, 112,  576,  576;
  0, 1, 128,  612,  576;
  0, 1, 200, 1116, 1152;
  ...
The T(5,1) = 1 pattern is 11111.
The T(5,2) = 8 patterns are 12222, 11222, 11122, 11112, 21111, 22111, 22211, 22221.

Links

Crossrefs

Row sums are A351292.
Cf. A000217, A001044, A003056, A008289, A351637, A351640.

Programs

  • PARI
    P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
R(u, k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
T(n)={my(u=P(n), v=concat([1], sum(k=1, n, R(u, k)*sum(r=k, n, y^r*binomial(r, k)*(-1)^(r-k)) ))); [Vecrev(p) | p<-v]}
{ my(A=T(16)); for(n=1, #A, print(A[n])) }

Formula

T(n,k) = Sum_{j=1..k} R(n,j)*binomial(k, j)*(-1)^(k-j) for n > 0, where R(n,k) = Sum_{j=1..A003056(n)} k*(k-1)^(j-1) * j! * A008289(n,j).
T(n,k) = k! * A351637(n,k).
T(A000217(n),n) = A001044(n). - Alois P. Heinz, Feb 15 2022

A351639 Number of length n word structures with all distinct run-lengths using at most 3 symbols.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 17, 19, 31, 45, 153, 167, 287, 397, 613, 1695, 2007, 3185, 4565, 6799, 9235, 24057, 27645, 44483, 61619, 92089, 122857, 179355, 385995, 468605, 713849, 996331, 1441447, 1947813, 2766657, 3659135, 7467623, 8930629, 13471885, 18283575, 26484639
Offset: 0

Views

Author

Andrew Howroyd, Feb 15 2022

Keywords

Links

Crossrefs

Column k=3 of A351637.
Cf. A351638.

Programs

  • PARI
    \\ See A351637 for P, R.
seq(n)={my(u=P(n)); concat([1], sum(k=1, n, R(u, k)*sum(r=k, 3, binomial(r, k)*(-1)^(r-k)/r!) ))}

A351645 Triangle read by rows: T(n,k) is the number of length n word structures using exactly k different symbols with all distinct run-lengths and the first run length of a symbol less than that of previous symbols, n >= 0, k = 0..floor(sqrt(8*n+1)-1/2).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 5, 1, 0, 1, 6, 1, 0, 1, 9, 2, 0, 1, 13, 3, 0, 1, 28, 16, 1, 0, 1, 32, 17, 1, 0, 1, 50, 31, 2, 0, 1, 66, 44, 3, 0, 1, 96, 70, 5, 0, 1, 175, 224, 36, 1, 0, 1, 217, 262, 39, 1, 0, 1, 308, 428, 71, 2, 0, 1, 425, 619, 105, 3
Offset: 0

Views

Author

Andrew Howroyd, Feb 16 2022

Keywords

Comments

Permuting the symbols will not change the structure.
The k-th column of A351637 is divisible by k!.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1;
  0, 1,  1;
  0, 1,  1;
  0, 1,  2;
  0, 1,  5,  1;
  0, 1,  6,  1;
  0, 1,  9,  2;
  0, 1, 13,  3;
  0, 1, 28, 16, 1;
  0, 1, 32, 17, 1;
  0, 1, 50, 31, 2;
  0, 1, 66, 44, 3;
  0, 1, 96, 70, 5;
  ...
The T(8,1) = 1 word is 11111111.
The T(8,2) = 9 words are 11111112, 11111122, 11111211, 11111221, 11111222, 11112111, 11112221, 11121111, 11211111.
The T(8,3) = 2 words are 11111223, 11112223.
In the last example, the word 11111223 corresponds with 6 words in A351637 which are 11111223, 11111233, 11222223, 11233333, 12222233, 12233333.

Links

Crossrefs

Row sums are A351732.
Cf. A000217, A003056, A008289, A350824, A351637.

Programs

  • PARI
    P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
R(u, k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
T(n)={my(u=P(n), v=concat([1], sum(k=1, n, R(u, k)*sum(r=k, n, y^r*binomial(r, k)*(-1)^(r-k)/(r!)^2) ))); [Vecrev(p) | p<-v]}
{ my(A=T(16)); for(n=1, #A, print(A[n])) }

Formula

T(n,k) = Sum_{j=1..k} R(n,j)*binomial(k, j)*(-1)^(k-j)/(k!)^2 for n > 0, where R(n,k) = Sum_{j=1..A003056(n)} k*(k-1)^(j-1) * j! * A008289(n,j).
T(n,k) = A351637(n,k)/k! = A350824(n,k)/(k!)^2.
T(A000217(n),n) = 1.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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