A027540 -id:A027540 - cp's OEIS frontend

A334218 Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.

1, 1, 0, 0, 2, 0, 0, 3, 3, 0, 0, 4, 16, 4, 0, 0, 5, 55, 55, 5, 0, 0, 6, 156, 396, 156, 6, 0, 0, 7, 399, 2114, 2114, 399, 7, 0, 0, 8, 960, 9528, 19328, 9528, 960, 8, 0, 0, 9, 2223, 38637, 140571, 140571, 38637, 2223, 9, 0, 0, 10, 5020, 146080, 882340, 1561900, 882340, 146080, 5020, 10, 0

Offset: 0

Andrew Howroyd, May 04 2020

			Triangle begins:
  1;
  1, 0;
  0, 2,   0;
  0, 3,   3,    0;
  0, 4,  16,    4,     0;
  0, 5,  55,   55,     5,    0;
  0, 6, 156,  396,   156,    6,   0;
  0, 7, 399, 2114,  2114,  399,   7, 0;
  0, 8, 960, 9528, 19328, 9528, 960, 8, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns k=2..9 are A027540(n-1), A151576, A151577, A151578, A151579, A151580, A151581, A151582.
Row sums are A000142.
Cf. A008292.

T(n, k) = {if(n==0, k==0, n*sum(j=0, k, (-1)^j * (k-j)^(n-1) * binomial(n, j)))}

T(n, k) = n*A008292(n-1, k) for n > 1.
T(n, k) = T(n, n-k) for n > 1.
T(n, k) = n*Sum_{j=0..k} (-1)^j * (k-j)^(n-1) * binomial(n, j) for n > 0.

A151576 Number of permutations of 1..n arranged in a circle with exactly 3 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 4, 55, 396, 2114, 9528, 38637, 146080, 526240, 1831644, 6217523, 20716164, 68059710, 221195824, 712856665, 2282058360, 7266358556, 23035517940, 72760054815, 229112753980, 719545590010, 2254604460264, 7050252659525, 22006821057936, 68581455012504, 213411502891468

Offset: 3

R. H. Hardin, May 21 2009

Exactly 2 adjacent element pairs in decreasing order gives A027540(n-1).

Andrew Howroyd, Table of n, a(n) for n = 3..500
Index entries for linear recurrences with constant coefficients, signature (16,-111,438,-1083,1740,-1817,1190,-444,72).

Column k=3 of A334218.
Related sequences: A151577-A151610.
Cf. A000460.

a(n)={n*(3^(n-1) - n*2^(n-1) + n*(n-1)/2)} \\ Andrew Howroyd, May 05 2020

From Andrew Howroyd, May 05 2020: (Start)
a(n) = n*A000460(n-1).
a(n) = n*(3^(n-1) - n*2^(n-1) + n*(n-1)/2).
a(n) = 16*a(n-1) - 111*a(n-2) + 438*a(n-3) - 1083*a(n-4) + 1740*a(n-5) - 1817*a(n-6) + 1190*a(n-7) - 444*a(n-8) + 72*a(n-9).
G.f.: x^4*(4 - 9*x - 40*x^2 + 131*x^3 - 98*x^4)/((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)^2).
(End)

Terms a(18) and beyond from Andrew Howroyd, May 05 2020

A126136 Binomial transform of A107430.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 8, 7, 9, 3, 16, 15, 28, 16, 6, 32, 31, 75, 55, 40, 10, 64, 63, 186, 156, 165, 75, 20, 128, 127, 441, 399, 546, 336, 175, 35, 256, 255, 1016, 960, 1596, 1176, 896, 336, 70, 512, 511, 2295, 2223, 4320, 3564, 3528, 1848, 756, 126, 1024, 1023, 5110, 5020, 11115, 9855, 11880, 7680, 4620, 1470, 252

Offset: 0

Gary W. Adamson, Dec 18 2006

Row sums = powers of 3.

			First few rows of the triangle are:
1;
2, 1;
4, 3, 2;
8, 7, 9, 3;
16, 15, 28, 16, 6;
32, 31, 75, 55, 40, 10;
...

Cf. A107430.

Columns : A000079, A000225, A058877, A027540

tabl(nn) = {p = matrix(nn+1, nn+1, n, k, binomial(n-1, k-1)); m = matrix(nn+1, nn+1, n, k, if (k<=n, binomial(n-1, (k-1)\2), 0)); r = p*m; for (n=0, nn, for (k=0, n, print1(r[n+1,k+1], ", ");); print(););} \\ Michel Marcus, Jul 03 2017

Given M = A107430 as an infinite lower triangular matrix and P = Pascal's triangle, A126136 = P*M.

More terms from Philippe Deléham, Jul 02 2017

A164238 Number of binary strings of length n with equal numbers of 00100 and 01101 substrings.

Original entry on oeis.org

1, 2, 4, 8, 16, 30, 56, 104, 194, 363, 682, 1286, 2432, 4612, 8766, 16701, 31887, 61017, 117001, 224812, 432789, 834689, 1612548, 3120361, 6047274, 11736581, 22809338, 44385205, 86473743, 168662839, 329316660, 643634073, 1259124723, 2465346235

Offset: 0

R. H. Hardin Aug 11 2009

nonn

Exactly 2 adjacent element pairs in decreasing order gives A027540

R. H. Hardin, Table of n, a(n) for n=0..500

cp's OEIS Frontend

A334218 Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A151576 Number of permutations of 1..n arranged in a circle with exactly 3 adjacent element pairs in decreasing order.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A126136 Binomial transform of A107430.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A164238 Number of binary strings of length n with equal numbers of 00100 and 01101 substrings.

Views

Author

Keywords

Comments

Links