A384893 -id:A384893 - cp's OEIS frontend

A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 4, 1, 1, 8, 10, 7, 5, 1, 1, 13, 18, 16, 9, 6, 1, 1, 21, 33, 31, 23, 11, 7, 1, 1, 34, 59, 62, 47, 31, 13, 8, 1, 1, 55, 105, 119, 101, 66, 40, 15, 9, 1, 1, 89, 185, 227, 205, 151, 88, 50, 17, 10, 1, 1, 144, 324, 426, 414, 321, 213, 113, 61, 19, 11, 1, 1

Offset: 0

Wouter Meeussen, May 23 2001

Riordan array (1/(1-x-x^2), x(1-x)/(1-x-x^2)). Row sums are A000079. Diagonal sums are A006053(n+2). - Paul Barry, Nov 01 2006
Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 05 2012
Mirror image of triangle in A208342. - Philippe Deléham, Mar 05 2012
A053538 is jointly generated with A076791 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1, for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x).  See the Mathematica section at A076791. - Clark Kimberling, Mar 08 2012
The matrix inverse starts
    1;
   -1,   1;
   -1,  -1,   1;
    1,  -2,  -1,  1;
    3,   1,  -3, -1,  1;
    1,   6,   1, -4, -1,  1;
   -7,   4,  10,  1, -5, -1,  1;
  -13, -13,   8, 15,  1, -6, -1,  1;
    3, -31, -23, 13, 21,  1, -7, -1, 1; - R. J. Mathar, Mar 15 2013
Also appears to be the number of subsets of {1..n} containing n with k maximal anti-runs of consecutive elements increasing by more than 1. For example, the subset {1,3,6,7,11,12} has maximal anti-runs ((1,3,6),(7,11),(12)) so is counted under a(12,3). For runs instead of anti-runs we get A202064. - Gus Wiseman, Jun 26 2025

			n=4; Table[binomial[k, j]binomial[n-k, k-j], {k, 0, n}, {j, 0, n}] splits {1, 4, 6, 4, 1} into {{1, 0, 0, 0, 0}, {3, 1, 0, 0, 0}, {1, 4, 1, 0, 0}, {0, 0, 3, 1, 0}, {0, 0, 0, 0, 1}} and this gives summed by columns {5, 5, 4, 1, 1}
Triangle begins :
   1;
   1,  1;
   2,  1,  1;
   3,  3,  1, 1;
   5,  5,  4, 1, 1;
   8, 10,  7, 5, 1, 1;
  13, 18, 16, 9, 6, 1, 1;
...
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, ...) begins :
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,  1;
  0,  3,  3,  1, 1;
  0,  5,  5,  4, 1, 1;
  0,  8, 10,  7, 5, 1, 1;
  0, 13, 18, 16, 9, 6, 1, 1;

Alois P. Heinz, Rows n = 0..140, flattened
R. P. Grimaldi, Extraordinary subsets: a generalization, Fib. Quart., 55 (No. 3, 2017), 114-122. See Table 1.

Column k = 1 is A000045.
Row sums are A000079.
Column k = 2 is A010049.
For runs instead of anti-runs we have A202064.
For integer partitions see A268193, strict A384905, runs A116674.
A034839 counts subsets by number of maximal runs.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs.
Cf. A000071, A001629, A010027, A208342, A384177, A384879, A384889, A384890.

Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j->  Binomial(j,k)*Binomial(n-j,j-k)) ))); # G. C. Greubel, May 16 2019

[[(&+[Binomial(j,k)*Binomial(n-j,j-k): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 16 2019

a:= (n, m)-> add(binomial(k, m)*binomial(n-k, k-m), k=0..n):
seq(seq(a(n,m), m=0..n), n=0..12);  # Alois P. Heinz, Sep 19 2013

Table[Sum[Binomial[k, m]*Binomial[n-k, k-m], {k,0,n}], {n,0,12}, {m,0,n}]

{T(n,k) = sum(j=0,n, binomial(j,k)*binomial(n-j,j-k))}; \\ G. C. Greubel, May 16 2019

[[sum(binomial(j,k)*binomial(n-j,j-k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 16 2019

From Philippe Deléham, Mar 05 2012: (Start)
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n.
G.f.: 1/(1-(1+y)*x-(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A077957(n), A000045(n+1), A000079(n), A001906(n+1), A007070(n), A116415(n), A084326(n+1), A190974(n+1), A190978(n+1), A190984(n+1), A190990(n+1), A190872(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively. (End)

A384889 Number of subsets of {1..n} with all equal lengths of maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 4, 8, 14, 23, 37, 59, 93, 146, 230, 365, 584, 940, 1517, 2450, 3959, 6404, 10373, 16822, 27298, 44297, 71843, 116429, 188550, 305200, 493930, 799422, 1294108, 2095291, 3392736, 5493168, 8892148, 14390372, 23282110, 37660759, 60914308, 98528312, 159386110

Offset: 0

Gus Wiseman, Jun 18 2025

nonn

			The subset {3,6,7,9,10,12} has maximal anti-runs ((3,6),(7,9),(10,12)), with lengths (2,2,2), so is counted under a(12).
The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {1,3}    {1,2}
                  {2,3}    {1,3}
                  {1,2,3}  {1,4}
                           {2,3}
                           {2,4}
                           {3,4}
                           {1,2,3}
                           {2,3,4}
                           {1,2,3,4}

Christian Sievers, Table of n, a(n) for n = 0..1000

For runs instead of anti-runs we have A243815, distinct A384175, complement A384176.
For distinct instead or equal lengths we have A384177, ranks A384879.
For partitions instead of subsets we have A384888.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A047966 counts uniform partitions (equal multiplicities), ranks A072774.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A010027, A044813, A047993, A356606, A384880, A384886, A384890.

Table[Length[Select[Subsets[Range[n]],SameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]

lista(n)=Vec(sum(i=1,(n+1)\2,1/(1-x^(2*i-1)/(1-x)^(i-1))-1,1-x+O(x*x^n))/(1-x)^2) \\ Christian Sievers, Jun 20 2025

G.f.: ( Sum_{i>=1} (1/(1-x^(2*i-1)/(1-x)^(i-1))-1) + 1-x ) / (1-x)^2. - Christian Sievers, Jun 21 2025

a(21) and beyond from Christian Sievers, Jun 20 2025

A385816 The number k such that the k-th composition in standard order lists the maximal anti-run lengths of the binary indices of n. Standard composition number of row n of A384877.

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 3, 7, 1, 2, 2, 6, 3, 5, 7, 15, 1, 2, 2, 6, 2, 4, 6, 14, 3, 5, 5, 13, 7, 11, 15, 31, 1, 2, 2, 6, 2, 4, 6, 14, 2, 4, 4, 12, 6, 10, 14, 30, 3, 5, 5, 13, 5, 9, 13, 29, 7, 11, 11, 27, 15, 23, 31, 63, 1, 2, 2, 6, 2, 4, 6, 14, 2, 4, 4, 12, 6, 10, 14

Offset: 0

Gus Wiseman, Jul 15 2025

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
If the k-th composition in standard order is y, then the standard composition number of y is defined to be k.

			The binary indices of 181 are {1,3,5,6,8}, with maximal anti-runs ((1,3,5),(6,8)), with lengths (3,2), which is the 18th composition in standard order, so a(181) = 18.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

The reverse version is A209859.
Sorted positions of first appearances are A247648.
These are standard composition numbers of rows of A384877 (duplicates removed A385886).
For runs instead of anti-runs the reverse is A385887 (duplicates removed A232559).
For runs instead of anti-runs we have A385889 (duplicates removed A385818).
A245563 lists run lengths of binary indices (ranks A246029), reverse A245562.
Cf. A000120, A029931, A044813, A048793, A052499, A069010, A348366, A384175, A384878, A384879, A384893.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
stcinv/@Table[Length/@Split[bpe[n],#2!=#1+1&],{n,0,100}]

A210034 Triangle of coefficients of polynomials v(n,x) jointly generated with A210033; see the Formula section.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 7, 5, 2, 1, 12, 10, 6, 2, 1, 20, 20, 13, 7, 2, 1, 33, 38, 29, 16, 8, 2, 1, 54, 71, 60, 39, 19, 9, 2, 1, 88, 130, 122, 86, 50, 22, 10, 2, 1, 143, 235, 241, 187, 116, 62, 25, 11, 2, 1, 232, 420, 468, 392, 267, 150, 75, 28, 12, 2, 1, 376, 744, 894, 806

Offset: 1

Clark Kimberling, Mar 16 2012

For a discussion and guide to related arrays, see A208510.
From Gus Wiseman, Jun 29 2025: (Start)
This appears to be the number of subsets of {1..n} with k>0 maximal anti-runs (sequences of consecutive elements increasing by more than 1). For example, the subset {1,2,4,5} has maximal anti-runs ((1),(2,4),(5)) so is counted under T(5,3). Row n = 5 counts the following:
  {1}      {1,2}    {1,2,3}    {1,2,3,4}  {1,2,3,4,5}
  {2}      {2,3}    {2,3,4}    {2,3,4,5}
  {3}      {3,4}    {3,4,5}
  {4}      {4,5}    {1,2,3,5}
  {5}      {1,2,4}  {1,2,4,5}
  {1,3}    {1,2,5}  {1,3,4,5}
  {1,4}    {1,3,4}
  {1,5}    {1,4,5}
  {2,4}    {2,3,5}
  {2,5}    {2,4,5}
  {3,5}
  {1,3,5}
For runs instead of anti-runs we have A034839, with n A202064. For reversed partitions instead of subsets we have A268193. (End)

			First five rows:
  1
  2    1
  4    2    1
  7    5    2   1
  12   10   6   2   1
First three polynomials v(n,x): 1, 2 + x, 4 + 2*x + x^2.

Cf. A210033, A208510.
Column k = 1 is A000071.
Row sums are A000225.
Column k = 2 is A001629.
Column k = 3 is A055243.
The version including k = 0 is A384893.
A034839 counts subsets by number of maximal runs, see also A202023, A202064.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.
Cf. A053538, A116674, A119900, A268193, A384177, A384879, A384889, A384905.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210033 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210034 *)

u(n,x)=u(n-1,x)+v(n-1,x)+1,
v(n,x)=u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A385886 Irregular triangle read by rows listing the lengths of maximal anti-runs (sequences of distinct consecutive elements increasing by more than 1) of binary indices, duplicate rows removed.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1

Offset: 0

Gus Wiseman, Jul 14 2025

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

This is the triangle A384877, except all duplicates after the first instance of each composition are removed. It lists all compositions in order of their first appearance as a row of A384877.

			The binary indices of 27 are {1,2,4,5}, with maximal anti-runs ((1),(2,4),(5)), with lengths (1,2,1). After removing duplicates, this is our row 10.
The binary indices of 53 are {1,3,5,6}, with maximal anti-runs ((1,3,5),(6)), with lengths (3,1). After removing duplicates, this is our row 16.
Triangle begins:
   0: .
   1: 1
   2: 1 1
   3: 2
   4: 1 1 1
   5: 1 2
   6: 2 1
   7: 1 1 1 1
   8: 3
   9: 1 1 2
  10: 1 2 1
  11: 2 1 1
  12: 1 1 1 1 1
  13: 1 3
  14: 2 2
  15: 1 1 1 2
  16: 3 1
  17: 1 1 2 1
  18: 1 2 1 1
  19: 2 1 1 1
  20: 1 1 1 1 1 1

In the following references, "before" is short for "before removing duplicate rows".
Positions of singleton rows appear to be A001906 = A055588 - 1.
Positions of rows of the form (1,1,...) appear to be A001911-2, before A023758.
Row sums appear to be A200648, before A000120.
Row lengths appear to be A200649, before A384890.
Standard composition numbers of each row appear to be A348366.
Before we had A384877, ranks A385816, firsts A052499.
For runs instead of anti-runs we have A385817, see A245563, A245562, A246029.
Cf. A044813, A048793, A069010, A164707, A242882, A328592, A384175, A384177, A384878, A384879, A384893.

DeleteDuplicates[Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2!=#1+1&],{n,0,100}]]

A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 4, 0, 0, 5, 10, 1, 0, 0, 6, 20, 6, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 8, 56, 56, 8, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 10, 120, 252, 120, 10, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (x/(1-x)^2, x^2/(1-x)^2).
Mirror image of triangle in A119900.
A203322*A130595 as infinite lower triangular matrices. - Philippe Deléham, Jan 05 2011
From Gus Wiseman, Jul 07 2025: (Start)
Also the number of subsets of {1..n} containing n with k maximal runs (sequences of consecutive elements increasing by 1). For example, row n = 5 counts the following subsets:
  {5}          {1,5}      {1,3,5}
  {4,5}        {2,5}
  {3,4,5}      {3,5}
  {2,3,4,5}    {1,2,5}
  {1,2,3,4,5}  {1,4,5}
               {2,3,5}
               {2,4,5}
               {1,2,3,5}
               {1,2,4,5}
               {1,3,4,5}
For anti-runs instead of runs we have A053538.
Without requiring n see A210039, A202023, reverse A098158, A109446.
(End)

			Triangle begins :
1
2, 0
3, 1, 0
4, 4, 0, 0
5, 10, 1, 0, 0
6, 20, 6, 0, 0, 0
7, 35, 21, 1, 0, 0, 0
8, 56, 56, 8, 0, 0, 0, 0

Cf. A007318, A005314 (antidiagonal sums), A119900, A084938, A130595, A203322.
Column k = 1 is A000027.
Row sums are A000079.
Column k = 2 is A000292.
Without zeros we have A034867.
Last nonzero term in each row appears to be A124625.
A034839 counts subsets by number of maximal runs, for anti-runs A384893.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
Cf. A000045, A000071, A001629, A010027, A053538, A208342, A210034, A245563, A268193, A384177, A384890.
Cf. A098158, A109446, A202023, A210039.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Length[Split[#,#2==#1+1&]]==k&]],{n,12},{k,n}] (* Gus Wiseman, Jul 07 2025 *)

G.f.: 1/((1-x)^2-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12, 13 respectively.
T(n,k) = binomial(n+1,2k+1).
T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 15 2012

A209859 Rewrite the binary expansion of n from the most significant end, 1 -> 1, 0+1 (one or more zeros followed by one) -> 0, drop the trailing zeros of the original n.

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 3, 7, 1, 2, 2, 5, 3, 6, 7, 15, 1, 2, 2, 5, 2, 4, 5, 11, 3, 6, 6, 13, 7, 14, 15, 31, 1, 2, 2, 5, 2, 4, 5, 11, 2, 4, 4, 9, 5, 10, 11, 23, 3, 6, 6, 13, 6, 12, 13, 27, 7, 14, 14, 29, 15, 30, 31, 63, 1, 2, 2, 5, 2, 4, 5, 11, 2, 4, 4, 9, 5, 10, 11, 23, 2, 4, 4, 9, 4, 8, 9, 19, 5, 10, 10, 21, 11, 22, 23, 47, 3, 6, 6, 13, 6, 12, 13, 27, 6, 12, 12, 25, 13

Offset: 0

Antti Karttunen, Mar 24 2012

This is the number k such that the k-th composition in standard order is the reversed sequence of lengths of the maximal anti-runs of the binary indices of n. Here, the binary indices of n are row n of A048793, and the k-th composition in standard order is row k of A066099. For example, the binary indices of 100 are {3,6,7}, with maximal anti-runs ((3,6),(7)), with reversed lengths (1,2), which is the 6th composition in standard order, so a(100) = 6. - Gus Wiseman, Jul 27 2025

			102 in binary is 1100110, we rewrite it from the left so that first two 1's stay same ("11"), then "001" is rewritten to "0", the last 1 to "1", and we ignore the last 0, thus getting 1101, which is binary expansion of 13, thus a(102) = 13.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

This is an "inverse" of A071162, i.e. a(A071162(n)) = n for all n. Bisection: A209639. Used to construct permutation A209862.
Removing duplicates appears to give A358654.
Sorted positions of firsts appearances appear to be A247648+1.
A245563 lists run-lengths of binary indices (ranks A246029), reverse A245562.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
Cf. A000120, A023758, A029931, A044813, A048793, A052499, A069010, A348366, A384878, A384879, A384893.
Cf. A384877, A385816, A385887, A385889.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Reverse[Length/@Split[bpe[n],#2!=#1+1&]]],{n,0,100}] (* Gus Wiseman, Jul 25 2025 *)

import re
def a(n): return int(re.sub("0+1", "0", bin(n)[2:].rstrip("0")), 2) if n else 0
print([a(n) for n in range(109)])  # Michael S. Branicky, Jul 25 2025

(define (A209859 n) (let loop ((n n) (s 0) (i (A053644 n))) (cond ((zero? n) s) ((> i n) (if (> (/ i 2) n) (loop n s (/ i 2)) (loop (- n (/ i 2)) (* 2 s) (/ i 4)))) (else (loop (- n i) (+ (* 2 s) 1) (/ i 2))))))

a(n) = a(A000265(n)).

A245564 a(n) = Product_{i in row n of A245562} Fibonacci(i+2).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 4, 4, 6, 3, 6, 5, 8, 2, 4, 4, 6, 4, 8, 6, 10, 3, 6, 6, 9, 5, 10, 8, 13, 2, 4, 4, 6, 4, 8, 6, 10, 4, 8, 8, 12, 6, 12, 10, 16, 3, 6, 6, 9, 6, 12, 9, 15, 5, 10, 10, 15, 8, 16, 13, 21, 2, 4, 4, 6, 4, 8, 6, 10, 4, 8, 8, 12, 6, 12, 10, 16, 4, 8, 8, 12, 8, 16, 12, 20, 6, 12, 12, 18

Offset: 0

N. J. A. Sloane, Aug 10 2014; revised Sep 05 2014

This is the Run Length Transform of S(n) = Fibonacci(n+2).
The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).
Also the number of sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference. The maximal case is A384883. For prime instead of binary indices we have A166469. - Gus Wiseman, Jul 05 2025

			From _Gus Wiseman_, Jul 05 2025: (Start)
The binary indices of 11 are {1,2,4}, with sparse subsets {{},{1},{2},{4},{1,4},{2,4}}, so a(11) = 6.
The maximal runs of binary indices of 11 are ((1,2),(4)), with lengths (2,1), so a(11) = F(2+2)*F(1+2) = 6.
The a(0) = 1 through a(12) = 3 sparse subsets are:
  0    1    2    3    4    5    6    7    8    9    10    11    12
  ------------------------------------------------------------------
  {}   {}   {}   {}   {}   {}   {}   {}   {}   {}    {}    {}    {}
       {1}  {2}  {1}  {3}  {1}  {2}  {1}  {4}  {1}   {2}   {1}   {3}
                 {2}       {3}  {3}  {2}       {4}   {4}   {2}   {4}
                           {1,3}     {3}       {1,4} {2,4} {4}
                                     {1,3}                 {1,4}
                                                           {2,4}
The greatest number whose set of binary indices is a member of column n above is A374356(n).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..8191
Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.

Cf. A245562, A000045, A001045, A071053, A245565, A246028.
A034839 counts subsets by number of maximal runs, strict partitions A116674.
A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000071, A001629, A010049, A053538, A166469, A202023, A202064, A208342, A374356, A384883.

with(combinat); ans:=[];
for n from 0 to 100 do lis:=[]; t1:=convert(n,base,2); L1:=nops(t1); out1:=1; c:=0;
for i from 1 to L1 do
   if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
   elif out1 = 0 and t1[i] = 1 then c:=c+1;
   elif out1 = 1 and t1[i] = 0 then c:=c;
   elif out1 = 0 and t1[i] = 0 then lis:=[c,op(lis)]; out1:=1; c:=0;
   fi;
   if i = L1 and c>0 then lis:=[c,op(lis)]; fi;
                   od:
a:=mul(fibonacci(i+2), i in lis);
ans:=[op(ans),a];
od:
ans;

a[n_] := Sum[Mod[Binomial[3k, k] Binomial[n, k], 2], {k, 0, n}];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 29 2020, after Chai Wah Wu *)
spars[S_]:=Select[Subsets[S],FreeQ[Differences[#],1]&];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[spars[bpe[n]]],{n,0,30}] (* Gus Wiseman, Jul 05 2025 *)

a(n)=my(s=1,k); while(n, n>>=valuation(n,2); k=valuation(n+1,2); s*=fibonacci(k+2); n>>=k); s \\ Charles R Greathouse IV, Oct 21 2016

# use RLT function from A278159
from sympy import fibonacci
def A245564(n): return RLT(n,lambda m: fibonacci(m+2)) # Chai Wah Wu, Feb 04 2022

a(n) = Sum_{k=0..n} ({binomial(3k,k)*binomial(n,k)} mod 2). - Chai Wah Wu, Oct 19 2016

A384891 Number of permutations of {1..n} with all distinct lengths of maximal runs (increasing by 1).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 23, 25, 43, 63, 345, 365, 665, 949, 1513, 8175, 9003, 15929, 23399, 36949, 51043, 293715, 314697, 570353, 826817, 1318201, 1810393, 2766099, 14180139, 15600413, 27707879, 40501321, 63981955, 88599903, 134362569, 181491125, 923029217

Offset: 0

Gus Wiseman, Jun 19 2025

nonn

			The permutation (1,2,6,7,8,9,3,4,5) has maximal runs ((1,2),(6,7,8,9),(3,4,5)), with lengths (2,4,3), so is counted under a(9).
The a(0) = 1 through a(7) = 25 permutations:
  ()  (1)  (12)  (123)  (1234)  (12345)  (123456)  (1234567)
                 (231)  (2341)  (23451)  (123564)  (1234675)
                 (312)  (4123)  (34512)  (123645)  (1234756)
                                (45123)  (124563)  (1245673)
                                (51234)  (126345)  (1273456)
                                         (145623)  (1456723)
                                         (156234)  (1672345)
                                         (231456)  (2314567)
                                         (234156)  (2345167)
                                         (234561)  (2345671)
                                         (312456)  (3124567)
                                         (345126)  (3456127)
                                         (345612)  (3456712)
                                         (412356)  (4567123)
                                         (451236)  (4567231)
                                         (456231)  (4567312)
                                         (456312)  (5123467)
                                         (561234)  (5612347)
                                         (562341)  (5671234)
                                         (564123)  (6712345)
                                         (612345)  (6723451)
                                         (634512)  (6751234)
                                         (645123)  (7123456)
                                                   (7345612)
                                                   (7561234)

Christian Sievers, Table of n, a(n) for n = 0..5000

Counting by number of maximal anti-runs gives A010027, for runs A123513.
For subsets instead of permutations we have A384175, complement A384176.
For partitions we have A384884 (anti-runs A384885), strict A384178 (anti-runs A384880).
For equal instead of distinct lengths we have A384892.
For anti-runs instead of runs we have A384907.
A000041 counts integer partitions, strict A000009.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A356606 counts strict partitions without a neighborless part, complement A356607.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000255, A044813, A072574, A242882, A287170, A325324, A325325, A328592, A329739, A351202, A384177, A384886.

Table[Length[Select[Permutations[Range[n]],UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

lista(n)=my(b(n)=sum(i=0,n-1,(-1)^i*(n-i)!*binomial(n-1,i)), d=floor(sqrt(2*n)), p=prod(i=1,n,1+x*y^i,1+O(y*y^n)*((1-x^(n+1))/(1-x))+O(x*x^d))); Vec(1+sum(i=1,d,i!*b(i)*polcoef(p,i))) \\ Christian Sievers, Jun 22 2025

a(n) = Sum_{k=1..n} ( T(n,k) * A000255(k-1) ) for n>=1, where T(n,k) is the number of compositions of n into k distinct parts (cf. A072574). - Christian Sievers, Jun 22 2025

a(11) and beyond from Christian Sievers, Jun 22 2025

A384892 Number of permutations of {1..n} with all equal lengths of maximal runs (increasing by 1).

Original entry on oeis.org

1, 1, 2, 4, 13, 54, 314, 2120, 16700, 148333, 1468512, 16019532, 190899736, 2467007774, 34361896102, 513137616840, 8178130784179, 138547156531410, 2486151753462260, 47106033220679060, 939765362754015750, 19690321886243848784, 432292066866187743954

Offset: 0

Gus Wiseman, Jun 19 2025

nonn

			The permutation (1,2,5,6,3,4,7,8) has maximal runs ((1,2),(5,6),(3,4),(7,8)), with lengths (2,2,2,2), so is counted under a(8).
The a(0) = 1 through a(4) = 13 permutations:
  ()  (1)  (12)  (123)  (1234)
           (21)  (132)  (1324)
                 (213)  (1432)
                 (321)  (2143)
                        (2413)
                        (2431)
                        (3142)
                        (3214)
                        (3241)
                        (3412)
                        (4132)
                        (4213)
                        (4321)

Christian Sievers, Table of n, a(n) for n = 0..450

For subsets instead of permutations we have A243815, for anti-runs A384889.
For strict partitions and distinct lengths we have A384178, anti-runs A384880.
For integer partitions and distinct lengths we have A384884, anti-runs A384885.
For distinct lengths we have A384891, for anti-runs A384907.
For partitions we have A384904, strict A384886.
A010027 counts permutations by maximal anti-runs, for runs A123513.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000255, A044813, A047993, A242882, A325325, A329739, A351202, A384175, A384176, A384177.

Table[Length[Select[Permutations[Range[n]],SameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

a(n)=if(n,sumdiv(n,d,sum(i=0,d-1,(-1)^i*(d-i)!*binomial(d-1,i))),1) \\ Christian Sievers, Jun 22 2025

a(n) = Sum_{d|n} A000255(d-1). - Christian Sievers, Jun 22 2025

a(11) and beyond from Christian Sievers, Jun 22 2025

cp's OEIS Frontend

A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).

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A384889 Number of subsets of {1..n} with all equal lengths of maximal anti-runs (increasing by more than 1).

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A385816 The number k such that the k-th composition in standard order lists the maximal anti-run lengths of the binary indices of n. Standard composition number of row n of A384877.

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A210034 Triangle of coefficients of polynomials v(n,x) jointly generated with A210033; see the Formula section.

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A385886 Irregular triangle read by rows listing the lengths of maximal anti-runs (sequences of distinct consecutive elements increasing by more than 1) of binary indices, duplicate rows removed.

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A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A209859 Rewrite the binary expansion of n from the most significant end, 1 -> 1, 0+1 (one or more zeros followed by one) -> 0, drop the trailing zeros of the original n.

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