A005183 -id:A005183 - cp's OEIS frontend

A049513 Array T by antidiagonals: T(k,n) = kn2^(n-1) + 1, n >= 0, k >= 0.

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 9, 13, 1, 1, 5, 13, 25, 33, 1, 1, 6, 17, 37, 65, 81, 1, 1, 7, 21, 49, 97, 161, 193, 1, 1, 8, 25, 61, 129, 241, 385, 449, 1, 1, 9, 29, 73, 161, 321, 577, 897, 1025, 1, 1, 10, 33, 85, 193, 401, 769, 1345, 2049, 2305, 1, 1, 11, 37, 97, 225, 481

Offset: 0

Michael Somos, Sep 25 1999

			Antidiagonals: 1; 1,1; 1,2,1; 1,3,5,1; 1,4,9,13,1; ...

Cf. A000337, A002064, A005183, A016813, A017533, A048474, A049069, A048472.

Essentially the same as A049069.

PARI
```
{T(k, n) = k * n * 2^(n-1) + 1}
```

A005183(n) = T(1, n), A002064(n) = T(2, n), A048474(n) = T(3, n), A000337(n) = T(4, n), A016813(n) = T(n, 2), A017533(n) = T(n, 3).

A196273 Primes of the form n*2^(n-1)+1.

Original entry on oeis.org

2, 5, 13, 193, 449, 114689, 1811939329, 2434970217729660813313, 4943727411754159833089, 6731298963614255244763987969, 2388456554926020709124028311441244161

Offset: 1

Juri-Stepan Gerasimov, Sep 29 2011

nonn

Generated by n = 1, 2, 3, 6, 7, 14, 27, 66, 67, 87, 115, .. = A029544(n)+1.

See also A236752 for primes of the form k*2^(k-1) - 1, and A230769 for the corresponding indices k (minus 1). - M. F. Hasler, Mar 01 2014

Cf. A005183.

Select[Table[n*2^(n-1)+1,{n,150}],PrimeQ] (* Harvey P. Dale, Jul 17 2018 *)

lista(nn) = {for (n=1, nn, if (isprime(p = n*2^(n-1)+1), print1(p, ", ")););} \\ Michel Marcus, Nov 09 2013

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 2, 3, 1, 2, 3, 4, 1, 4, 2, 4, 3, 4, 1, 2, 4, 1, 3, 4, 2, 3, 4, 1, 2, 3, 4, 5, 1, 5, 2, 5, 3, 5, 4, 5, 1, 2, 5, 1, 3, 5, 1, 4, 5, 2, 3, 5, 2, 4, 5, 3, 4, 5, 1, 2, 3, 5, 1, 2, 4, 5, 1, 3, 4, 5, 2, 3, 4, 5, 1, 2, 3, 4, 5

Offset: 1

Gus Wiseman, May 11 2021

			The sets are the columns below:
  1 2 1 3 1 2 1 4 1 2 3 1 1 2 1 5 1 2 3 4 1 1 1 2 2 3 1
      2   3 3 2   4 4 4 2 3 3 2   5 5 5 5 2 3 4 3 4 4 2
              3         4 4 4 3           5 5 5 5 5 5 3
                              4                       5
As a tetrangle, the first four triangles are:
  {1}
  {2},{1,2}
  {3},{1,3},{2,3},{1,2,3}
  {4},{1,4},{2,4},{3,4},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}

Wikiversity, Lexicographic and colexicographic order

Triangle lengths are A000079.
Triangle sums are A001793.
Positions of first appearances are A005183.
Set maxima are A070939.
Set lengths are A124736.
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A296774, A299755, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A036043, A049085, A115623, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Mathematica
```
SortBy[Rest[Subsets[Range[5]]],Last]
```

A386254 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times.

Original entry on oeis.org

1, 1, 2, 6, 18, 60, 240, 1085, 5012, 23730, 121440, 685707, 4144668, 25614589, 159141892, 1012740885, 6805631232, 48872707006, 369227821608, 2853779791619, 22131042288980, 172055270717463, 1362017827326860, 11208504802237327, 96939147303239304, 875473007351905045

Offset: 0

John Tyler Rascoe, Jul 16 2025

			a(3) = 6 counts: (1,1,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2), (3,3,3).

Alois P. Heinz, Table of n, a(n) for n = 0..674

Cf. A002999, A005183, A007837, A032299, A347006, A386255.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-j, min(n-j, i-1))/j!, j=i..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 17 2025

terms=26; CoefficientList[Series[Product[1+Sum[x^j/j!, {j,k,terms}],{k,terms}],{x,0,terms-1}],x]Range[0,terms-1]! (* Stefano Spezia, Jul 17 2025 *)

D_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + sum(i=k,N, x^i/i!))))}

E.g.f.: Product_{k>=1} (1 + Sum_{j>=k} x^j / j!).

A386255 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times and exactly one of each kind of letter is marked.

Original entry on oeis.org

1, 1, 4, 15, 64, 325, 1776, 11179, 72640, 489969, 3435580, 26495491, 221599104, 1893705697, 16145571820, 138299146665, 1241234863936, 12033569772769, 124055067568788, 1303750295285563, 13577876900409280, 139418829477000801, 1441311794301705964, 15537427948684769425

Offset: 0

John Tyler Rascoe, Jul 16 2025

			a(3) = 15 counts: (1#,1,1), (1,1#,1), (1,1,1#), (1#,2#,2), (1#,2,2#), (2#,1#,2), (2,1#,2#), (2#,2,1#), (2,2#,1#), (2#,2,2), (2,2#,2), (2,2,2#), (3#,3,3), (3,3#,3), (3,3,3#) where # denotes a mark.

Alois P. Heinz, Table of n, a(n) for n = 0..650

Cf. A002999, A005183, A007837, A032299, A347006, A386254.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-j, min(n-j, i-1))/(j-1)!, j=i..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 17 2025

terms=24; CoefficientList[Series[Product[1+Sum[x^j/(j-1)!, {j,k,terms}],{k,terms}],{x,0,terms-1}],x]Range[0,terms-1]! (* Stefano Spezia, Jul 17 2025 *)

E_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + sum(i=k,N, x^i/((i-1)!)))))}

E.g.f.: Product_{k>=1} (1 + Sum_{j>=k} x^j / (j-1)!).

A134399 Matrix product of Binomial triangle A007318 and triangle with (1, 1, 2, 3, 4, 5, ...) in the main diagonal and the rest zeros.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 4, 12, 12, 4, 1, 5, 20, 30, 20, 5, 1, 6, 30, 60, 60, 30, 6, 1, 7, 42, 105, 140, 105, 42, 7, 1, 8, 56, 168, 280, 280, 168, 56, 8, 1, 9, 72, 252, 504, 630, 504, 252, 72, 9

Offset: 0

Gary W. Adamson, Oct 23 2007

Row sums = A005183: (1, 2, 5, 13, 33, 81, 193, ...).

			First few rows of the triangle:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  6,   3;
  1, 4, 12,  12,   4;
  1, 5, 20,  30,  20,   5;
  1, 6, 30,  60,  60,  30,  6;
  1, 7, 42, 105, 140, 105, 42, 7;
  ...

Cf. A005183, A007318.

Table[Max[k, 1]*Binomial[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, May 03 2024 *)

T(n,0) = 1; T(n,k) = k*binomial(n,k) for 1 <= k <= n. [corrected by Paolo Xausa, May 03 2024]

Corrected and edited by Olivier Gérard, Aug 01 2013

A145033 T(n,k) is the number of amenable quasi-idempotent order-decreasing partial one-one transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 14, 10, 1, 1, 9, 25, 30, 15, 1, 1, 11, 39, 65, 55, 21, 1, 1, 13, 56, 119, 140, 91, 28, 1, 1, 15, 76, 196, 294, 266, 140, 36, 1, 1, 17, 99, 300, 546, 630, 462, 204, 45, 1, 1, 19, 125, 435, 930, 1302, 1218, 750, 285, 55, 1

Offset: 0

Abdullahi Umar, Sep 30 2008

T(n,k) is also the rank of the semigroup of order-decreasing partial one-one transformations (of an n-chain) of height <= k.
The matrix inverse starts:
1;
-1,1;
2,-3,1;
-8,13,-6,1;
58,-95,46,-10,1;
-672,1101,-535,120,-15,1;
11374,-18635,9056,-2035,260,-21,1; - R. J. Mathar, Mar 29 2013

			T(3,2) = 6 because there are exactly 6 amenable quasi-idempotent order-decreasing partial one-one transformations (on a 3- chain) of height 2, namely: (1,2)->(1,2), (1,3)->(1,2), (1,3)->(1,3), (2,3)->(1,3), (2,3)->(2,1), (2,3)->(2,3).
   1;
   1,   1;
   1,   3,   1;
   1,   5,   6,   1;
   1,   7,  14,  10,   1;
   1,   9,  25,  30,  15,   1;
   1,  11,  39,  65,  55,  21,   1;
   1,  13,  56, 119, 140,  91,  28,   1;
   1,  15,  76, 196, 294, 266, 140,  36,   1;
   1,  17,  99, 300, 546, 630, 462, 204,  45,   1;
   1,  19, 125, 435, 930,1302,1218, 750, 285,  55,   1;

A. Umar, On the ranks of certain finite semigroups of order-decreasing transformations Portugaliae Math. 53, (1996), 23-34.

Row sums of T(n, k) is A005183.

T(n,k) = binomial(n,k)*((n-k)*(k+1)+1)/(n-k+1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Apr 23 2018

T(n,k) = C(n,k)*((n-k)*(k+1)+1)/(n-k+1), (n>=k>=0).

More terms from Jinyuan Wang, Feb 26 2020

A343291 a(n) = (n-2)*2^(n-1) + n + 2.

Original entry on oeis.org

1, 2, 4, 9, 22, 55, 136, 329, 778, 1803, 4108, 9229, 20494, 45071, 98320, 213009, 458770, 983059, 2097172, 4456469, 9437206, 19922967, 41943064, 88080409, 184549402, 385875995, 805306396, 1677721629, 3489660958, 7247757343, 15032385568, 31138512929, 64424509474

Offset: 0

Alois P. Heinz, Apr 10 2021

a(n) is the cardinality of set s(n), where s(0) = {0} and s(n+1) = s(n) union {(i+j+1)/2 : i,j in s(n)}. s(4) = {0, 1/2, 3/4, 7/8, 15/16, 1, 17/16, 9/8, 19/16, 5/4, 21/16, 11/8, 23/16, 3/2, 25/16, 13/8, 27/16, 7/4, 29/16, 15/8, 31/16, 2} has cardinality a(4) = 22.

Total number of 0-bits in all numbers <= 2^n and for n >= 1 the total number of bits in all numbers <= 2^(n-1); similar to A048495. - Ruud H.G. van Tol, Apr 28 2025

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Partial differences give A005183 (shifted).

Cf. A048495, A343264.

a:= n-> (n-2)*2^(n-1)+n+2:
seq(a(n), n=0..35);

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

A179257 Number of permutations of length n which avoid the patterns 321 and 1324.

Original entry on oeis.org

1, 1, 2, 5, 13, 32, 72, 148, 281, 499, 838, 1343, 2069, 3082, 4460, 6294, 8689, 11765, 15658, 20521, 26525, 33860, 42736, 53384, 66057, 81031, 98606, 119107, 142885, 170318, 201812, 237802, 278753, 325161, 377554, 436493, 502573, 576424, 658712, 750140, 851449

Offset: 0

Vincent Vatter, Jul 05 2010

			There are 13 permutations of length 4 which avoid these two patterns, so a(4)=13.

M. D. Atkinson, Restricted permutations, Discrete Math., 195 (1999), 27-38.
Christian Bean, Bjarki Gudmundsson, Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.
J. West, Generating trees and forbidden subsequences, Discrete Math., 157 (1996), 363-374.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A116699, A116701, A116702, A088921, A005183, A116703, A001519.

LinearRecurrence[{6,-15,20,-15,6,-1},{1,1,2,5,13,32},50] (* Harvey P. Dale, May 19 2024 *)

a(n) = 1+binomial(n,2)+binomial(n+2,5).
G.f.: 1-x*(x^5-4*x^4+7*x^3-8*x^2+4*x-1)/(x-1)^6. - Colin Barker, Aug 02 2012
a(n) = 1+A027658(n-2). - R. J. Mathar, Aug 19 2022

a(0)=1 prepended by Alois P. Heinz, Jul 05 2018

cp's OEIS Frontend

A049513 Array T by antidiagonals: T(k,n) = k*n*2^(n-1) + 1, n >= 0, k >= 0.

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PARI

Formula

A196273 Primes of the form n*2^(n-1)+1.

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PARI

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

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Author

Keywords

Examples

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Programs

Mathematica

A386254 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times.

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Maple

Mathematica

PARI

Formula

A386255 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times and exactly one of each kind of letter is marked.

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Maple

Mathematica

PARI

Formula

A134399 Matrix product of Binomial triangle A007318 and triangle with (1, 1, 2, 3, 4, 5, ...) in the main diagonal and the rest zeros.

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Mathematica

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Extensions

A145033 T(n,k) is the number of amenable quasi-idempotent order-decreasing partial one-one transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|).

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Author

Keywords

Comments

Examples

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Programs

PARI

Formula

Extensions

A343291 a(n) = (n-2)*2^(n-1) + n + 2.

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A049513 Array T by antidiagonals: T(k,n) = kn2^(n-1) + 1, n >= 0, k >= 0.