A005183 -id:A005183 - cp's OEIS frontend

A344089 Flattened tetrangle of reversed strict integer partitions, sorted first by length and then colexicographically.

1, 2, 3, 1, 2, 4, 1, 3, 5, 2, 3, 1, 4, 6, 2, 4, 1, 5, 1, 2, 3, 7, 3, 4, 2, 5, 1, 6, 1, 2, 4, 8, 3, 5, 2, 6, 1, 7, 1, 3, 4, 1, 2, 5, 9, 4, 5, 3, 6, 2, 7, 1, 8, 2, 3, 4, 1, 3, 5, 1, 2, 6, 10, 4, 6, 3, 7, 2, 8, 1, 9, 2, 3, 5, 1, 4, 5, 1, 3, 6, 1, 2, 7, 1, 2, 3, 4

Offset: 0

Gus Wiseman, May 12 2021

First differs from the revlex (instead of colex) version for partitions of 12.
The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (3)(12)
  4: (4)(13)
  5: (5)(23)(14)
  6: (6)(24)(15)(123)
  7: (7)(34)(25)(16)(124)
  8: (8)(35)(26)(17)(134)(125)
  9: (9)(45)(36)(27)(18)(234)(135)(126)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724 plus one.
Taking lex instead of colex gives A026793 (non-reversed: A118457).
Triangle sums are A066189.
Reversing all partitions gives A344090.
The non-strict version is A344091.
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
Cf. A005117, A014466, A209862, A325683, A325859.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Table[Reverse/@Sort[Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,30}]

A108730 Triangle read by rows: row n gives the list of the number of zeros following each 1 in the binary representation of n.

Original entry on oeis.org

0, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 2, 0, 1, 1, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 4, 3, 0, 2, 1, 2, 0, 0, 1, 2, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 3, 0, 2, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 5, 4, 0, 3, 1, 3, 0, 0, 2, 2, 2, 1, 0, 2, 0, 1, 2, 0, 0, 0, 1, 3, 1, 2, 0

Offset: 1

Franklin T. Adams-Watters, Jun 22 2005

This is probably the simplest method for putting the nonnegative integers into one-to-one correspondence with the finite sequences of nonnegative integers and is the standard ordering for such sequences in this database.
This sequence contains every finite sequence of nonnegative integers.
This can be regarded as a table in two ways: with each weak composition as a row, or with the weak compositions of each integer as a row. The first way has A000120 as row lengths and A080791 as row sums; the second has A001792 as row lengths and A001787 as row sums. - Franklin T. Adams-Watters, Nov 06 2006
Concatenate the base-two positive integers (A030190 less the initial zero). Left to right and disallowing leading zeros, reorganize the digits into the smallest possible numbers. These will be the base-two powers-of-two of A108730. - Hans Havermann, Nov 14 2009
T(2^(n-1),0) = n-1 and T(m,k) < n-1 for all m < 2^n, k <= A000120(m). When the triangle is seen as a flattened list, each n occurs first at position n*2^(n-1)+1, cf. A005183. - Reinhard Zumkeller, Feb 26 2012
Equals A066099-1, elementwise. - Andrey Zabolotskiy, May 18 2018

			Triangle begins:
  0
  1
  0,0
  2
  1,0
  0,1
  0,0,0
  3
For example, 25 = 11001_2; following the 1's are 0, 2 and 0 zeros, so row 25 is 0, 2, 0.

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..5120 (through 10 bit numbers)

Cf. A066099 (main entry for compositions), A007088, A000120, A080791, A001792, A001787, A124735.

import Data.List (unfoldr, group)
a108730 n k = a108730_tabf !! (n-1) !! (k-1)
a108730_row = f . group . reverse . unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2) where
   f [] = []
   f [os] = replicate (length os) 0
   f (os:zs:dss) = replicate (length os - 1) 0 ++ [length zs] ++ f dss
a108730_tabf = map a108730_row [1..]
a108730_list = concat a108730_tabf
-- Reinhard Zumkeller, Feb 26 2012

row[n_] := (id = IntegerDigits[n, 2]; sp = Split[id]; f[run_List] := If[First[run] == 0, run, Sequence @@ Table[{}, {Length[run] - 1}]]; len = Length /@ f /@ sp; If[Last[id] == 0, len, Append[len, 0]]); Flatten[ Table[row[n], {n, 1, 41}]] (* Jean-François Alcover, Jul 13 2012 *)

row(n)=my(v=vector(hammingweight(n)),t=n); for(i=0,#v-1,v[#v-i] = valuation(t,2); t>>=v[#v-i]+1); v \\ Charles R Greathouse IV, Sep 14 2015

Edited by Franklin T. Adams-Watters, Nov 06 2006

A344085 Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 77, 66, 110, 154, 165, 231, 385, 330, 462, 770, 1155, 2310, 13, 26, 39, 65, 91, 143, 78, 130, 182, 195, 273, 286, 429, 455, 715, 1001, 390, 546, 858, 910, 1365, 1430, 2002, 2145, 3003, 5005, 2730, 4290, 6006, 10010, 15015, 30030

Offset: 1

Gus Wiseman, May 11 2021

Differs from A339195 in having 77 before 66.

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70 105 210

Row lengths are A000079.
Grouping by greatest prime factor only gives A339195.
Row sums are 1 and A339360.
Cf. A001221, A005117, A005183, A014466, A019565, A187769, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344086, A344087, A344088, A344089.
Partition/composition applications: A001793, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124.

nn=4;
GatherBy[SortBy[Select[Range[Times@@Prime/@Range[nn]],SquareFreeQ[#]&&PrimePi[FactorInteger[#][[-1,1]]]<=nn&],PrimeOmega],FactorInteger[#][[-1,1]]&]

A344087 Flattened tetrangle of strict integer partitions sorted first by sum, then colexicographically.

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 1, 4, 4, 1, 3, 2, 5, 3, 2, 1, 5, 1, 4, 2, 6, 4, 2, 1, 6, 1, 5, 2, 4, 3, 7, 5, 2, 1, 4, 3, 1, 7, 1, 6, 2, 5, 3, 8, 6, 2, 1, 5, 3, 1, 8, 1, 4, 3, 2, 7, 2, 6, 3, 5, 4, 9, 4, 3, 2, 1, 7, 2, 1, 6, 3, 1, 5, 4, 1, 9, 1, 5, 3, 2, 8, 2, 7, 3, 6, 4, 10

Offset: 0

Gus Wiseman, May 11 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (21)(3)
  4: (31)(4)
  5: (41)(32)(5)
  6: (321)(51)(42)(6)
  7: (421)(61)(52)(43)(7)
  8: (521)(431)(71)(62)(53)(8)
  9: (621)(531)(81)(432)(72)(63)(54)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
Taking revlex instead of colex gives A118457.
The not necessarily strict version is A211992.
Taking lex instead of colex gives A344086.
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080576, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344088, A344089, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 4, 2, 3, 1, 4, 5, 1, 2, 3, 2, 4, 1, 5, 6, 1, 2, 4, 3, 4, 2, 5, 1, 6, 7, 1, 3, 4, 1, 2, 5, 3, 5, 2, 6, 1, 7, 8, 2, 3, 4, 1, 3, 5, 4, 5, 1, 2, 6, 3, 6, 2, 7, 1, 8, 9, 1, 2, 3, 4, 2, 3, 5, 1, 4, 5, 1, 3, 6, 4, 6, 1, 2, 7, 3, 7, 2, 8, 1, 9, 10

Offset: 0

Gus Wiseman, May 12 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (12)(3)
  4: (13)(4)
  5: (23)(14)(5)
  6: (123)(24)(15)(6)
  7: (124)(34)(25)(16)(7)
  8: (134)(125)(35)(26)(17)(8)
  9: (234)(135)(45)(126)(36)(27)(18)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
The non-strict version is A080576.
Taking lex instead of colex gives A246688 (non-reversed: A344086).
The non-reversed version is A344087.
Taking revlex instead of colex gives A344089 (non-reversed: A118457).
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Reverse/@Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

A003013 E.g.f. 1 + xexp(x) + x^2exp(2*x).

Original entry on oeis.org

1, 1, 4, 15, 52, 165, 486, 1351, 3592, 9225, 23050, 56331, 135180, 319501, 745486, 1720335, 3932176, 8912913, 20054034, 44826643, 99614740, 220200981, 484442134, 1061158935, 2315255832, 5033164825

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-25,38,-28,8).

Cf. A048482, A001787, A005183.

Cf. A002999, A001815.

With[{nn=30},CoefficientList[Series[1+x Exp[x]+x^2 Exp[2x],{x,0,nn}],x] Range[0,nn]!] (* or *) Join[{1},LinearRecurrence[{8,-25,38,-28,8},{1,4,15,52,165},30]] (* Harvey P. Dale, Nov 01 2011 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 8,-28,38,-25,8]^n*[1;1;4;15;52])[1,1] \\ Charles R Greathouse IV, Jun 23 2020

From Ralf Stephan, Sep 02 2003: (Start)
a(0) = 1, a(n) = (n^2 - n)*2^n/4 + n.
a(n) = A002999(n) - n = A001815(n) + n. (End)
O.g.f.: 1+x*(-1+4*x-8*x^2+6*x^3) / ( (x-1)^2*(2*x-1)^3 ). - R. J. Mathar, Mar 22 2011
a(n) = 8*a(n-1) - 25*a(n-2) + 38*a(n-3) - 28*a(n-4) + 8*a(n-5); a(0)=1, a(1)=1, a(2)=4, a(3)=15, a(4)=52, a(5)=165. - Harvey P. Dale, Nov 01 2011

A048482 a(n) = T(n,n), array T given by A048472.

Original entry on oeis.org

1, 3, 13, 49, 161, 481, 1345, 3585, 9217, 23041, 56321, 135169, 319489, 745473, 1720321, 3932161, 8912897, 20054017, 44826625, 99614721, 220200961, 484442113, 1061158913, 2315255809, 5033164801, 10905190401, 23555211265, 50734301185, 108984795137

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Cf. A002999, A003013, A001787, A005183, A001788.

[(n^2+n)*2^(n-1) + 1: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

LinearRecurrence[{7,-18,20,-8},{1,3,13,49},30] (* Harvey P. Dale, Feb 02 2015 *)

Vec(-(8*x^3-10*x^2+4*x-1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Nov 26 2014

a(n) = 2 * A001788(n) + 1.
a(n) = (n^2+n)*2^(n-1) + 1. - Ralf Stephan, Sep 02 2003
G.f.: -(8*x^3-10*x^2+4*x-1) / ((x-1)*(2*x-1)^3). - Colin Barker, Nov 26 2014
a(0)=1, a(1)=3, a(2)=13, a(3)=49, a(n)=7*a(n-1)-18*a(n-2)+ 20*a(n-3)- 8*a(n-4). - Harvey P. Dale, Feb 02 2015

A128133 Binomial transform of A128132.

Original entry on oeis.org

1, 0, 2, -1, 3, 3, -2, 3, 8, 4, -3, 2, 14, 15, 5, -4, 0, 20, 35, 24, 6, -5, -3, 25, 65, 69, 35, 7, -6, -7, 28, 105, 154, 119, 48, 8, -7, -12, 28, 154, 294, 308, 188, 63, 9, -8, -18, 24, 210, 504, 672, 552, 279, 80, 10

Offset: 1

Gary W. Adamson, Feb 15 2007

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

A128132 * A007318 = A128134.

			First few rows of the triangle:
   1;
   0,  2;
  -1,  3,  3;
  -2,  3,  8,  4;
  -3,  2, 14, 15,  5;
  -4,  0, 20, 35, 24,  6;
  -5, -3, 25, 65, 69, 35,  7;
  ...

Cf. A128132, A007318, A128134, A005183.

A007318 * A128132 as infinite lower triangular matrices.

A046688 Antidiagonals of square array in which k-th row (k>0) is an arithmetic progression of difference 2^(k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 5, 1, 1, 5, 7, 9, 9, 1, 1, 6, 9, 13, 17, 17, 1, 1, 7, 11, 17, 25, 33, 33, 1, 1, 8, 13, 21, 33, 49, 65, 65, 1, 1, 9, 15, 25, 41, 65, 97, 129, 129, 1, 1, 10, 17, 29, 49, 81, 129, 193, 257, 257, 1, 1, 11, 19, 33, 57, 97, 161, 257, 385, 513, 513, 1

Offset: 0

N. J. A. Sloane

			From _Gus Wiseman_, May 08 2021: (Start):
Array A(m,n) = 1 + n*2^(m-1) begins:
       n=0: n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
  m=0:   1    1    1    1    1    1    1    1    1    1
  m=1:   1    2    3    5    9   17   33   65  129  257
  m=2:   1    3    5    9   17   33   65  129  257  513
  m=3:   1    4    7   13   25   49   97  193  385  769
  m=4:   1    5    9   17   33   65  129  257  513 1025
  m=5:   1    6   11   21   41   81  161  321  641 1281
  m=6:   1    7   13   25   49   97  193  385  769 1537
  m=7:   1    8   15   29   57  113  225  449  897 1793
  m=8:   1    9   17   33   65  129  257  513 1025 2049
  m=9:   1   10   19   37   73  145  289  577 1153 2305
Triangle T(n,k) = 1 + (n-k)*2^(k-1) begins:
   1
   1   1
   1   2   1
   1   3   3   1
   1   4   5   5   1
   1   5   7   9   9   1
   1   6   9  13  17  17   1
   1   7  11  17  25  33  33   1
   1   8  13  21  33  49  65  65   1
   1   9  15  25  41  65  97 129 129   1
   1  10  17  29  49  81 129 193 257 257   1
   1  11  19  33  57  97 161 257 385 513 513   1
(End)

G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers, Vol. VII, p. 430.

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

Row sums are A000079.
Diagonal n = m + 1 of the array is A002064.
Diagonal n = m of the array is A005183.
Column m = 1 of the array is A094373.
Diagonal n = m - 1 of the array is A131056.
A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
A009998(k,n) = n^k.
A009999(n,k) = n^k.
A057156 = (2^n)^(2^n).
A062319 counts divisors of n^n.
Cf. A000169, A000272, A000312, A036289, A343656, A343658.

Table[If[k==0,1,n*2^(k-1)+1],{n,0,9},{k,0,9}] (* ARRAY, Gus Wiseman, May 08 2021 *)
Table[If[k==0,1,1+(n-k)*2^(k-1)],{n,0,10},{k,0,n}] (* TRIANGLE, Gus Wiseman, May 08 2021 *)

A(m,n)={if(m>0, 1+n*2^(m-1), 1)}
{ for(m=0, 10, for(n=0, 10, print1(A(m,n), ", ")); print) } \\ Andrew Howroyd, Mar 07 2020

A(m,n) = 1 + n*2^(m-1) for m > 1. - Andrew Howroyd, Mar 07 2020

As a triangle, T(n,k) = A(k,n-k) = 1 + (n-k)*2^(k-1). - Gus Wiseman, May 08 2021

More terms from Larry Reeves (larryr(AT)acm.org), Apr 06 2000

A049069 Array T read by antidiagonals: T(k,n) = kn2^(n-1) + 1, n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 9, 13, 1, 5, 13, 25, 33, 1, 6, 17, 37, 65, 81, 1, 7, 21, 49, 97, 161, 193, 1, 8, 25, 61, 129, 241, 385, 449, 1, 9, 29, 73, 161, 321, 577, 897, 1025, 1, 10, 33, 85, 193, 401, 769, 1345, 2049, 2305, 1, 11, 37, 97, 225, 481, 961, 1793, 3073, 4609, 5121

Offset: 0

N. J. A. Sloane, Clark Kimberling, Michael Somos

			Antidiagonals: {1}; {1,2}; {1,3,5}; ...

Transpose of the array in A048472.
Row 1 = (1, 2, 5, 13, 33, ...) = A005183.
Row 2 = (1, 3, 9, 25, 65, ...) = A002064.
Cf. A049513.
Essentially the same as A049513.

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

cp's OEIS Frontend

A344089 Flattened tetrangle of reversed strict integer partitions, sorted first by length and then colexicographically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A108730 Triangle read by rows: row n gives the list of the number of zeros following each 1 in the binary representation of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A344085 Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A344087 Flattened tetrangle of strict integer partitions sorted first by sum, then colexicographically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A003013 E.g.f. 1 + x*exp(x) + x^2*exp(2*x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A048482 a(n) = T(n,n), array T given by A048472.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A128133 Binomial transform of A128132.

Views

Author

Keywords

Comments

Examples

A003013 E.g.f. 1 + xexp(x) + x^2exp(2*x).

A049069 Array T read by antidiagonals: T(k,n) = kn2^(n-1) + 1, n >= 0, k >= 1.