A007051 -id:A007051 - cp's OEIS frontend

A201730 Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

1, 2, 0, 5, 1, 0, 14, 6, 0, 0, 41, 26, 1, 0, 0, 122, 100, 10, 0, 0, 0, 365, 363, 63, 1, 0, 0, 0, 1094, 1274, 322, 14, 0, 0, 0, 0, 3281, 4372, 1462, 116, 1, 0, 0, 0, 0, 9842, 14760, 6156, 744, 18, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 04 2011

Riordan array ((1-2x)/(1-4x+3x^2),x^2/(1-4x+3x^2)).

A007318*A201701 as lower triangular matrices.

			Triangle begins:
1
2, 0
5, 1, 0
14, 6, 0, 0
41, 26, 1, 0, 0
122, 100, 10, 0, 0, 0
365, 363, 63, 1, 0, 0, 0

Cf. A007051 (1st column), A261064 (2nd column).

A201730 := proc(n,k)
    (1-2*x)/(1-4*x+(3-y)*x^2) ;
    coeftayl(%,y=0,k) ;
    coeftayl(%,x=0,n) ;
end proc:
seq(seq(A201730(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 06 2011

m = 13;
(* DELTA is defined in A084938 *)
DELTA[Join[{2, 1/2, 3/2}, Table[0, {m}]], Join[{0, 1/2, -1/2}, Table[0, {m}]], m] // Flatten (* Jean-François Alcover, Feb 19 2020 *)

G.f.: (1-2x)/(1-4x+(3-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A139011(n), A000079(n), A007051(n), A006012(n), A001075(n), A081294(n), A001077(n), A084059(n), A108851(n), A084128(n), A081340(n), A084132(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
Sum_{k, k>+0} T(n+k,k) = A081704(n) .
T(n,k) = 3*T(n-1,k)+ Sum_{j>0} T(n-1-j,k-1).
T(n,k) = 4*T(n-1,k)+ T(n-2,k-1) - 3*T(n-2,k) with T(0,0)=1, T(1,0)= 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if n

A209100 T(n,k)=Number of nXk 0..2 arrays with new values 0..2 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

1, 2, 2, 5, 11, 5, 14, 76, 82, 14, 41, 520, 1326, 612, 41, 122, 3552, 20928, 23248, 4568, 122, 365, 24256, 329064, 849548, 407832, 34096, 365, 1094, 165632, 5171088, 30836932, 34538488, 7154944, 254496, 1094, 3281, 1131008, 81254376, 1118366188

Offset: 1

R. H. Hardin Mar 05 2012

Table starts
....1.......2..........5............14...............41.................122
....2......11.........76...........520.............3552...............24256
....5......82.......1326.........20928...........329064.............5171088
...14.....612......23248........849548.........30836932..........1118366188
...41....4568.....407832......34538488.......2896732704........242632290432
..122...34096....7154944....1404480904.....272236743760......52675800891748
..365..254496..125526240...57113932788...25586970618660...11437585318270860
.1094.1899584.2202232576.2322577420320.2404896311723064.2483525516352937680

			Some solutions for n=4 k=3
..0..0..0....0..0..0....0..0..1....0..0..0....0..0..1....0..0..1....0..0..0
..1..1..1....1..1..2....1..2..1....1..1..1....2..0..2....2..0..2....1..1..2
..2..0..0....2..0..0....1..0..1....2..2..0....0..2..0....1..0..2....2..0..1
..1..2..2....1..1..1....2..1..2....1..2..1....2..0..2....1..2..1....0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..254

Column 1 and row 1 are A007051(n-1)

A222986 T(n,k)=Number of nXk 0..2 arrays with no more than floor(nXk/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..2 order.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 4, 7, 14, 14, 5, 25, 43, 70, 41, 14, 86, 314, 584, 326, 122, 17, 332, 946, 4572, 2903, 1630, 365, 70, 1172, 10417, 42607, 68385, 45718, 8058, 1094, 89, 4566, 36987, 414907, 601407, 1037326, 227569, 40598, 3281, 326, 16562, 431995, 4405097

Offset: 1

R. H. Hardin Mar 11 2013

Table starts
.....1........1...........1..............4.................5
.....2........4...........7.............25................86
.....5.......14..........43............314...............946
....14.......70.........584...........4572.............42607
....41......326........2903..........68385............601407
...122.....1630.......45718........1037326..........29158784
...365.....8058......227569.......15904038.........433151066
..1094....40598.....3706702......245900395.......21273422658
..3281...204914....18721501.....3827698634......321537852228
..9842..1041934...309335666....59908716776....15962124223493
.29525..5315674..1578160193...941887270541...243618393075038
.88574.27228710.26302908158.14864346253334.12191944518185381

			Some solutions for n=3 k=4
..0..0..0..1....0..0..1..0....0..1..1..1....0..0..1..1....0..0..0..0
..0..0..0..0....2..1..1..1....1..1..1..2....0..2..2..2....0..0..0..1
..0..0..1..1....1..1..1..1....1..1..1..2....2..2..2..2....0..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..159

Column 1 is A007051(n-1)

Row 1 is A222364

A223126 T(n,k)=Number of nXk 0..2 arrays with no more than floor(nXk/2) elements equal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..2 order.

Original entry on oeis.org

1, 1, 2, 2, 10, 5, 10, 64, 72, 14, 24, 489, 1154, 664, 41, 72, 3599, 28907, 33604, 5376, 122, 168, 27803, 444555, 1780641, 622568, 47968, 365, 664, 212771, 11814919, 95995609, 112106293, 18539512, 406400, 1094, 1632, 1656545, 186184618

Offset: 1

R. H. Hardin Mar 15 2013

Table starts
.....1...........1................2....................10
.....2..........10...............64...................489
.....5..........72.............1154.................28907
....14.........664............33604...............1780641
....41........5376...........622568.............112106293
...122.......47968.........18539512............7157970340
...365......406400........354680696..........461597847603
..1094.....3585408......10596201904........29989116614529
..3281....30987264.....206498940320......1959625907209239
..9842...272565760....6177157242496....128645318481467647
.29525..2378450944..121870726469408...8477443608456769792
.88574.20921604096.3647201710914880.560422268684443059564

			Some solutions for n=3 k=4
..0..1..0..2....0..1..0..0....0..1..0..2....0..1..2..0....0..1..0..2
..2..1..1..2....0..1..2..2....0..1..2..2....2..0..1..1....2..0..2..0
..0..0..1..0....0..1..0..0....2..2..1..2....0..2..2..0....1..2..1..0

R. H. Hardin, Table of n, a(n) for n = 1..125

Column 1 is A007051(n-1)

Row 1 is A222848

A241370 T(n,k)=Number of nXk 0..2 arrays with no element equal to fewer vertical neighbors than horizontal neighbors, with new values 0..2 introduced in row major order.

Original entry on oeis.org

1, 1, 2, 2, 7, 5, 4, 28, 47, 14, 8, 121, 460, 326, 41, 16, 523, 4617, 7376, 2284, 122, 32, 2261, 46245, 169982, 118488, 16026, 365, 64, 9775, 463567, 3910194, 6280325, 1904096, 112458, 1094, 128, 42261, 4646421, 90008909, 332185927, 232173463

Offset: 1

R. H. Hardin, Apr 20 2014

Table starts
....1.......1..........2..............4.................8...................16
....2.......7.........28............121...............523.................2261
....5......47........460...........4617.............46245...............463567
...14.....326.......7376.........169982...........3910194.............90008909
...41....2284.....118488........6280325.........332185927..........17583615124
..122...16026....1904096......232173463.......28238828935........3437694358689
..365..112458...30598800.....8582759752.....2400505507498......672068364873884
.1094..789166..491723328...317280724429...204061855414167...131390467341043995
.3281.5537942.7902006144.11729003927933.17346886991310331.25687100469219790719

			Some solutions for n=4 k=4
..0..1..0..1....0..1..0..2....0..1..0..2....0..1..0..2....0..1..0..2
..1..1..2..0....1..0..2..1....0..2..0..0....2..0..0..1....0..1..2..1
..1..2..2..0....2..0..0..1....1..2..0..0....1..0..2..0....2..0..1..0
..1..2..2..1....0..1..0..2....0..1..0..1....2..0..1..2....1..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..144

Column 1 is A007051(n-1)

Row 1 is A000079(n-2)

Empirical for column k:
k=1: a(n) = 4*a(n-1) -3*a(n-2)
k=2: a(n) = 7*a(n-1) +2*a(n-3) -8*a(n-4) for n>5
k=3: [order 8]
k=4: [order 31]
k=5: [order 94]
Empirical for row n:
n=1: a(n) = 2*a(n-1) for n>2
n=2: a(n) = 5*a(n-1) -2*a(n-2) -4*a(n-3) for n>5
n=3: a(n) = 9*a(n-1) +16*a(n-2) -50*a(n-3) -72*a(n-4) -32*a(n-5) -32*a(n-6) for n>8
n=4: [order 21] for n>23
n=5: [order 65] for n>67

A049455 Triangle read by rows: T(n,k) = numerator of fraction in k-th term of n-th row of variant of Farey series.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9

Offset: 1

N. J. A. Sloane

Stern's diatomic array read by rows (version 4, the 0,1 version).
This sequence divided by A049456 gives another version of the Stern-Brocot tree.
Row n has length 2^n + 1.
Define mediant of a/b and c/d to be (a+c)/(b+d). We get A006842/A006843 if we omit terms from n-th row in which denominator exceeds n.
Largest term of n-th row = A000045(n), Fibonacci numbers. - Reinhard Zumkeller, Apr 02 2014

			0/1, 1/1; 0/1, 1/2, 1/1; 0/1, 1/3, 1/2, 2/3, 1/1; 0/1, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1/1; 0/1, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, ... = A049455/A049456
The 0,1 version of Stern's diatomic array (cf. A002487) begins:
0,1,
0,1,1,
0,1,1,2,1,
0,1,1,2,1,3,2,3,1,
0,1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,1,
0,1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,1,5,4,7,3,8,5,7,2,7,5,3,3,7,4,5,1,
...

Martin Gardner, Colossal Book of Mathematics, Classic Puzzles, Paradoxes, and Problems, Chapter 25, Aleph-Null and Aleph-One, p. 328, W. W. Norton & Company, NY, 2001.
J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 8204 terms from Reinhard Zumkeller)
C. Giuli and R. Giuli, A primer on Stern's diatomic sequence, Fib. Quart., 17 (1979), 103-108, 246-248 and 318-320 (but beware errors).
Jennifer Lansing, Largest Values for the Stern Sequence, J. Integer Seqs., 17 (2014), #14.7.5.
M. Shrader-Frechette, Modified Farey sequences and continued fractions, Math. Mag., 54 (1981), 60-63.
N. J. A. Sloane, Stern-Brocot or Farey Tree
Index entries for sequences related to Stern's sequences

Cf. A049456. Also A007305, A007306, A006842, A006843, A070878, A070879.
Row sums are A007051.
Cf. A000051 (row lengths), A293165 (distinct terms).

import Data.List (transpose)
import Data.Ratio ((%), numerator, denominator)
a049455 n k = a049455_tabf !! (n-1) !! (k-1)
a049455_row n = a049455_tabf !! (n-1)
a049455_tabf = map (map numerator) $ iterate
   (\row -> concat $ transpose [row, zipWith (+/+) row $ tail row]) [0, 1]
   where u +/+ v = (numerator u + numerator v) %
                   (denominator u + denominator v)
-- Reinhard Zumkeller, Apr 02 2014

f[l_List] := Block[{k = Length@l, j = l}, While[k > 1, j = Insert[j, j[[k]] + j[[k - 1]], k]; k--]; j]; NestList[f, {0, 1}, 6] // Flatten (* Robert G. Wilson v, Nov 10 2019 *)

mediant(x, y) = (numerator(x)+numerator(y))/(denominator(x)+denominator(y));
newrow(rowa) = {my(rowb = []); for (i=1, #rowa-1, rowb = concat(rowb, rowa[i]); rowb = concat(rowb, mediant(rowa[i], rowa[i+1]));); concat(rowb, rowa[#rowa]);}
rows(nn) = {my(rowa); for (n=1, nn, if (n==1, rowa = [0, 1], rowa = newrow(rowa)); print(apply(x->numerator(x), rowa)););} \\ Michel Marcus, Apr 03 2019

Row 1 is 0/1, 1/1. Obtain row n from row n-1 by inserting mediants between each pair of terms.

More terms from Larry Reeves (larryr(AT)acm.org), Apr 12 2000

A083066 5th row of number array A083064.

Original entry on oeis.org

1, 5, 29, 173, 1037, 6221, 37325, 223949, 1343693, 8062157, 48372941, 290237645, 1741425869, 10448555213, 62691331277, 376147987661, 2256887925965, 13541327555789, 81247965334733, 487487792008397, 2924926752050381

Offset: 0

Paul Barry, Apr 21 2003

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=8, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). - Milan Janjic, Feb 21 2010

An Engel expansion of 3/2 to the base b := 6/5 as defined in A181565, with the associated series expansion 3/2 = b + b^2/5 + b^3/(5*29) + b^4/(5*29*173) + .... Cf. A007051. - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 [math.CO], 2014, eq (43).
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A083064, A083065, A083067, A007051, A007583.

[(4*6^n+1)/5: n in [0..30]]; // Vincenzo Librandi, Nov 06 2011

f[n_]:=6^n; lst={}; Do[a=f[n]; Do[a-=f[m],{m,n-1,1,-1}]; AppendTo[lst,a/6],{n,1,30}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *)

a(n) = (4*6^n+1)/5.
G.f.: (1-2*x)/((1-6*x)*(1-x)).
E.g.f.: (4*exp(6*x)+exp(x))/5.
a(n) = 6*a(n-1)-1 with n>0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 7*a(n-1)-6*a(n-2). - Vincenzo Librandi, Nov 04 2011
a(n) = 6^n - Sum_{i=0..n-1} 6^i for n>0. - Bruno Berselli, Jun 20 2013

A164864 Number of ways of placing n labeled balls into 10 indistinguishable boxes; word structures of length n using a 10-ary alphabet.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678569, 4213530, 27641927, 190829797, 1381367941, 10448276360, 82285618467, 672294831619, 5676711562593, 49344452550230, 439841775811967, 4005444732928641, 37136385907400125, 349459367068932740

Offset: 0

Alois P. Heinz, Aug 28 2009

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See pp. 3-4, 16-18.
N. Moreira and R. Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71.
Eric Weisstein's World of Mathematics, Set Partition
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, signature (46,-906,9996,-67809,291774,-790964,1290824,-1136160,403200).

Cf. A000110, A048993, A008291, A098825, A000012, A000079, A007051, A007581, A124303, A056272, A056273, A099262, A099263, A164863.

A row of the array in A278984.

# First program:
a:= n-> ceil(2119/11520*2^n +103/1680*3^n +53/3456*4^n +11/3600*5^n +6^n/1920 +7^n/15120 +8^n/80640 +10^n/3628800): seq(a(n), n=0..25);
# second program:
a:= n-> add(Stirling2(n, k), k=0..10): seq(a(n), n=0..25);

Table[Sum[StirlingS2[n,k],{k,0,10}],{n,0,30}] (* Harvey P. Dale, Nov 22 2023 *)

a(n) = Sum_{k=0..10} Stirling2 (n,k).
a(n) = ceiling(2119/11520*2^n +103/1680*3^n +53/3456*4^n +11/3600*5^n +6^n/1920 +7^n/15120 +8^n/80640 +10^n/3628800).
G.f.: (148329*x^9 -613453*x^8 +855652*x^7 -596229*x^6 +240065*x^5 -59410*x^4 +9177*x^3 -862*x^2 +45*x-1) / ((10*x-1) *(8*x-1) *(7*x-1) *(6*x-1) *(5*x-1) *(4*x-1) *(3*x-1) *(2*x-1) *(x-1)).
a(n) <= A000110(n) with equality only for n <= 10.

A208392 T(n,k)=Number of nXk 0..2 arrays with new values 0..2 introduced in row major order and no element equal to more than two of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

1, 2, 2, 5, 14, 5, 14, 117, 122, 14, 41, 1017, 3042, 1094, 41, 122, 8838, 76806, 79092, 9842, 122, 365, 76806, 1937736, 5800644, 2056392, 88574, 365, 1094, 667476, 48890520, 424785708, 438083928, 53466192, 797162, 1094, 3281, 5800644

Offset: 1

R. H. Hardin Feb 25 2012

Table starts
....1.......2...........5..............14.................41
....2......14.........117............1017...............8838
....5.....122........3042...........76806............1937736
...14....1094.......79092.........5800644..........424785708
...41....9842.....2056392.......438083928........93120350760
..122...88574....53466192.....33085555344.....20413586117376
..365..797162..1390120992...2498731184736...4475009970818208
.1094.7174454.36143145792.188712490047552.980999326809336384

			Some solutions for n=4 k=3
..0..0..1....0..0..0....0..0..0....0..0..0....0..0..0....0..0..1....0..0..1
..2..0..2....0..1..2....1..1..1....0..1..2....1..2..1....2..0..2....2..0..2
..1..0..2....0..0..0....1..0..0....2..0..0....0..2..2....2..2..0....0..2..2
..1..2..1....1..2..0....1..2..2....1..0..1....0..0..2....0..2..1....2..0..2

R. H. Hardin, Table of n, a(n) for n = 1..301

Column 1 is A007051(n-1)
Column 2 is A199560(n-1)
Row 1 is A007051(n-1)

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cp's OEIS Frontend

A243506 Permutation of natural numbers: a(n) = A048673(A122111(n)).

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A201730 Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) 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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A209100 T(n,k)=Number of nXk 0..2 arrays with new values 0..2 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

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A222986 T(n,k)=Number of nXk 0..2 arrays with no more than floor(nXk/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..2 order.

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A223126 T(n,k)=Number of nXk 0..2 arrays with no more than floor(nXk/2) elements equal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..2 order.

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A241370 T(n,k)=Number of nXk 0..2 arrays with no element equal to fewer vertical neighbors than horizontal neighbors, with new values 0..2 introduced in row major order.

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A049455 Triangle read by rows: T(n,k) = numerator of fraction in k-th term of n-th row of variant of Farey series.

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A083066 5th row of number array A083064.

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