A033484 -id:A033484 - cp's OEIS frontend

A169722 a(n) = (32^(n-1)-1)(18*2^(n-1)-7).

1, 22, 145, 715, 3151, 13207, 54055, 218695, 879751, 3528967, 14135815, 56583175, 226412551, 905809927, 3623559175, 14494875655, 57980780551, 231925678087, 927707824135, 3710841520135, 14843386527751, 59373587005447, 237494429810695, 949977882820615

Offset: 0

Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alice V. Kleeva, Grid for this sequence
Alice V. Kleeva, Illustration of initial terms
Robert Munafo, Sequence A169720, and two others by Alice V. Kleeva
Robert Munafo, Sequence A169720, and two others by Alice V. Kleeva [Cached copy, in pdf format, included with permission]
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A169720-A169727, A033484.

I:=[1,22,145]; [n le 3 select I[n] else 7*Self(n-1)-14*Self(n-2)+8*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 04 2012

LinearRecurrence[{7, -14, 8}, {1, 22, 145}, 30] (* Vincenzo Librandi, Dec 04 2012 *)

makelist(coeff(taylor((1+15*x+5*x^2)/(1-7*x+14*x^2-8*x^3), x, 0, n), x, n), n, 0, 23); /* Bruno Berselli, Dec 04 2012 */

G.f.: (1+15*x+5*x^2)/(1-7*x+14*x^2-8*x^3). - Bruno Berselli, Dec 04 2012

a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3). - Vincenzo Librandi, Dec 04 2012

A182659 A canonical permutation designed to thwart a certain naive attempt to guess whether sequences are permutations.

Original entry on oeis.org

0, 2, 3, 1, 5, 6, 7, 8, 9, 4, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 10, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 22, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 0

Sam Alexander, Nov 26 2010

A naive way to guess whether a function f:N->N is a permutation, based on just an initial subsequence (f(0),...,f(n)), is to guess "no" if (f(0),...,f(n)) contains a repeated entry or if there is some i in {0,...,n} such that i is not in {f(0),...,f(n)} and 2 i<=n; and guess "yes" otherwise. a(n) thwarts that method, causing it to change its mind infinitely often as n->infinity.
a(0)=0. Suppose a(0),...,a(n) have been defined.
1. If the above method guesses that (a(0),...,a(n)) is NOT an initial subsequence of a permutation, then unmark any "marked" numbers.
2. If the above method guesses that (a(0),...,a(n)) IS an initial subsequence of a permutation, then "mark" the smallest number not in {a(0),...,a(n)}.
3. Let a(n+1) be the least unmarked number not in {a(0),...,a(n)}.
A030301 can be derived by a similar method, where instead of trying to guess whether sequences are permutations, the naive victim is trying to guess whether sequences contain infinitely many 0s.

S. Alexander, On Guessing Whether A Sequence Has A Certain Property, arXiv:1011.6626 [math.LO], 2010-2012, J. Int. Seq. 14 (2011) # 11.4.4.

Cf. A030301, A082691, A182660, A068156, A033484.

a(0) = 0; if n = A068156(k+1) = 6*2^k - 3 for some k >= 0 then a(n) = A033484(k) = (n-1)/2; otherwise, a(n) = n+1. - Andrey Zabolotskiy, Feb 27 2025

a(22) corrected and further terms added by Andrey Zabolotskiy, Feb 27 2025

A227341 Triangular array: Number of partitions of the vertex set of a forest of two trees on n vertices into k nonempty independent sets.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 2, 4, 1, 0, 2, 10, 7, 1, 0, 2, 22, 31, 11, 1, 0, 2, 46, 115, 75, 16, 1, 0, 2, 94, 391, 415, 155, 22, 1, 0, 2, 190, 1267, 2051, 1190, 287, 29, 1, 0, 2, 382, 3991, 9471, 8001, 2912, 490, 37, 1, 0, 2, 766, 12355, 41875, 49476, 25473, 6342, 786, 46, 1

Offset: 1

Peter Bala, Jul 10 2013

For a graph G and a positive integer k, the graphical Stirling number S(G;k) is the number of set partitions of the vertex set of G into k nonempty independent sets (an independent set in G is a subset of the vertices, no two elements of which are adjacent).

Here we take the graph G to be a forest of two trees on n vertices. The corresponding graphical Stirling numbers S(G;k) do not depend on the choice of the trees. See Galvin and Thanh. If the graph G is a single tree on n vertices then the graphical Stirling numbers S(G;k) are the classical Stirling numbers of the second kind A008277. See also A105794.

			Triangle begins
n\k | 1 2  3   4   5  6  7
= = = = = = = = = = = = =
  1 | 1
  2 | 1 1
  3 | 0 2  1
  4 | 0 2  4   1
  5 | 0 2 10   7   1
  6 | 0 2 22  31  11  1
  7 | 0 2 46 115  75 16  1
Connection constants: Row 5: 2*x*(x-1) + 10*x*(x-1)*(x-2) + 7*x*(x-1)*(x-2)*(x-3) + x*(x-1)*(x-2)*(x-3)*(x-4) = x^2*(x-1)^3.

B. Duncan and R. Peele, Bell and Stirling Numbers for Graphs, Journal of Integer Sequences 12 (2009), article 09.7.1.
D. Galvin and D. T. Thanh, Stirling numbers of forests and cycles, Electr. J. Comb. Vol. 20(1): P73 (2013)

A008277, A011968 (row sums), A033484 (col. 3), A091344 (col. 4), A105794.

T(n,k) = Stirling2(n-1,k-1) + Stirling2(n-2,k-1), n,k >= 1.
Recurrence equation: T(n,k) = (k-1)*T(n-1,k) + T(n-1,k-1). Cf. A105794.
k-th column o.g.f.: x^k*(1+x)/((1-x)*(1-2*x)*...*(1-(k-1)*x)).
For n >= 2, sum {k = 0..n} T(n,k)*x_(k) = x^2*(x-1)^(n-2), where x_(k) = x*(x-1)*...*(x-k+1) is the falling factorial.
Column 3: A033484; Column 4: A091344; Row sums are essentially A011968.

A296953 Number of bisymmetric, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n}.

Original entry on oeis.org

0, 1, 4, 10, 22, 46, 94, 190, 382, 766, 1534, 3070, 6142, 12286, 24574, 49150, 98302, 196606, 393214, 786430, 1572862, 3145726, 6291454, 12582910, 25165822, 50331646, 100663294, 201326590, 402653182, 805306366, 1610612734, 3221225470, 6442450942, 12884901886

Offset: 0

J. Devillet, Dec 22 2017

Apart from the offset the same as A033484. - R. J. Mathar, Alois P. Heinz, Jan 02 2018

J. Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA] (2017).
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Nest[Append[#, 2 Last@ # + 2] &, {0, 1}, 32] (* or *)
Array[3*2^(# - 1) - 2 + Boole[# == 0]/2 &, 34, 0] (* or *)
CoefficientList[Series[x (1 + x)/((1 - x) (1 - 2 x)), {x, 0, 33}], x] (* Michael De Vlieger, Dec 22 2017 *)

concat(0, Vec(x*(1 + x) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Dec 22 2017

a(0)=0, a(1)=1, a(n+1)-2*a(n) = 2.
From Colin Barker, Dec 22 2017: (Start)
G.f.: x*(1 + x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*2^(n-1) - 2 for n>0.
a(n) = 3*a(n-1) - 2*a(n-2) for n>2.
(End)

G.f. replaced by a better g.f. by Colin Barker, Dec 23 2017

A365863 a(0) = 1; thereafter a(n) = nSum_{k = 0..n-1} binomial(n, k)(-1)^(1+n+k)*a(k).

Original entry on oeis.org

1, 1, 2, 12, 156, 3380, 108930, 4876242, 289111032, 21916777752, 2067208751790, 237380181141950, 32601704893973556, 5276471519805880836, 993835167745129599162, 215520207875112312124890, 53311353846240820033325040, 14919977169758349265112350256, 4690364757880376663319746737926

Offset: 0

Thomas Scheuerle, Nov 09 2023

nonn

Let P_k(x) be the polynomial of order k which satisfies a(m) = P_k(m) for m = 0..k, then a(k+1) = k * P_k(k+1).
This sequence is a member of a family of sequences with related properties. Here are some examples:
With b(k+1) = 1 + P_k(k+1) we get b(k) = A000079(k).
With b(k+1) = 2 + P_k(k+1) we get b(k) = A000225(k).
With b(k+1) = 3 + P_k(k+1) we get b(k) = A033484(k).
With b(k+1) = 2 * P_k(k+1) we get b(k) = A000629(k).
With b(k+1) = 1 + 2 * P_k(k+1) we get b(k) = A007047(k).
With b(k+1) = 3 * P_k(k+1) we get b(k) = A201339(k).
With b(k+1) = 5 * P_k(k+1) we get b(k) = A201365(k).
With b(k+1) = -1 * P_k(k+1) we get b(k) = A000670(k)*(-1)^k.
With b(k+1) = -2 * P_k(k+1) we get b(k) = A004123(k+1)*(-1)^k.
With b(k+1) = -3 * P_k(k+1) we get b(k) = A032033(k)*(-1)^k.
With b(k+1) = -4 * P_k(k+1) we get b(k) = A094417(k)*(-1)^k.
With b(k+1) = -m * P_k(k+1) we get b(k) = Bo(m, k)*(-1)^k, Bo(m, k) are Generalized ordered Bell numbers.

Cf. A000079, A000225, A000629, A000670.
Cf. A004123, A007047, A032033, A033484.
Cf. A094417, A201339, A201365, A367308.

a[n_] := a[n] = If[n == 0, 1, n*Sum[Binomial[n, k]*(-1)^(1 + n + k)*a[k], {k, 0, n - 1}]]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Nov 12 2023 *)

a(n) = if(n == 0, 1,sum(k = 0,n-1, n*binomial(n, k)*(-1)^(1+n+k)*a(k)))

a(n) ~ c * n^(2*n + 1/2) / exp(2*n), where c = 2.9711739498821842863440481701659942323709511474486414... - Vaclav Kotesovec, Nov 12 2023

A368826 Square array T(n,k) = 3*2^k - n read by ascending antidiagonals.

Original entry on oeis.org

3, 2, 6, 1, 5, 12, 0, 4, 11, 24, -1, 3, 10, 23, 48, -2, 2, 9, 22, 47, 96, -3, 1, 8, 21, 46, 95, 192, -4, 0, 7, 20, 45, 94, 191, 384, -5, -1, 6, 19, 44, 93, 190, 383, 768, -6, -2, 5, 18, 43, 92, 189, 382, 767, 1536, -7, -3, 4, 17, 42, 91, 188, 381, 766, 1535, 3072

Offset: 0

Paul Curtz, Jan 07 2024

Similar to A367559.

			Table begins:
   3  6 12 24 48 96 ...
   2  5 11 23 47 95 ...
   1  4 10 22 46 94 ...
   0  3  9 21 45 93 ...
  -1  2  8 20 44 92 ...
  -2  1  7 19 43 91 ...
  ...

Cf. A000225, A000918, A007283, A033484, A048488, A083329, A165751, A367559.

Cf. diagonals: A123720, A079583, A095151, A101945 (after -1).

T[n_, k_] := 3*2^k - n; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jan 15 2024 *)

T(0,k) = 3*2^k     =   A007283(k).
T(1,k) = 3*2^k - 1 =   A083329(k+1).
T(2,k) = 3*2^k - 2 =   A033484(k).
T(3,k) = 3*2^k - 3 = 3*A000225(k).
T(4,k) = 3*2^k - 4 =  -A165751(k).
T(5,k) = 3*2^k - 5 =   A048488(k-1)
T(6,k) = 3*2^k - 6 = 3*A000918(k).

A067763 Square array read by antidiagonals of base n numbers written as 122...222 with k 2's (and a suitable interpretation for n=0, 1 or 2).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 7, 10, 5, 1, 2, 9, 22, 17, 6, 1, 2, 11, 46, 53, 26, 7, 1, 2, 13, 94, 161, 106, 37, 8, 1, 2, 15, 190, 485, 426, 187, 50, 9, 1, 2, 17, 382, 1457, 1706, 937, 302, 65, 10, 1, 2, 19, 766, 4373, 6826, 4687, 1814, 457, 82, 11, 1, 2, 21, 1534, 13121

Offset: 0

Henry Bottomley, Feb 06 2002

Start with a node; step one is to connect that node to n+1 new nodes so that it is of degree n+1; further steps are to connect each existing node of degree 1 to n new nodes so that it is of degree n+1; T(n,k) is the total number of nodes after k steps.

			Rows start: 1,2,2,2,2,2,...; 1,3,5,7,9,11,...; 1,4,10,22,46,94,...; 1,5,17,53,161,485,... T(3,2) =122 base 3 =17.

Rows include A040000, A005408, A033484, A048473, A020989, A057651, A061801 etc. For negative n (not shown) absolute values of rows would effectively include A000012, A014113, A046717.

T(n, k) =((n+1)*n^k-2)/(n-1) [with T(1, k)=2k+1] =n*T(n, k-1)+2 =(n+1)*T(n, k-1)-n*T(n, k-2) =T(n, k-1)+(1+1/n)*n^k =A055129(k, n)+A055129(k-1, n). Coefficient of x^k in expansion of (1+x)/((1-x)(1-nx)).

A130453 A097806 * A059268.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 2, 4, 8, 8, 2, 4, 8, 16, 16, 2, 4, 8, 16, 32, 32, 2, 4, 8, 16, 32, 64, 64, 2, 4, 8, 16, 32, 64, 128, 128

Offset: 1

Gary W. Adamson, May 26 2007

Row sums = A033484: (1, 4, 10, 22, 46, 94, ...).

A130459 = A059268 * A097806.

			First few rows of the triangle:
  1;
  2, 2;
  2, 4, 4;
  2, 4, 8,  8;
  2, 4, 8, 16, 16;
  2, 4, 8, 16, 32, 32;
  ...

Cf. A097806, A059268, A033484, A130459.

A097806 * A059268 as infinite lower triangular matrices.

A134063 a(n) = (1/2)*(3^n - 2^(n+1) + 3).

Original entry on oeis.org

1, 1, 2, 7, 26, 91, 302, 967, 3026, 9331, 28502, 86527, 261626, 788971, 2375102, 7141687, 21457826, 64439011, 193448102, 580606447, 1742343626, 5228079451, 15686335502, 47063200807, 141197991026, 423610750291, 1270865805302, 3812664524767, 11438127792026

Offset: 0

Ross La Haye, Jan 11 2008

Let P(A) be the power set of an n-element set A. Then a(n-1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 2) x = y.

The inverse binomial transform yields A033484 with another leading 1. - R. J. Mathar, Jul 06 2009

			a(3) = 7 because for P(A) = {{},{1},{2},{1,2}} we have: case 0 {{1},{2}}, case 1 {{1},{1,2}}, {{2},{1,2}}, case 2 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}.

Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [_Ross La Haye_, Feb 22 2009]
Index entries for linear recurrences with constant coefficients, signature (6,-11,6)

Cf. A000392, A028243, A000079.

Maple
```
f := n -> (1/2)*(3^n - 2^(n+1) + 3);
```

Table[(3^n-2^(n+1)+3)/2,{n,0,30}] (* or *) LinearRecurrence[{6,-11,6},{1,1,2},30] (* Harvey P. Dale, May 05 2020 *)

a(n) = 3*StirlingS2(n,3) + StirlingS2(n,2) + 1.
a(n) = StirlingS2(n+1,3) + 1. - Ross La Haye, Jan 21 2008
a(n) = 6 a(n-1)-11 a(n-2) +6 a(n-3) (n >= 3). Also a(n) = 4 a(n-1)-3 a(n-2)+ 2^{n-2} (n >= 3). - Tian-Xiao He (the(AT)iwu.edu), Jul 02 2009
G.f.: -(1-4*x+6*x^2)/((x-1)*(3*x-1)*(2*x-1)). a(n+1)-a(n)=A001047(n+1). [R. J. Mathar, Jul 06 2009]

Edited by N. J. A. Sloane, Jul 06 2009

A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 0

Philippe Deléham, Oct 07 2009

			Triangle begins :
1 ;
0,2 ;
0,1,2 ;
0,1,1,2 ;
0,1,1,1,2 ;
0,1,1,1,1,2 ;
0,1,1,1,1,1,2 ; ...

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k)= A166122(n), A166114(n), A084222(n), A084247(n), A000034(n), A040000(n), A000027(n+1), A000079(n), A007051(n), A047849(n), A047850(n), A047851(n), A047852(n), A047853(n), A047854(n), A047855(n), A047856(n) for x= -5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^k= A000007(n), A000027(n+1), A033484(n), A134931(n), A083597(n) for x= 0,1,2,3,4 respectively.
T(n,k)= A166065(n,k)/2^(n-k).
G.f.: (1-x+x*y)/(1-x-x*y+x^2*y). - Philippe Deléham, Nov 09 2013
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 09 2013

cp's OEIS Frontend

A169722 a(n) = (3*2^(n-1)-1)*(18*2^(n-1)-7).

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A182659 A canonical permutation designed to thwart a certain naive attempt to guess whether sequences are permutations.

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A227341 Triangular array: Number of partitions of the vertex set of a forest of two trees on n vertices into k nonempty independent sets.

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A296953 Number of bisymmetric, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n}.

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Mathematica

PARI

Formula

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A365863 a(0) = 1; thereafter a(n) = n*Sum_{k = 0..n-1} binomial(n, k)*(-1)^(1+n+k)*a(k).

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A368826 Square array T(n,k) = 3*2^k - n read by ascending antidiagonals.

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Mathematica

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A067763 Square array read by antidiagonals of base n numbers written as 122...222 with k 2's (and a suitable interpretation for n=0, 1 or 2).

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A130453 A097806 * A059268.

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A134063 a(n) = (1/2)*(3^n - 2^(n+1) + 3).

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A169722 a(n) = (32^(n-1)-1)(18*2^(n-1)-7).

A365863 a(0) = 1; thereafter a(n) = nSum_{k = 0..n-1} binomial(n, k)(-1)^(1+n+k)*a(k).