A033484 -id:A033484 - cp's OEIS frontend

A347738 A variant of the inventory sequence: record the number of terms >= 0 thus far in the sequence, then the number of terms >= 1 thus far, then the number of terms >= 2 thus far, and so on, until a zero is recorded; the inventory then starts again, recording the number of terms >= 0, etc.

0, 1, 1, 0, 4, 3, 2, 2, 1, 0, 10, 8, 6, 5, 5, 5, 3, 2, 2, 1, 1, 0, 22, 19, 15, 12, 11, 11, 9, 9, 10, 10, 9, 6, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 0, 46, 42, 35, 28, 24, 23, 21, 20, 21, 21, 19, 17, 16, 16, 17, 18, 18, 17, 15, 13, 11, 10, 7, 6, 5, 4, 4, 4, 4, 3, 3

Offset: 0

David James Sycamore, Sep 12 2021

Sequence starts off as A342585 but diverges after a(4). The effect is to introduce some numbers earlier in this sequence than in the original, and to stretch out the incidences of zero terms by the fact that the term immediately following a zero is now the total number of prior terms, rather than the total number of prior zero terms.

In A342585 zeros occur at positions 1,4,8,14,20,28,... (see A343880) whereas in this version they occur at positions 1,4,10,22,46,... (which is A033484, as is easily proved by induction).

			As an irregular triangle this begins:
   0;
   1,  1,  0;
   4,  3,  2,  2,  1,  0;
  10,  8,  6,  5,  5,  5, 3, 2,  2,  1, 1, 0;
  22, 19, 15, 12, 11, 11, 9, 9, 10, 10, 9, 6, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 0;
  46, ...
(for row lengths see A003945)

Michael De Vlieger, Table of n, a(n) for n = 0..24573 (rows 0 <= k <= 12 when considered as an irregular triangle)
Michael De Vlieger, log-log scatterplot of a(n) for 1 <= n <= 49150 (ignoring zeros).

Cf: A342585, A033484, A003945, A343880, A003945 (row lengths), A347324 (row sums).

A347326 has a version of this in which the rows have been normalized.

a[n_] := a[n] = Block[{t}, t = If[a[n - 1] == 0, 0, b[n - 1] + 1]; b[n] = t; Sum[If[a[j] >= t, 1, 0], {j, n - 1}]]; b[1] = a[1] = 0; Array[a, 77] (* Michael De Vlieger, Sep 12 2021, after Jean-François Alcover at A342585 *)

def aupton(nn):
    num, gte_inventory, alst, bigc = 0, [1], [0], 0
    while len(alst) < nn+1:
        c = gte_inventory[num] if num <= bigc else 0
        num = 0 if c == 0 else num + 1
        for i in range(min(c, bigc)+1):
            gte_inventory[i] += 1
        for i in range(bigc+1, c+1):
            gte_inventory.append(1)
        bigc = len(gte_inventory) - 1
        alst.append(c)
    return alst
print(aupton(76)) # Michael S. Branicky, Sep 19 2021

Offset changed to 0 by N. J. A. Sloane, Sep 12 2021

A169720 a(n) = (32^(n-1)-1)(3*2^n-1).

Original entry on oeis.org

1, 10, 55, 253, 1081, 4465, 18145, 73153, 293761, 1177345, 4713985, 18865153, 75479041, 301953025, 1207885825, 4831690753, 19327057921, 77308821505, 309236465665, 1236948221953, 4947797606401, 19791199862785, 79164818325505, 316659311050753, 1266637319700481

Offset: 0

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

A subsequence of the triangular numbers A000217.

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. A169721-A169727, A033484, A000217.

I:=[1, 10, 55]; [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 03 2012

CoefficientList[Series[(1 + 3*x - x^2)/((1-x)*(1-2*x)*(1-4*x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{7, -14, 8}, {1, 10, 55}, 30] (* Vincenzo Librandi, Dec 03 2012 *)

a(n)=polcoeff((1+3*x-x^2)/((1-x)*(1-2*x)*(1-4*x)+x*O(x^n)),n) \\ Paul D. Hanna, Apr 29 2010

G.f.: (1 + 3*x - x^2)/((1-x)*(1-2*x)*(1-4*x)). - Paul D. Hanna, Apr 29 2010
a(n) = A000217(A033484(n)). - Mitch Harris, Dec 02 2012
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - Vincenzo Librandi, Dec 03 2012
a(n) = (3*A169726(n)-1)/2. - L. Edson Jeffery, Dec 03 2012
a(n) = A006095(n+2) +3*A006095(n+1) - A006905(n). - R. J. Mathar, Dec 04 2016

A083416 Add 1, double, add 1, double, etc.

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 22, 23, 46, 47, 94, 95, 190, 191, 382, 383, 766, 767, 1534, 1535, 3070, 3071, 6142, 6143, 12286, 12287, 24574, 24575, 49150, 49151, 98302, 98303, 196606, 196607, 393214, 393215, 786430, 786431, 1572862, 1572863, 3145726, 3145727, 6291454

Offset: 1

N. J. A. Sloane, Jun 10 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A033484, A081026, A153893.

a083416 n = a083416_list !! (n-1)
a083416_list = 1 : f 2 1 where
   f x y = z : f (x+1) z where z = (1 + x `mod` 2) * y + 1 - x `mod` 2
-- Reinhard Zumkeller, Feb 27 2012

[Floor(3*2^((2*n-(-1)^n-3)/4)+((-1)^n-3)/2): n in [1..50]]; // Vincenzo Librandi, Aug 17 2011

A083416 := proc(n) if type(n,'even') then 3*2^(n/2-1)-1 ; else 3*2^((n-1)/2)-2 ; end if; end proc: # R. J. Mathar, Feb 16 2011

a=0; b=0; lst={a,b}; Do[z=a+b+1; AppendTo[lst,z]; a=b; b=z; z=b+1; AppendTo[lst,z]; a=b; b=z,{n,50}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 16 2010 *)
LinearRecurrence[{0,3,0,-2},{1,2,4,5},40] (* Harvey P. Dale, Nov 18 2014 *)

G.f.: x*(1+2*x+x^2-x^3)/(1-x^2)/(1-2*x^2).
a(2*n) = 3*2^(n-1)-1, a(2*n+1) = 3*2^n-2.
a(n) = A081026(n+1)-1.
a(n) = 3*2^((2*n-(-1)^n-3)/4)+((-1)^n-3)/2. - Bruno Berselli, Feb 17 2011
For n > 1: a(n) = (1 + n mod 2) * a(n-1) + 1 - n mod 2. - Reinhard Zumkeller, Feb 27 2012
a(2n+1) = A033484(n), a(2n) = A153893(n). - Philippe Deléham, Apr 14 2013
E.g.f.: (3*cosh(sqrt(2)*x) - 4*sinh(x) + 3*sqrt(2)*sinh(sqrt(2)*x) - 2*cosh(x) - 1)/2. - Stefano Spezia, Jul 11 2023

More terms from Donald Sampson (marsquo(AT)hotmail.com), Dec 04 2003

Corrected by T. D. Noe, Nov 02 2006

A089143 a(n) = 9*2^n - 6.

Original entry on oeis.org

3, 12, 30, 66, 138, 282, 570, 1146, 2298, 4602, 9210, 18426, 36858, 73722, 147450, 294906, 589818, 1179642, 2359290, 4718586, 9437178, 18874362, 37748730, 75497466, 150994938, 301989882, 603979770, 1207959546, 2415919098, 4831838202, 9663676410, 19327352826

Offset: 0

Parthasarathy Nambi, Sep 21 2004

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A033484.

a=3; lst={a}; k=9; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 16 2008 *)
9*2^Range[0,40]-6 (* Harvey P. Dale, Sep 15 2013 *)

my(x='x+O('x^32)); Vec(3*(1+x)/(1-3*x+2*x^2)) \\ Elmo R. Oliveira, May 24 2025

a(n) = 3*A033484(n).
From Elmo R. Oliveira, May 24 2025: (Start)
G.f.: 3*(x+1)/((x-1)*(2*x-1)).
E.g.f.: 3*exp(x)*(3*exp(x) - 2).
a(n) = 3*a(n-1) - 2*a(n-2). (End)

A131110 A000012 * A133084.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 10, 6, 5, 1, 15, 10, 15, 5, 1, 21, 15, 35, 15, 7, 1, 28, 21, 70, 35, 28, 7, 1, 36, 28, 126, 70, 84, 28, 9, 1, 45, 36, 210, 126, 210, 84, 45, 9, 1, 55, 45, 330, 210, 462, 210, 165, 45, 11, 1

Offset: 1

Gary W. Adamson, Sep 08 2007

Row sums give A033484.

Duplicate of A133093. - Georg Fischer, Oct 10 2021

			First few rows of the triangle are:
1;
3, 1;
6, 3, 1;
10, 6, 5, 1;
15, 10, 15, 5, 1;
21, 15, 35, 15, 7, 1;
28, 21, 70, 35, 28, 7, 1;
...

Cf. A133084, A033484.

T4(n, k) = if(k == n, 1, (1  - (1 + (-1)^k)/2 )*binomial(n, k) + ((1 + (-1)^k)/2)*binomial(n - 1, k - 1)); \\ A133084
N=10; matrix(N, N, n, k, if(n>=k, 1))*matrix(N, N, n, k, T4(n,k)) \\ Michel Marcus, Oct 11 2021

A000012 * A133084 as infinite lower triangular matrices.

a(46) corrected by Georg Fischer, Oct 10 2021

A204201 Triangle based on (0,1/3,1) averaging array.

Original entry on oeis.org

1, 1, 4, 1, 5, 10, 1, 6, 15, 22, 1, 7, 21, 37, 46, 1, 8, 28, 58, 83, 94, 1, 9, 36, 86, 141, 177, 190, 1, 10, 45, 122, 227, 318, 367, 382, 1, 11, 55, 167, 349, 545, 685, 749, 766, 1, 12, 66, 222, 516, 894, 1230, 1434, 1515, 1534, 1, 13, 78, 288, 738, 1410

Offset: 1

Clark Kimberling, Jan 12 2012

For a1, let
t(n,1)=[a+t(n-1,1)]/2,
t(n,n)=[b+t(n-1,n-1)]/2,
t(n,k)=[t(n-1,k-1)+t(n-1,k)]/2 for 2<=k<=n-1.
We call (t(n,k)) the (a,r,b) averaging array.  If a and b
are integers and r is a rational number, then multiplying
row n of (t(n,k)) by the LCM of its denominators yields a
triangle of integers; A204201 arises in this manner from
(a,r,b)=(0,1/3,1).
...
Guide to related arrays:
(a,r,b).........triangle
(0,1/2,1).......A054143
(0,1/3,1).......A204201
(0,2/3,1).......A204202
(0,1/4,1).......A204203
(0,3/4,1).......A204204
(0,1/5,1).......A204205
(1,3/2,2).......A204206
(1,2,3).........A204207

			The (0,1/3,1) averaging array has these first four rows:
1/3
1/6....2/3
1/12...5/12...5/6
1/24...1/4....5/8...11/12.
Multiplying those rows by 3,6,12,24, respectively:
1
1...4
1...5...10
1...6...15...22
The first nine rows:
1
1...4
1...5...10
1...6...15...22
1...7...21...37...46
1...8...28...58...83...94
1...9...36...86...141..177..190
1...10..45...122..227..318..367..382
1...11..55...167..349..545..685..749..766

Cf. A204202.

a = 0; r = 1/3; b = 1;
t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]   (* averaging array *)
u = Table[(1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]             (* A204102 triangle *)
Flatten[u]               (* A204201 sequence *)

From Philippe Deléham, Dec 24 2013: (Start)
T(n,n) = A033484(n-1).
Sum{k=1..n} T(n,k) = A053220(n).
T(n,k) = T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=1, T(2,1)=1, T(2,2)=4, T(n,k)=0 if k<1 or if k>n. (End)

A362824 Array read by antidiagonals: T(n,k) is the number of k-tuples of involutions on [n] that pairwise commute.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 1, 1, 8, 10, 10, 1, 1, 1, 16, 22, 52, 26, 1, 1, 1, 32, 46, 232, 196, 76, 1, 1, 1, 64, 94, 976, 1016, 1216, 232, 1, 1, 1, 128, 190, 4000, 4576, 12496, 5944, 764, 1, 1, 1, 256, 382, 16192, 19376, 111376, 73648, 42400, 2620, 1

Offset: 0

Andrew Howroyd, May 06 2023

Two involutions x,y on [n] commute if x*y = y*x.

			Array begins:
===========================================================
n/k| 0   1    2     3      4       5        6         7 ...
---+-------------------------------------------------------
0  | 1   1    1     1      1       1        1         1 ...
1  | 1   1    1     1      1       1        1         1 ...
2  | 1   2    4     8     16      32       64       128 ...
3  | 1   4   10    22     46      94      190       382 ...
4  | 1  10   52   232    976    4000    16192     65152 ...
5  | 1  26  196  1016   4576   19376    79696    323216 ...
6  | 1  76 1216 12496 111376  936976  7680016  62177296 ...
7  | 1 232 5944 73648 716416 6289312 52647904 430723168 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).

Columns k=0..3 are A000012, A000085, A362819, A362825.
Rows n=2..3 are A000079, A033484.
Main diagonal is A362823.
Cf. A022166, A362648.

\\ B(n,k) is A022166.
B(n,k)={polcoef(x^k/prod(j=0, k, 1-2^j*x + O(x*x^n)), n)}
T(n,k)={if(n==0, 1, n!*polcoef(exp(sum(j=0, min(k,logint(n,2)), B(k,j)*x^(2^j)/2^j, O(x*x^n))), n))}

T(0,k) = T(1,k) = 1.

A097813 a(n) = 32^n - 2n - 2.

Original entry on oeis.org

1, 2, 6, 16, 38, 84, 178, 368, 750, 1516, 3050, 6120, 12262, 24548, 49122, 98272, 196574, 393180, 786394, 1572824, 3145686, 6291412, 12582866, 25165776, 50331598, 100663244, 201326538, 402653128, 805306310, 1610612676, 3221225410, 6442450880, 12884901822, 25769803708

Offset: 0

Paul Barry, Aug 25 2004

An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 58, 154, 178 and 184, lead to this sequence. For the central square these vectors lead to the companion sequence A033484. - Johannes W. Meijer, Aug 15 2010

a(n) is also the number of order-preserving partial isometries of an n-chain, i.e., the row sums of A183153 and A183154. - Abdullahi Umar, Dec 28 2010

Harvey P. Dale, Table of n, a(n) for n = 0..1000
F. Al-Kharousi, R. Kehinde and A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain, The Australasian Journal of Combinatorics, Volume 58 (3) (2014), 363-375.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000295, A033484, A079583, A175654, A183153, A183154.

[3*2^n -2*(n+1): n in [0..40]]; // G. C. Greubel, Dec 30 2021

Table[3 2^n-2n-2,{n,0,40}] (* or *) LinearRecurrence[{4,-5,2},{1,2,6},40] (* Harvey P. Dale, Oct 25 2011 *)

a(n)=3*2^n-2*n-2 \\ Charles R Greathouse IV, Oct 07 2015

[3*2^n -2*(n+1) for n in (0..40)] # G. C. Greubel, Dec 30 2021

G.f.: (1 - 2*x + 3*x^2)/((1-x)^2*(1-2*x)).
a(n) = 2*a(n-1) + 2*n - 2, for n>0, with a(0)=1.
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
From G. C. Greubel, Dec 30 2021: (Start)
a(n) = 2^n + 2*A000295(n).
E.g.f.: 3*exp(2*x) - 2*(1 + x)*exp(x). (End)

A101945 a(n) = 6*2^n - n - 5.

Original entry on oeis.org

1, 6, 17, 40, 87, 182, 373, 756, 1523, 3058, 6129, 12272, 24559, 49134, 98285, 196588, 393195, 786410, 1572841, 3145704, 6291431, 12582886, 25165797, 50331620, 100663267, 201326562, 402653153, 805306336, 1610612703, 3221225438

Offset: 0

Gary W. Adamson, Dec 22 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A033484, A101946, A135855.

[6*2^n -n-5: n in [0..40]]; // G. C. Greubel, Feb 06 2022

a[0]=1; a[1]=6; a[2]=17; a[n_]:= a[n]= 4a[n-1] -5a[n-2] +2a[n-3];
Table[a[n], {n, 0, 30}] (* Robert G. Wilson v, Jan 12 2005 *)

a(n)=if(n==1,1,if(n==2,6,if(n==3,17,4*a(n-1)-5*a(n-2)+2*a(n-3)))) \\ (Klasen)

[3*2^(n+1) -(n+5) for n in (0..40)] # G. C. Greubel, Feb 06 2022

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n >= 3.
Row sums of triangle A135855. - Gary W. Adamson, Dec 01 2007
From G. C. Greubel, Feb 06 2022: (Start)
G.f.: (1 + 2*x - 2*x^2)/((1-x)^2*(1-2*x)).
E.g.f.: 6*exp(2*x) - (5+x)*exp(x). (End)

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 06 2005

New definition from Ralf Stephan, May 17 2007

A232642 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 2 are in S, and duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 8, 7, 14, 12, 11, 22, 9, 18, 16, 15, 30, 13, 26, 24, 23, 46, 20, 19, 38, 17, 34, 32, 31, 62, 28, 27, 54, 25, 50, 48, 47, 94, 21, 42, 40, 39, 78, 36, 35, 70, 33, 66, 64, 63, 126, 29, 58, 56, 55, 110, 52, 51, 102, 49, 98, 96, 95, 190, 44

Offset: 1

Clark Kimberling, Nov 28 2013

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 2 are in S. Then S is the set of positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2,4), g(3) = (3,6,5,10), etc. Concatenating these gives A232642, a permutation of the positive integers. For n > 1, the number of numbers in g(n) is 2*F(n+1), where F = A000045, the Fibonacci numbers. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to x + 1 if x + 1 has not already occurred, and an edge from x to 2*x + 2 if 2*x + 2 has not already occurred.

Seen as triangle read by rows: A082560 with duplicates removed. - Reinhard Zumkeller, May 14 2015

			Each x begets x + 1 and 2*x + 2, but if either has already occurred it is deleted. Thus, 1 begets 2 and 4; then 2 begets 3 and 6, and 4 begets 5 and 10, so that g(3) = (3,6,5,10).
First 5 generations, also showing the places where duplicates were removed:
.  1:                                1
.  2:                2                               4
.  3:        3              6               5                10
.  4:    _       8      7       14      _       12       11       22
.  5:  _  __   9  18  _  16  15   30  _  __  13   26  __   24  23   46
These are the corresponding complete rows of triangle A082560:
.  1:                                1
.  2:                2                               4
.  3:        3              6               5                10
.  4:    4       8      7       14      6       12       11       22
.  5:  5  10   9  18  8  16  15   30  7  14  13   26  12   24  23   46

Clark Kimberling and Reinhard Zumkeller, Rows n = 1..17 of triangle, flattened, first 13 rows from Clark Kimberling
Index entries for sequences that are permutations of the natural numbers

Cf. A232559, A232639, A232643, A000045.

Cf. A128588 (row lengths), A033484 (right edges), A257956 (row sums), A082560.

import Data.List.Ordered (member); import Data.List (sort)
a232642 n k = a232642_tabf !! (n-1) !! (k-1)
a232642_row n = a232642_tabf !! (n-1)
a232642_tabf = f a082560_tabf [] where
   f (xs:xss) zs = ys : f xss (sort (ys ++ zs)) where
     ys = [v | v <- xs, not $ member v zs]
a232642_list = concat a232642_tabf
-- Reinhard Zumkeller, May 14 2015

z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] + 2]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]]  (* A232642 *)
Table[Length[g1[n]], {n, 1, z}]  (* A000045 *)
Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232643 *)

Keyword tabf added, to bring out function g, by Reinhard Zumkeller, May 14 2015

cp's OEIS Frontend

A347738 A variant of the inventory sequence: record the number of terms >= 0 thus far in the sequence, then the number of terms >= 1 thus far, then the number of terms >= 2 thus far, and so on, until a zero is recorded; the inventory then starts again, recording the number of terms >= 0, etc.

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A169720 a(n) = (3*2^(n-1)-1)*(3*2^n-1).

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A083416 Add 1, double, add 1, double, etc.

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A089143 a(n) = 9*2^n - 6.

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A131110 A000012 * A133084.

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A204201 Triangle based on (0,1/3,1) averaging array.

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A362824 Array read by antidiagonals: T(n,k) is the number of k-tuples of involutions on [n] that pairwise commute.

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A169720 a(n) = (32^(n-1)-1)(3*2^n-1).

A097813 a(n) = 32^n - 2n - 2.