A003992 -id:A003992 - cp's OEIS frontend

A292741 Number A(n,k) of partitions of n with k sorts of part 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 2, 1, 4, 10, 11, 5, 2, 1, 5, 17, 31, 24, 7, 4, 1, 6, 26, 69, 95, 50, 11, 4, 1, 7, 37, 131, 278, 287, 104, 15, 7, 1, 8, 50, 223, 657, 1114, 865, 212, 22, 8, 1, 9, 65, 351, 1340, 3287, 4460, 2599, 431, 30, 12, 1, 10, 82, 521, 2459, 8042, 16439, 17844, 7804, 870, 42, 14

Offset: 0

Alois P. Heinz, Sep 22 2017

			A(1,3) = 3: 1a, 1b, 1c.
A(2,3) = 10: 2, 1a1a, 1a1b, 1a1c, 1b1a, 1b1b, 1b1c, 1c1a, 1c1b, 1c1c.
A(3,2) = 11: 3, 21a, 21b, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,     1,      1,      1, ...
  0,  1,   2,    3,     4,     5,      6,      7, ...
  1,  2,   5,   10,    17,    26,     37,     50, ...
  1,  3,  11,   31,    69,   131,    223,    351, ...
  2,  5,  24,   95,   278,   657,   1340,   2459, ...
  2,  7,  50,  287,  1114,  3287,   8042,  17215, ...
  4, 11, 104,  865,  4460, 16439,  48256, 120509, ...
  4, 15, 212, 2599, 17844, 82199, 289540, 843567, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-2 give: A002865, A000041, A090764.
Rows n=0-2 give: A000012, A001477, A002522, A071568.
Main diagonal gives A292462.
Cf. A003992, A004248, A009998, A051129, A292508, A292622, A292745.

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, k^n,
      add(b(n-i*j, i-1, k), j=0..iquo(n, i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[0, , ] = 1; b[n_, i_, k_] := b[n, i, k] = If[i < 2, k^n, Sum[b[n - i*j, i - 1, k], {j, 0, Quotient[n, i]}]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 19 2018, translated from Maple *)

G.f. of column k: 1/(1-k*x) * 1/Product_{j>=2} (1-x^j).

A(n,k) = Sum_{j=0..n} A002865(j) * k^(n-j).

A321391 Array read by antidiagonals: T(n,k) is the number of achiral rows of n colors using up to k colors.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 3, 4, 1, 0, 1, 5, 4, 9, 4, 1, 0, 1, 6, 5, 16, 9, 8, 1, 0, 1, 7, 6, 25, 16, 27, 8, 1, 0, 1, 8, 7, 36, 25, 64, 27, 16, 1, 0, 1, 9, 8, 49, 36, 125, 64, 81, 16, 1, 0, 1, 10, 9, 64, 49, 216, 125, 256, 81, 32, 1, 0

Offset: 0

Robert A. Russell, Nov 08 2018

The antidiagonals go from top-right to bottom-left.

			The array begins with T(0,0):
1 1  1   1    1     1     1      1      1      1       1       1 ...
0 1  2   3    4     5     6      7      8      9      10      11 ...
0 1  2   3    4     5     6      7      8      9      10      11 ...
0 1  4   9   16    25    36     49     64     81     100     121 ...
0 1  4   9   16    25    36     49     64     81     100     121 ...
0 1  8  27   64   125   216    343    512    729    1000    1331 ...
0 1  8  27   64   125   216    343    512    729    1000    1331 ...
0 1 16  81  256   625  1296   2401   4096   6561   10000   14641 ...
0 1 16  81  256   625  1296   2401   4096   6561   10000   14641 ...
0 1 32 243 1024  3125  7776  16807  32768  59049  100000  161051 ...
0 1 32 243 1024  3125  7776  16807  32768  59049  100000  161051 ...
0 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 ...
For T(3,3)=9, the rows are AAA, ABA, ACA, BAB, BBB, BCB, CAC, CBC, and CCC.

Columns 0-6 are A000007, A000012, A060546, A056449, A056450, A056451, A056452.

Cf. A003992 (oriented), A277504 (unoriented), A293500 (chiral).

Table[If[n>0, (n-k)^Ceiling[k/2], 1], {n, 0, 12}, {k, 0, n}] // Flatten

T(n,k) = [n==0] + [n>0] * k^ceiling(n/2).

The generating function for column k is (1+k*x) / (1-k*x^2).

A095884 Triangle read by rows: T(n,k) = (n-k)^k, n>=1, 1<=k<=n.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 4, 9, 8, 1, 0, 5, 16, 27, 16, 1, 0, 6, 25, 64, 81, 32, 1, 0, 7, 36, 125, 256, 243, 64, 1, 0, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0, 11, 100

Offset: 1

Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004

			0
1 0
2 1 0
3 4 1 0
4 9 8 1 0
5 16 27 16 1 0
6 25 64 81 32 1 0
7 36 125 256 243 64 1 0
8 49 216 625 1024 729 128 1 0
9 64 343 1296 3125 4096 2187 256 1 0

Cf. A051128, A003992, A009999.

A062275 Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 16, 3, 0, 0, 4, 72, 72, 4, 0, 0, 5, 256, 729, 256, 5, 0, 0, 6, 800, 5184, 5184, 800, 6, 0, 0, 7, 2304, 30375, 65536, 30375, 2304, 7, 0, 0, 8, 6272, 157464, 640000, 640000, 157464, 6272, 8, 0, 0, 9, 16384, 750141, 5308416, 9765625

Offset: 0

Henry Bottomley, Jul 02 2001

Here 0^0 is defined to be 1. - Wolfdieter Lang, May 27 2018

			A(3, 2) = 3^2 * 2^3 = 9*8 = 72.
The array A(n, k) begins:
n\k 0 1   2   3    4     5      6      7       8        9       10 ...
0:  1 0   0   0    0     0      0      0       0        0        0 ...
1:  0 1   2   3    4     5      6      7       8        9       10 ...
2:  0 2  16  72  256   800   2304   6272   16384    41472   102400 ...
3:  0 3  72 729 5184 30375 157464 750141 3359232 14348907 59049000 ...
...
The triangle T(n, k) begins:
n\k  0  1    2      3      4      5      6    7  8  9 ...
0:   1
1:   0  0
2:   0  1    0
3:   0  2    2      0
4:   0  3   16      3      0
5:   0  4   72     72      4      0
6:   0  5  256    729    256      5      0
7:   0  6  800   5184   5184    800      6    0
8:   0  7 2304  30375  65536  30375   2304    7  0
9:   0  8 6272 157464 640000 640000 157464 6272  8  0
... - _Wolfdieter Lang_, May 22 2018

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)

Columns and rows of A, or columns and diagonals of T, include A000007, A001477, A007758, A062074, A062075 etc. Diagonals of A include A062206, A051443, A051490. Sum of rows of T are A062817(n), for n >= 1

Cf. A055651, A055652, A303990.

{{1}}~Join~Table[(#^k k^#) &[n - k], {n, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 24 2018 *)

t1(n)=n-binomial(round(sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
a(n)=t1(n)^t2(n)*t2(n)^t1(n) \\ Eric Chen, Jun 09 2018

From Wolfdieter Lang, May 22 2018: (Start)
As a sequence: a(n) = A003992(n)*A004248(n).
As a triangle: T(n, k) = (n-k)^k * k^(n-k), for n >= 1 and k = 1..n. (End)

A099555 Triangle, read by rows, where T(n,k) = (n-floor(k/2))^k for k = 0..2*n - 1, with T(0,0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 8, 1, 1, 1, 4, 9, 27, 16, 32, 1, 1, 1, 5, 16, 64, 81, 243, 64, 128, 1, 1, 1, 6, 25, 125, 256, 1024, 729, 2187, 256, 512, 1, 1, 1, 7, 36, 216, 625, 3125, 4096, 16384, 6561, 19683, 1024, 2048, 1, 1, 1, 8, 49, 343, 1296, 7776, 15625, 78125, 65536

Offset: 0

Paul D. Hanna, Oct 22 2004

Row functions in y are given by: R_n(y) = Sum_{k=0..2n} (n-floor(k/2))^k*y^k/k!. Evaluated at y=1, the asymptotic behavior of the rows is given by: R_n(1) ~ c*r^n where c = (r+sqrt(r))/(1+2*sqrt(r)) = 0.8957126... and r = 2.0207473586... satisfies r = exp(1/sqrt(r)) -- see A099554 for the decimal expansion of this constant.

			The asymptotic behavior can be demonstrated at the 4th row function:
R_4(y) = 1 + 4*y + 9*y^2/2! + 27*y^3/3! + 16*y^4/4! + 32*y^5/5! + y^6/6! + y^7/7!;
R_4(1) = 14.93492... = (0.895684...)*r^4, where r = 2.0207473586...
Rows begin:
  [1]
  [1, 1],
  [1, 2,  1,   1],
  [1, 3,  4,   8,   1,    1],
  [1, 4,  9,  27,  16,   32,    1,     1],
  [1, 5, 16,  64,  81,  243,   64,   128,    1,     1],
  [1, 6, 25, 125, 256, 1024,  729,  2187,  256,   512,    1,    1],
  [1, 7, 36, 216, 625, 3125, 4096, 16384, 6561, 19683, 1024, 2048, 1, 1],
  ...
which can be derived from the square array A003992:
  [1, 0,  0,   0,   0,    0,     0, ...],
  [1, 1,  1,   1,   1,    1,     1, ...],
  [1, 2,  4,   8,  16,   32,    64, ...],
  [1, 3,  9,  27,  81,  243,   729, ...],
  [1, 4, 16,  64, 256, 1024,  4096, ...],
  [1, 5, 25, 125, 625, 3125, 15625, ...],
  ...
by shifting each column k down by floor(k/2) rows, and omitting the zeros coming from row 0 of A003992.

Cf. A003992, A099554, A099556.

seq(print(`if`(n=0, 1, seq((n - floor(k/2))^k, k=0..2*n-1))), n=0..10); # Georg Fischer, Nov 21 2024

PARI
```
T(n,k)=(n-k\2)^k
```

E.g.f.: ((1-x*cosh(sqrt(x)*y)) + sqrt(x)*sinh(sqrt(x)*y))/(1+x^2-2*x*cosh(sqrt(x)*y)).

Definition corrected by Georg Fischer, Nov 21 2024

A361475 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)/(k - 1), with k >= 2.

Original entry on oeis.org

0, 1, 0, 3, 1, 0, 7, 4, 1, 0, 15, 13, 5, 1, 0, 31, 40, 21, 6, 1, 0, 63, 121, 85, 31, 7, 1, 0, 127, 364, 341, 156, 43, 8, 1, 0, 255, 1093, 1365, 781, 259, 57, 9, 1, 0, 511, 3280, 5461, 3906, 1555, 400, 73, 10, 1, 0, 1023, 9841, 21845, 19531, 9331, 2801, 585, 91, 11, 1, 0

Offset: 0

Stefano Spezia, Mar 13 2023

			The array begins:
   0,  0,  0,   0,   0, ...
   1,  1,  1,   1,   1, ...
   3,  4,  5,   6,   7, ...
   7, 13, 21,  31,  43, ...
  15, 40, 85, 156, 259, ...
  ...

Cf. A003992, A361291 (k=2*n+1), A361476 (antidiagonal sums).

Cf. A000225 (k=2), A003462 (k=3), A002450 (k=4), A003463 (k=5), A003464 (k=6), A023000 (k=7), A023001 (k=8), A002452 (k=9), A002275 (k=10), A016123 (k=11).

A[n_,k_]:=(k^n-1)/(k-1); Flatten[Table[A[n-k+2,k],{n,0,10},{k,2,n+2}]]

E.g.f. of column k: exp(x)*(exp((k-1)*x) - 1)/(k - 1).
E.g.f. of column k: 2*exp((k+1)*x/2)*sinh((k-1)*x/2)/(k - 1).
A(n, k) = Sum_{i=0..n-1} k^i.

A361870 Array read by downward antidiagonals: A(n,k) is the number of nonequivalent 2-colorings of the cells of an n-dimensional hypercube with edges k cells long under action of symmetry.

Original entry on oeis.org

2, 2, 1, 2, 2, 1, 2, 3, 2, 1, 2, 6, 6, 2, 1, 2, 10, 102, 22, 2, 1, 2, 20, 8548, 2852288, 402, 2, 1, 2, 36, 4211744, 384307306807269376, 6296489398464125698304, 1228158, 2, 1

Offset: 0

Natalia L. Skirrow, May 28 2023

A(0,0) = 2 by the convention that 0^0 = 1, in spirit with A003992.
A(n,2) = A000616(n), because the Boolean functions' truth tables are n-dimensional edge-length-2 hypercubes, and are considered equivalent under action of input permutation (transposition of dimensions) and applications of NOT to inputs (reflection in dimensions) that compose for the hypercube's symmetry group.
A(n,3) is the number of transitions in an isotropic non-totalistic Life-like cellular automaton with an n-dimensional Moore neighborhood.
A(n,k) ~ 2^(k^n)/(n!*2^n) for large k where n >= 1, and large n where k >= 2, and converges from above. Proof: When computing it with the Pólya enumeration theorem (for each action, count colorings of cycles under it instead of cells, average result over actions), the asymptotic form describes the number of states contributed by the identity action. Over k, the values contributed by the other actions are at most O(2^(k^n/2)), so the proportion that they contribute may be made arbitrarily small by choosing a large enough n. Over n, there are no more than n!*2^n such non-identity actions, assuming that they are all of order 2 (as an upper bound). Each one may have no more than a hyperplane (2^k^(n-1)) of fixed points, which (if it is of order 2) will multiply its colorings by 2^(k^(n-1)/2). The ratio of the identity term to the others is at least O(2^k^n/(n!*2^n*2^(k^(n-1)/2))), and its base-2 logarithm, by Stirling's approximation, is O(k^n-n*log(n)-n-k^(n-1)/2), so the 2^(k^n) term will dominate.
A(1,k) is the only row >= 1 that is linear-recurrent over k (and has a rational generating function), all other nontrivial rows and columns grow faster than any linear-recurrent function.
A000120(A(n,k)) is eventually periodic over k if and only if n <= 2.

			n\k|0, 1,       2,                      3,                  4,       5,  6, 7, ...
---+------------------------------------------------------------------------------
 0 |2, 2,       2,                      2,                  2,       2,  2, 2, ...
 1 |1, 2,       3,                      6,                 10,      20, 36, ...
 2 |1, 2,       6,                    102,               8548, 4211744, ...
 3 |1, 2,      22,                2852288, 384307306807269376, ...
 4 |1, 2,     402, 6296489398464125698304, ...
 5 |1, 2, 1228158, ...
 6 |1, 2, ...
 7 |1, ...
...

Natalia L. Skirrow, Table of n, a(n) for n = 0..59
Adam P. Goucher, Combinatorics I, Mathematical Olympiad Dark Arts (2005), 8-11.
LifeWiki, Isotropic non-totalistic rule, Pólya enumeration theorem.
Natalia L. Skirrow, dimensional_INT_enumerator.py (unreduced version of Python program below, that also finds equations for fixed n).

Cf. A000616, A005418, A054247, A003992.

from functools import reduce
from itertools import accumulate
from math import isqrt,lcm,factorial as fact
tap=lambda f,*i:tuple(map(f,*i))
redumulate=lambda f,l,i=None: accumulate(l,f,initial=i)
expumulate=lambda f,l: lambda i: accumulate(range(l),lambda x,i: f(x),initial=i)
factorise=lambda m: tuple(filter(lambda n: not m%n,range(1,m//2+1)))+(m,)
def cycleLengths(dims,size):
    convert=(lambda m,i,a: (lambda d,n,i: ('('*bool(i)+str(size)+'+~(')*n+a+("//"+str(size**d))*bool(d)+('%'+str(size))*(dd for d,m in p[:i]))*fact(len(p)+~i)*2**dims+2**i*n for i,(e,n) in enumerate(p)))
    matrices=sorted((tuple((j,n>>i&1) for i,j in enumerate((lambda t: tuple(reduce(lambda e,n: e+(e>=n),t[i-1::-1],e)%dims for i,e in enumerate(t)))(tuple(a[0]%(dims-i) for i,a in enumerate(redumulate(lambda m,i: divmod(m[0],i),range(dims,1,-1),(m,0))))))) for m in range(fact(dims)) for n in range(2**dims)),key=matrindex) if dims else []
    exps=tap(lambda m: tap(matrindex,expumulate(lambda i: tap(lambda j: (lambda k,l: (k,l^j[1]))(*i[j[0]]),m),lcm(4,fact(dims)))(matrices[0])),matrices)
    lambdas=tap(lambda m: eval("lambda s: ["+','.join('s['+str(eval('+'.join(map(lambda i: convert(m,i,str(j)),range(dims)))))+']' for j in range(boardCells))+']'),matrices)
    test=list(range(1,boardCells+1))
    factors=tap(lambda e: factorise(e[1:].index(0)+1),exps)
    subperiods=tuple(tuple(sum(map(int._eq_,test,lambdas[exps[i][a]](test))) for a in f) for i,f in enumerate(factors))
    return((lambda t: tap(lambda t: reduce(lambda r,t: r+((t[0],t[1]-sum(i[1] for i in r if not t[0]%i[0])),),t,()),t))(tap(lambda a,b: tuple(zip(a,b)),factors,subperiods)))
specific=(lambda cycles: int(bool(cycles) and sum(2**sum(i[1]//i[0] for i in c) for c in cycles)//len(cycles)))
line=lambda k: (1,2)[k] if k<2 else 1<>1)
A054247=lambda n: (1,2,6)[n] if n<3 else 1<>1)+3**(~n&1)<<(n**2-5>>1)|1<<(n**2-5>>2)
cube=lambda k: (2**k**3+3*2**((k+1>>1)*k**2)+9*2**((k**2+1>>1)*k)+2**(k**3+1>>1)+6*2**(k**2*(k+1)>>1)+6*2**((k**2+3)//4*k)+6*2**((k**2+1>>1)*k+1>>1)+8*2**(k*(k**2+2)//3)+8*2**(k*(k**2+2)//3+1>>1))//48
tesseract=lambda k: (2**k**4+4*2**((k+1>>1)*k**3)+30*2**((k**2+1>>1)*k**2)+16*2**((k**3+1>>1)*k)+2**(k**4+1>>1)+12*2**(k**3*(k+1)>>1)+12*2**((k**2+3>>2)*k**2)+48*2**(((k**2+1>>1)*k+1>>1)*k)+12*2**((k**2+1>>1)*k**2+1>>1)+32*2**(k**2*(k**2+2)//3)+32*2**((k+1>>1)*k*(k**2+2)//3)+32*2**((k*(k**2-1)//3+k+1>>1)*k)+32*2**(k**2*(k**2+2)//3+1>>1)+12*2**(k**2*(k**2+1)>>1)+12*2**(k**4+3>>2)+48*2**(k*(k**3+k+2)>>2)+48*2**(k**4+7>>3))//384
nonequivalents=lambda n,k: (lambda k: 2,line,A054247,cube,tesseract)[n](k) if n<5 else 2**k if k<2 else specific(cycleLengths(n,k))
A002262=(lambda n: (lambda s: (lambda o: (o,s-o))(n-s*(s+1)//2))(isqrt((n<<3)+1)-1>>1))
print(tuple(map(lambda n: nonequivalents(*A002262(n)),range(28)))) # Natalia L. Skirrow, May 29 2023

Where an expression can be simplified by dividing a power of 2's coefficient by 2 and incrementing its exponent by 1, it is left as-is, so that the 2^ can be changed to c^ for general c-colorings.
A(n,2) = A000616(n).
A(0,k) = 2.
Herein, c(x) denotes the ceiling function.
A(1,k) = A005418(k+1) = (2^k + 2^c(k/2))/2.
A(2,k) = A054247(k) = (2^k^2 + 2*2^(k*(k+1)/2) + 2*2^(c(k/2)*k) + 2^c(k^2/2) + 2*2^c(k^2/4))/8.
A(3,k) = (2^k^3 + 3*2^(c(k/2)*k^2) + 9*2^(c(k^2/2)*k) + 2^c(k^3/2) + 6*2^(k^2*(k+1)/2) + 6*2^(c(k^2/4)*k) + 6*2^c(c(k^2/2)*k/2) + 8*2^(k*(k^2+2)/3) + 8*2^c(k*(k^2+2)/6))/48.
A(4,k) = (2^k^4 + 4*2^(c(k/2)*k^3) + 30*2^(c(k^2/2)*k^2) + 16*2^(c(k^3/2)*k) + 2^c(k^4/2) + 12*2^(k^3*(k+1)/2) + 12*2^(c(k^2/4)*k^2) + 48*2^(c(c(k^2/2)*k/2)*k) + 12*2^c((c(k^2/2)*k^2)/2) + 32*2^(k^2*(k^2+2)/3) + 32*2^(c(k/2)*k*(k^2+2)/3) + 32*2^(c((k*(k^2-1)/3+k)/2)*k) + 32*2^c((k^2*(k^2+2)/3)/2) + 12*2^(k^2*(k^2+1)/2) + 12*2^c(k^4/4) + 48*2^(k*(k^3+k+2)/4) + 48*2^c(k^4/8))/384.

A373341 Array read by ascending antidiagonals: A(n,k) is the number of acyclic de Bruijn sequences of order k and alphabet of size n, with k > 0.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 24, 216, 16, 1, 120, 331776, 10077696, 256, 1, 720, 24883200000, 12116574790945106558976, 1023490369077469249536, 65536, 1

Offset: 1

Stefano Spezia, Jun 01 2024

The 7th antidiagonal is too large to be inserted in Data.

			The array begins:
   1,      1,                       1, ...
   2,      4,                      16, ...
   6,    216,                10077696, ...
  24, 331776, 12116574790945106558976, ...
  ...

D. Condon, Yuxin Wang, and E. Yang, De Bruijn Polyominoes, arXiv:2405.18543 [math.CO], 2024. See page 5.
T. van Aardenne-Ehrenfest and N. G. de Brujin, Circuits and Trees in Oriented Linear Graphs. In: Simon Stevin 28 (1951), pp. 203-217.

Cf. A000012 (n=1), A000142 (k=1), A001146, A003992, A036740 (k=2), A373342 (antidiagonal sums), A373343 (cyclic).

A[n_,k_]:=(n!)^(n^(k-1)); Table[A[n-k+1,k],{n,6},{k,n}]//Flatten

A(n,k) = (n!)^(n^(k-1)).

A(2,n) = A001146(n-1).

A373343 Array read by ascending antidiagonals: A(n,k) is the number of cyclic de Bruijn sequences of order k and alphabet of size n, with k > 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 24, 2, 1, 24, 20736, 373248, 16, 1, 120, 995328000, 189321481108517289984, 12635683568857645056, 2048, 1

Offset: 1

Stefano Spezia, Jun 01 2024

The 7th antidiagonal is too large to be included in Data.

			The array begins:
  1,  1,      1,                    1, ...
  1,  1,      2,                   16, ...
  2, 24, 373248, 12635683568857645056, ...
  ...

D. Condon, Yuxin Wang, and E. Yang, De Bruijn Polyominoes, arXiv:2405.18543 [math.CO], 2024. See page 5.
T. van Aardenne-Ehrenfest and N. G. de Brujin, Circuits and Trees in Oriented Linear Graphs. In: Simon Stevin 28 (1951), pp. 203-217.

Cf. A000012 (n=1), A000142 (k=1), A003992, A016031 (n=2), A373341 (acyclic), A373344 (antidiagonal sums).

A[n_,k_]:=(n!)^(n^(k-1))/n^k; Table[A[n-k+1,k],{n,6},{k,n}]//Flatten

A(n,k) = (n!)^(n^(k-1))/n^k.

A(n,k) = A373341(n,k)/A003992(n,k).

A060508 Exponent of largest power of n < 2^n.

Original entry on oeis.org

2, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13

Offset: 2

Labos Elemer, Apr 11 2001

			For n = 10, 1000 < 1024 < 10000, so a(10) = 3; a(1) is indeterminate, since 1^j < 2^1.

Cf. A000079, A003992.

Table[Floor[n/Log[2, n]],{n,2,89}] (* Stefano Spezia, Dec 07 2021 *)

a(n) = floor(n/log(2, n)); n^a(n) <= 2^n <= n^(1+a(n)).

cp's OEIS Frontend

A292741 Number A(n,k) of partitions of n with k sorts of part 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A361475 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)/(k - 1), with k >= 2.

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A373341 Array read by ascending antidiagonals: A(n,k) is the number of acyclic de Bruijn sequences of order k and alphabet of size n, with k > 0.

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