Joerg Arndt - cp's OEIS frontend

A370606 Primes p such that valuation(p-1,2) is a record.

2, 3, 5, 17, 97, 193, 257, 7681, 12289, 40961, 65537, 786433, 5767169, 7340033, 23068673, 104857601, 167772161, 469762049, 2013265921, 3221225473, 75161927681, 77309411329, 206158430209, 2061584302081, 2748779069441, 6597069766657, 39582418599937, 79164837199873

Offset: 1

Joerg Arndt, Feb 23 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..1770

Cf. A084924 (valuation(p+1,2) is a record).

Cf. A370607 (corresponding 2-valuations).

r=-1;forprime(p=2,10^12,v=valuation(p-1,2);if(v>r,print1(p,", ");r=v))

from itertools import count, islice
from sympy import isprime
def A370606_gen(): # generator of terms
    a = 1
    while True:
        for q in count(a,a):
            if isprime(q+1):
                yield q+1
                a = (q&-q)<<1
                break
A370606_list = list(islice(A370606_gen(),30)) # Chai Wah Wu, Feb 23 2024

a(21)-a(28) from Chai Wah Wu, Feb 23 2024

A370607 2-valuations of terms of A370606.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 12, 13, 16, 18, 19, 20, 21, 22, 25, 26, 27, 30, 31, 33, 36, 37, 39, 41, 42, 43, 44, 45, 46, 47, 48, 50, 52, 55, 56, 57, 59, 60, 63, 66, 67, 68, 69, 70, 71, 75, 77, 78, 81, 82, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 98, 99, 100, 103, 104

Offset: 1

Joerg Arndt, Feb 23 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..2000

r=-1;forprime(p=2,10^12,v=valuation(p-1,2);if(v>r,print1(v,", ");r=v));

from itertools import count, islice
from sympy import isprime
def A370607_gen(): # generator of terms
    a = 1
    while True:
        for q in count(a,a):
            if isprime(q+1):
                yield (b:=q&-q).bit_length()-1
                a = b<<1
                break
A370607_list = list(islice(A370607_gen(),30)) # Chai Wah Wu, Feb 23 2024

a(21)-a(67) from Chai Wah Wu, Feb 23 2024

A370199 a(n) is the number of odd polyominoes with n cells.

Original entry on oeis.org

0, 1, 3, 11, 35, 108, 380, 1348, 5014, 18223, 67634, 252849, 950346, 3602437, 13697333, 52293534, 200399576, 770410271, 2970369338, 11482572252, 44492417777, 172766286339, 672186167762, 2619985274260, 10228841840226, 39996338183554, 156612016049122

Offset: 1

Joerg Arndt and Hugo Pfoertner, Feb 11 2024

Whether a polyomino is "odd" is determined by the fact that the permutation defined by assigning the positions from row- or column-wise enumeration of its cells on a square grid is an odd permutation. See the description in the 'Ponder This' challenge for the exact definition.

The terms a(1)-a(10) were given in this description, and a(11)-a(20) were in the solution. The larger terms are results of the program that the user "uau" provided in the Mersenne forum.

IBM Research, Counting odd polyominoes, Ponder This, June 2022 - Challenge.
IBM Research, Counting odd polyominoes, Ponder This, June 2022 - Solution.
User uau, C program, post in mersenneforum, Jul 22, 2022.

Cf. A001168.

C
```
See the 'user uau' link.
```

a(4*n+2) = A001168(4*n+2)/2.

a(4*n+3) = A001168(4*n+3)/2.

A369770 a(n) is the maximal coefficient in the expansion of Product_{k=1..n} (1+k*x)^k.

Original entry on oeis.org

1, 1, 8, 387, 192832, 1348952000, 142641794707200, 271057611231886800384, 10679112895658933205816311808, 9866210328276596971591655994333069312, 238373589086269734817383263830485997977600000000, 166142193793387680126634957823414405189312889036472320000000

Offset: 0

Joerg Arndt, Jan 31 2024

nonn

Cf. A065048 (maximal coefficient in Product_{k=1..n} (1+k*x) ).

b:= proc(n) b(n):= `if`(n=0, 1, expand(b(n-1)*(1+n*x)^n)) end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..11);  # Alois P. Heinz, Jan 31 2024

a(n)=vecmax(Vec(prod(k=1,n,(1+k*x)^k)));
vector(20,n,a(n-1))

from collections import Counter
from math import comb
def A369770(n):
    c = {0:1}
    for k in range(1,n+1):
        d = Counter(c)
        for j in c:
            a = c[j]
            for i in range(1,k+1):
                d[j+i] += comb(k,i)*k**i*a
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

A369425 Expansion of Sum_{n>=1} x^(n(n+1)/2)((1+x)/(1-x))^n.

Original entry on oeis.org

1, 2, 3, 6, 10, 15, 24, 40, 64, 97, 144, 216, 328, 496, 737, 1074, 1546, 2218, 3186, 4578, 6555, 9318, 13130, 18358, 25530, 35398, 49002, 67735, 93416, 128392, 175704, 239352, 324680, 438888, 591672, 795993, 1068984, 1433032, 1917128, 2558648, 3405784, 4520872, 5984872, 7903256, 10413561

Offset: 1

Joerg Arndt, Jan 23 2024

Cf. A369424.

my(N=66,x='x+O('x^N)); Vec(sum(n=1,N,x^(n*(n+1)/2)*((1+x)/(1-x))^n))

A369424 Expansion of Sum_{n>=1} x^(n^2)*((1+x)/(1-x))^n.

Original entry on oeis.org

1, 2, 2, 3, 6, 10, 14, 18, 23, 32, 48, 72, 104, 144, 192, 249, 320, 416, 552, 744, 1008, 1360, 1816, 2392, 3105, 3978, 5050, 6386, 8082, 10266, 13098, 16770, 21506, 27562, 35226, 44819, 56702, 71298, 89134, 110898, 137502, 170146, 210382, 260178, 321982, 398786, 494190, 612466, 758623, 938480, 1158768

Offset: 1

Joerg Arndt, Jan 23 2024

Cf. A369425.

my(N=66,x='x+O('x^N)); Vec(sum(n=1,N,x^(n^2)*((1+x)/(1-x))^n))

A369377 a(n) is the number of elements p(j) < j (left displacements) in the n-th permutation in lexicographic order.

Original entry on oeis.org

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

Offset: 0

Joerg Arndt, Jan 22 2024

nonn

			In the following dots are used for zeros in the permutations and their inverses.
   n:    permutation    inv. perm.   a(n)
   0:    [ . 1 2 3 ]    [ . 1 2 3 ]   0
   1:    [ . 1 3 2 ]    [ . 1 3 2 ]   1
   2:    [ . 2 1 3 ]    [ . 2 1 3 ]   1
   3:    [ . 2 3 1 ]    [ . 3 1 2 ]   1
   4:    [ . 3 1 2 ]    [ . 2 3 1 ]   2
   5:    [ . 3 2 1 ]    [ . 3 2 1 ]   1
   6:    [ 1 . 2 3 ]    [ 1 . 2 3 ]   1
   7:    [ 1 . 3 2 ]    [ 1 . 3 2 ]   2
   8:    [ 1 2 . 3 ]    [ 2 . 1 3 ]   1
   9:    [ 1 2 3 . ]    [ 3 . 1 2 ]   1
  10:    [ 1 3 . 2 ]    [ 2 . 3 1 ]   2
  11:    [ 1 3 2 . ]    [ 3 . 2 1 ]   1
  12:    [ 2 . 1 3 ]    [ 1 2 . 3 ]   2
  13:    [ 2 . 3 1 ]    [ 1 3 . 2 ]   2
  14:    [ 2 1 . 3 ]    [ 2 1 . 3 ]   1
  15:    [ 2 1 3 . ]    [ 3 1 . 2 ]   1
  16:    [ 2 3 . 1 ]    [ 2 3 . 1 ]   2
  17:    [ 2 3 1 . ]    [ 3 2 . 1 ]   2
  18:    [ 3 . 1 2 ]    [ 1 2 3 . ]   3
  19:    [ 3 . 2 1 ]    [ 1 3 2 . ]   2
  20:    [ 3 1 . 2 ]    [ 2 1 3 . ]   2
  21:    [ 3 1 2 . ]    [ 3 1 2 . ]   1
  22:    [ 3 2 . 1 ]    [ 2 3 1 . ]   2
  23:    [ 3 2 1 . ]    [ 3 2 1 . ]   2

Joerg Arndt, Table of n, a(n) for n = 0..40319

Cf. A369376, A055093, A034968.

a(n) + A369376(n) = A055093(n).

A369376 a(n) is the number of elements p(j) > j (right displacements) in the n-th permutation in lexicographic order.

Original entry on oeis.org

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

Offset: 0

Joerg Arndt, Jan 22 2024

nonn

			In the following dots are used for zeros in the permutations and their inverses.
   n:    permutation    inv. perm.   a(n)
   0:    [ . 1 2 3 ]    [ . 1 2 3 ]   0
   1:    [ . 1 3 2 ]    [ . 1 3 2 ]   1
   2:    [ . 2 1 3 ]    [ . 2 1 3 ]   1
   3:    [ . 2 3 1 ]    [ . 3 1 2 ]   2
   4:    [ . 3 1 2 ]    [ . 2 3 1 ]   1
   5:    [ . 3 2 1 ]    [ . 3 2 1 ]   1
   6:    [ 1 . 2 3 ]    [ 1 . 2 3 ]   1
   7:    [ 1 . 3 2 ]    [ 1 . 3 2 ]   2
   8:    [ 1 2 . 3 ]    [ 2 . 1 3 ]   2
   9:    [ 1 2 3 . ]    [ 3 . 1 2 ]   3
  10:    [ 1 3 . 2 ]    [ 2 . 3 1 ]   2
  11:    [ 1 3 2 . ]    [ 3 . 2 1 ]   2
  12:    [ 2 . 1 3 ]    [ 1 2 . 3 ]   1
  13:    [ 2 . 3 1 ]    [ 1 3 . 2 ]   2
  14:    [ 2 1 . 3 ]    [ 2 1 . 3 ]   1
  15:    [ 2 1 3 . ]    [ 3 1 . 2 ]   2
  16:    [ 2 3 . 1 ]    [ 2 3 . 1 ]   2
  17:    [ 2 3 1 . ]    [ 3 2 . 1 ]   2
  18:    [ 3 . 1 2 ]    [ 1 2 3 . ]   1
  19:    [ 3 . 2 1 ]    [ 1 3 2 . ]   1
  20:    [ 3 1 . 2 ]    [ 2 1 3 . ]   1
  21:    [ 3 1 2 . ]    [ 3 1 2 . ]   1
  22:    [ 3 2 . 1 ]    [ 2 3 1 . ]   2
  23:    [ 3 2 1 . ]    [ 3 2 1 . ]   2

Joerg Arndt, Table of n, a(n) for n = 0..40319

Cf. A369377, A055093, A034968.

a(n) + A369377(n) = A055093(n).

A369322 a(n) is the number of weak ascent sequences (of any length) with n weak ascents.

Original entry on oeis.org

1, 1, 3, 20, 285, 8498, 521549, 65149296, 16446593964, 8354292354562, 8517018874559019, 17400156347544892896, 71175200852044807325678, 582639858848549658827324726, 9542182685892187892079287210803, 312611431819035281373960038697247872

Offset: 0

Joerg Arndt, Jan 20 2024

nonn

Column sums of A369321.

A weak ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(1)=0, d(k)>=0, and d(k) <= 1 + asc([d(1), d(2), ..., d(k-1)]) and asc(.) counts the weak ascents d(j) >= d(j-1) of its argument.

Alois P. Heinz, Table of n, a(n) for n = 0..50
Beata Benyi, Anders Claesson, Mark Dukes, Weak ascent sequences and related combinatorial structures, arXiv:2111.03159 [math.CO], (4-November-2021).

Cf. A369321, A336070.

b:= proc(n, i, t, k) option remember;
     `if`(k<0, 0,  `if`(n=0, `if`(k=0, 1, 0), add((d->
        b(n-1, j, t+d, k-d))(`if`(j>=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> add(b(j, -1$2, n), j=n..n*(n+1)/2):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 25 2024

b[n_, i_, t_, k_] := b[n, i, t, k] = If[k < 0, 0, If[n == 0, If[k == 0, 1, 0], Sum[Function[d, b[n-1, j, t+d, k-d]][If[j >= i, 1, 0]], {j, 0, t+1}]]];
a[n_] := Sum[b[j, -1, -1, n], {j, n, n*(n+1)/2}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 07 2025, after Alois P. Heinz *)

A369321 T(n,k) is the number of length-n weak ascent sequences (prefixed with a zero) with k weak ascents, triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 5, 0, 0, 0, 9, 14, 0, 0, 0, 5, 59, 42, 0, 0, 0, 1, 92, 342, 132, 0, 0, 0, 0, 75, 1073, 1863, 429, 0, 0, 0, 0, 35, 1882, 10145, 9794, 1430, 0, 0, 0, 0, 9, 2131, 31345, 84977, 50380, 4862, 0, 0, 0, 0, 1, 1661, 64395, 417220, 658423, 255606, 16796

Offset: 0

Joerg Arndt, Jan 20 2024

A weak ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(1)=0, d(k)>=0, and d(k) <= 1 + asc([d(1), d(2), ..., d(k-1)]) and asc(.) counts the weak ascents d(j) >= d(j-1) of its argument.

			1,
0, 1,
0, 0, 2,
0, 0, 1, 5,
0, 0, 0, 9, 14,
0, 0, 0, 5, 59,   42,
0, 0, 0, 1, 92,  342,    132,
0, 0, 0, 0, 75, 1073,   1863,     429,
0, 0, 0, 0, 35, 1882,  10145,    9794,     1430,
0, 0, 0, 0,  9, 2131,  31345,   84977,    50380,     4862,
0, 0, 0, 0,  1, 1661,  64395,  417220,   658423,   255606,    16796,
0, 0, 0, 0,  0,  912,  95477, 1370141,  4818426,  4835924,  1285453,   58786,
0, 0, 0, 0,  0,  350, 107002, 3291589, 23507705, 50477693, 34184279, 6428798, 208012,
...

Alois P. Heinz, Rows n = 0..140, flattened
Beata Benyi, Anders Claesson, and Mark Dukes, Weak ascent sequences and related combinatorial structures, arXiv:2111.03159 [math.CO], 2021-2022.

Cf. A000108 (main diagonal), A336070 (row sums), A369322 (column sums).
T(2n,n) gives A373115.
Cf. A137251.

b:= proc(n, i, t) option remember; expand(`if`(n=0, 1, add(
      b(n-1, j, t+`if`(j>=i, 1, 0))*`if`(j>=i, x, 1), j=0..t+1)))
    end:
T:= (n, k)-> coeff(b(n, -1$2), x, k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jan 23 2024

b[n_, i_, t_] := b[n, i, t] = Expand[If[n == 0, 1, Sum[
   b[n - 1, j, t + If[j >= i, 1, 0]]*If[j >= i, x, 1], {j, 0, t + 1}]]];
T[n_, k_] := Coefficient[b[n, -1, -1], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 24 2024, after Alois P. Heinz *)

\\ see formula (5) on page 18 of the Benyi/Claesson/Dukes reference
N=40;
M=matrix(N, N, r, c, -1);  \\ memoization
a(n, k)=
{
    if ( n==0 && k==0, return(1) );
    if ( k==0, return(0) );
    if ( n==0, return(0) );
    if ( M[n, k] != -1 , return( M[n, k] ) );
    my( s );
    s = sum( i=0, n, sum( j=0, k-1,
         (-1)^j * binomial(k-j, i) * binomial(i, j) * a( n-i, k-j-1 )) );
    M[n, k] = s;
    return( s );
}
\\ for (n=0, N, print1( sum(k=1, n, a(n, k)), ", "); ); \\ A336070
for (n=0, N, for(k=0, n, print1(a(n, k), ", "); ); print(); );
\\ Joerg Arndt, Jan 20 2024

T(n,n) = A000108(n) (number of length-n weak ascent sequences with maximal number of weak ascents).

cp's OEIS Frontend

User: Joerg Arndt

A370606 Primes p such that valuation(p-1,2) is a record.

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A370607 2-valuations of terms of A370606.

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A370199 a(n) is the number of odd polyominoes with n cells.

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C

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A369770 a(n) is the maximal coefficient in the expansion of Product_{k=1..n} (1+k*x)^k.

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A369425 Expansion of Sum_{n>=1} x^(n*(n+1)/2)*((1+x)/(1-x))^n.

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A369424 Expansion of Sum_{n>=1} x^(n^2)*((1+x)/(1-x))^n.

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A369377 a(n) is the number of elements p(j) < j (left displacements) in the n-th permutation in lexicographic order.

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A369376 a(n) is the number of elements p(j) > j (right displacements) in the n-th permutation in lexicographic order.

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A369322 a(n) is the number of weak ascent sequences (of any length) with n weak ascents.

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A369425 Expansion of Sum_{n>=1} x^(n(n+1)/2)((1+x)/(1-x))^n.