Tom Edgar - cp's OEIS frontend

A331128 Number of ways to write n as n = h_11! + h_22! + ... + h_kk! where 0 <= h_i <= 2i for all i.

1, 1, 2, 1, 2, 1, 3, 2, 4, 2, 3, 1, 3, 2, 4, 2, 3, 1, 3, 2, 4, 2, 3, 1, 4, 3, 6, 3, 5, 2, 6, 4, 8, 4, 6, 2, 6, 4, 8, 4, 6, 2, 5, 3, 6, 3, 4, 1, 4, 3, 6, 3, 5, 2, 6, 4, 8, 4, 6, 2, 6, 4, 8, 4, 6, 2, 5, 3, 6, 3, 4, 1, 4, 3, 6, 3, 5, 2, 6, 4, 8, 4, 6, 2, 6

Offset: 0

Tom Edgar, Jan 10 2020

We call such a partition of n a hyperfactorial partition as these are in some sense analogous to hyperbinary partitions (A002487).

This sequence also counts the possible carry sequences when adding two numbers that sum to n using the traditional algorithm for adding two factorial-base representations.

			There are 6 ways to write n = 705 in the desired fashion:
705 = 1*1! + 1*2! + 1*3! + 4*4! + 5*5!;
705 = 1*1! + 1*2! + 5*3! + 3*4! + 5*5!;
705 = 1*1! + 4*2! + 4*3! + 3*4! + 5*5!;
705 = 1*1! + 4*2! + 4*3! + 8*4! + 4*5!;
705 = 1*1! + 1*2! + 5*3! + 8*4! + 4*5!;
705 = 1*1! + 4*2! + 0*3! + 4*4! + 5*5!.
Thus a(705) = 6.

Cf. A108731, A084558, A099563.

def factoradic(n):
    if n==0:
        return [0]
    L=[]
    i=2
    while n!=0:
        dm=divmod(n,i)
        L.append(dm[1])
        n=dm[0]
        i+=1
    return L
@cached_function
def carryseq(n):
    if n<0:
        return 0
    elif n==0:
        return 1
    else:
        L=factoradic(n)
        k=len(L)
        nk=L[-1]
        return carryseq(n-nk*factorial(k))+carryseq((nk+1)*factorial(k)-n-2)

a(n) = 0 if n<0; a(0) = 1; a(n) = a(n-n_k*k!) + a((n_k+1)*k!-n-2) for n > 0, where n_k is the most significant digit of the factorial-base representation of n (i.e., n_k = A099563(k)).

A273867 Degrees of fundamental invariants of Weyl group W(E_6).

Original entry on oeis.org

2, 5, 6, 8, 9, 12

Offset: 1

Tom Edgar, Jun 01 2016

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 141.

Cf. A005795, A005796.

A273338 Alternate version of A273317 with rows sorted in ascending order.

Original entry on oeis.org

2, 3, 6, 4, 7, 12, 42, 5, 8, 13, 20, 43, 56, 156, 1806, 6, 9, 14, 21, 30, 44, 57, 72, 157, 182, 420, 1807, 1892, 3192, 24492, 3263442, 7, 10, 15, 22, 31, 42, 45, 58, 73, 90, 158, 183, 210, 421, 462, 930, 1808, 1893, 1980, 3193, 3306, 5256, 24493, 24806, 33306, 176820, 3263443, 3267056, 3581556, 10192056

Offset: 0

Tom Edgar, May 20 2016

See the entry A273317 for more details.

Row n has 2^n entries.

			The triangle begins:
2,
3, 6,
4, 7, 12, 42,
5, 8, 13, 20, 43, 56, 156, 1806,
6, 9, 14, 21, 30, 44, 57, 72, 157, 182, 420, 1807, 1892, 3192, 24492, 3263442,

Cf. A273317.

A273317 Irregular table read by rows: T(0,0) = 2 and T(n,2k) = T(n-1,k)+1, T(n,2k+1) = T(n-1,k)*(T(n-1,k)+1) for 0 <= k < 2^(n-1).

Original entry on oeis.org

2, 3, 6, 4, 12, 7, 42, 5, 20, 13, 156, 8, 56, 43, 1806, 6, 30, 21, 420, 14, 182, 157, 24492, 9, 72, 57, 3192, 44, 1892, 1807, 3263442, 7, 42, 31, 930, 22, 462, 421, 176820, 15, 210, 183, 33306, 158, 24806, 24493, 599882556, 10, 90, 73, 5256, 58, 3306, 3193, 10192056

Offset: 0

Tom Edgar, May 19 2016

The first entry in row n is n+2.
The second entry in row n (n>0) is the A002378(n+2).
No number appears twice in the same row, so row n has 2^n distinct terms.
Row n and row n+1 have no elements in common.
There are infinitely many mutually disjoint rows; this fact can be used to show that the harmonic series diverges since the sum of reciprocals of entries in every row equals 1/2. This also allows a proof that every positive rational number is the sum of a finite number of distinct Egyptian fractions.
Let S(0) = {2} and for n>=1 define S(n) = {a | a = c+1 or a = c*(c+1) for some c in S(n-1)}; then row n contains the elements of S(n).

			The table begins:
2,
3, 6,
4, 12, 7, 42,
5, 20, 13, 156, 8, 56, 43, 1806,
6, 30, 21, 420, 14, 182, 157, 24492, 9, 72, 57, 3192, 44, 1892, 1807, 3263442,

Steven J. Kifowit, More Proofs of Divergence of the Harmonic Series.
J. C. Owings, Jr., Another Proof of the Egyptian Fraction Theorem, Amer. Math. Monthly, 75(7) (1968), 777-778.

Cf. A002378, A002061.

A273317 := proc(n,j)
    if n = 0 then
        2 ;
    elif type(j,'even') then
        1+procname(n-1,j/2) ;
    else
        procname(n-1,floor(j/2)) ;
        %*(%+1) ;
    end if;
end proc: # R. J. Mathar, May 20 2016

def T(n,j):
    if n==0:
        return 2
    if j%2==0:
        return T(n-1,floor(j/2))+1
    else:
        t=T(n-1,floor(j/2))
        return t*(t+1)
S=[[T(n,k) for k in [0..2^n-1]] for n in [0..10]]
[x for sublist in S for x in sublist]

T(0,0) = 2, and T(n,2k) = T(n-1,k)+1, T(n,2k+1) = T(n-1,k)*(T(n-1,k)+1) for 0 <= k < 2^(n-1).

Sum_{a in row(n)} 1/a = 1/2 for all n.

A273183 The H_3-Eulerian numbers.

Original entry on oeis.org

1, 59, 59, 1

Offset: 0

Tom Edgar, May 17 2016

a(n) is the number of elements of the Weyl group H_3 that have a descent set of size n.

Sum_{n=0..3} a(n) = 120, which is the order of the group H_3.

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Table 11.5.

Cf. A008292, A060187, A066094, A273000, A273035, A273036, A273181.

A273181 The F_4-Eulerian numbers.

Original entry on oeis.org

1, 236, 678, 236, 1

Offset: 0

Tom Edgar, May 17 2016

a(n) is the number of elements of the Weyl group F_4 that have a descent set of size n.

Sum_{n=0..4} a(n) = 1152, which is the order of the group F_4.

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Table 11.5.

Cf. A008292, A060187, A066094, A273000, A273035, A273036.

A273184 The H_4-Eulerian numbers.

Original entry on oeis.org

1, 2636, 9126, 2636, 1

Offset: 0

Tom Edgar, May 17 2016

a(n) is the number of elements of the Weyl group H_4 that have a descent set of size n.

Sum_{n=0..4} a(n) = 14400, which is the order of the group H_4.

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Table 11.5.

Cf. A008292, A060187, A066094, A273000, A273035, A273036, A273181, A273183.

A273035 The E_7-Eulerian numbers.

Original entry on oeis.org

1, 17635, 309969, 1123915, 1123915, 309969, 17635, 1

Offset: 0

Tom Edgar, May 13 2016

a(n) is the number of elements of the Weyl group E_7 that have a descent set of size n.

Sum_{n=0..7} a(n) = 2903040, which is the order of the group E_7.

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Table 11.5.

Cf. A008292, A060187, A066094, A273000.

A273036 The E_8-Eulerian numbers.

Original entry on oeis.org

1, 881752, 28336348, 169022824, 300247750, 169022824, 28336348, 881752, 1

Offset: 0

Tom Edgar, May 13 2016

a(n) is the number of elements of the Weyl group E_8 that have a descent set of size n.

Sum_{n=0..8} a(n) = 696729600, which is the order of the group E_8.

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Table 11.5.

Cf. A008292, A060187, A066094, A273000, A273035.

A273000 The E_6-Eulerian numbers.

Original entry on oeis.org

1, 1272, 12183, 24928, 12183, 1272, 1

Offset: 0

Tom Edgar, May 12 2016

a(n) is the number of elements of the Weyl group E_6 that have a descent set of size n.

Sum_{n=0..6} a(n) = 51840, which is the order of the group E_6.

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Table 11.5.

Cf. A008292, A060187, A066094.

cp's OEIS Frontend

User: Tom Edgar

A331128 Number of ways to write n as n = h_1*1! + h_2*2! + ... + h_k*k! where 0 <= h_i <= 2*i for all i.

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A273867 Degrees of fundamental invariants of Weyl group W(E_6).

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A273338 Alternate version of A273317 with rows sorted in ascending order.

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A273317 Irregular table read by rows: T(0,0) = 2 and T(n,2k) = T(n-1,k)+1, T(n,2k+1) = T(n-1,k)*(T(n-1,k)+1) for 0 <= k < 2^(n-1).

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A273183 The H_3-Eulerian numbers.

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A273181 The F_4-Eulerian numbers.

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A273184 The H_4-Eulerian numbers.

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A273035 The E_7-Eulerian numbers.

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A273036 The E_8-Eulerian numbers.

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A273000 The E_6-Eulerian numbers.

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A331128 Number of ways to write n as n = h_11! + h_22! + ... + h_kk! where 0 <= h_i <= 2i for all i.