A093659 -id:A093659 - cp's OEIS frontend

A093677 Self-convolution of A093659, which is the first column of triangle A093658.

1, 2, 3, 6, 7, 10, 14, 24, 27, 30, 42, 44, 58, 68, 104, 120, 147, 150, 230, 164, 254, 256, 436, 240, 346, 356, 604, 408, 680, 696, 1272, 720, 987, 990, 1694, 1004, 1786, 1720, 3316, 1080, 1886, 1888, 3696, 1872, 3772, 3576, 7608, 1560, 2530, 2540, 4828, 2592

Offset: 1

Paul D. Hanna, Apr 09 2004

nonn

A093659 consists only of factorial numbers, where A093659(2^n) = n!.

Cf. A093658, A093659.

a(2^n) = (n+1)!

A093658 Lower triangular matrix, read by rows, defined as the convergent of the concatenation of matrices using the iteration: M(n+1) = [[M(n),0*M(n)],[M(n)^2,M(n)]], with M(0) = [1].

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 6, 2, 2, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 6, 2, 2, 1, 0, 0, 0, 0, 2, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 6, 2, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 1, 1

Offset: 1

Paul D. Hanna, Apr 08 2004

Related to factorials, the incomplete gamma function (A010842) and the total number of arrangements of sets (A000522).
First column forms A093659, where A093659(2^n) = n! for n>=0.
Row sums form A093660, where A093660(2^n) = A000522(n) for n>=0.
Partial sums of the row sums form A093661, where A093661(2^n) = A010842(n) for n>=0.

			Let M(n) be the lower triangular matrix formed from the first 2^n rows.
To generate M(3) from M(2), take the matrix square of M(2):
[1,0,0,0]^2=[1,0,0,0]
[1,1,0,0]...[2,1,0,0]
[1,0,1,0]...[2,0,1,0]
[2,1,1,1]...[6,2,2,1]
and append M(2)^2 to the bottom left corner and M(2) to the bottom right:
[1],
[1,1],
[1,0,1],
[2,1,1,1],
.........
[1,0,0,0],[1],
[2,1,0,0],[1,1],
[2,0,1,0],[1,0,1],
[6,2,2,1],[2,1,1,1].
Repeating this process converges to triangle A093658.

Cf. A000522, A010842, A093655, A093662.

T(2^n, 1) = n! for n>=0.

A269221 Factorial of the sum of decimal digits of n.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 24, 120, 720, 5040

Offset: 0

M. F. Hasler, Mar 15 2016

Sequence A093659 is the binary (base 2) version.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000142, A007953, A113588, A166350, A093659.

Table[Total[IntegerDigits[n]]!,{n,0,50}] (* Harvey P. Dale, Dec 19 2021 *)

PARI
```
A269221(n)=sumdigits(n)!
```

a(n) = A000142(A007953(n)).

A139329 a(n) = (factorial of the number of 0's in the binary expansion of n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 6, 2, 2, 1, 2, 1, 1, 1, 24, 6, 6, 2, 6, 2, 2, 1, 6, 2, 2, 1, 2, 1, 1, 1, 120, 24, 24, 6, 24, 6, 6, 2, 24, 6, 6, 2, 6, 2, 2, 1, 24, 6, 6, 2, 6, 2, 2, 1, 6, 2, 2, 1, 2, 1, 1, 1, 720, 120, 120, 24, 120, 24, 24, 6, 120, 24, 24, 6, 24, 6, 6, 2, 120, 24, 24, 6, 24, 6, 6, 2, 24, 6, 6, 2, 6, 2, 2, 1, 120

Offset: 0

Max Sills, Apr 13 2008

Number of permutation symmetries of the 0's in the binary expansion of n. Consider the symmetric group that permutes floor(log_2(n)) elements acting on the 0's.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n.

Cf. A000142, A023416, A080791, A093659 (factorial of the number of 1's in binary expansion of n).

a[n_] := DigitCount[n, 2, 0]!; Array[a, 100, 0] (* Amiram Eldar, Jul 29 2023 *)

a(n) = (#binary(n)-hammingweight(n))!; \\ Michel Marcus, Oct 24 2017

a(n) = A000142(A023416(n)) = A000142(A080791(n)). - Antti Karttunen, Oct 24 2017

Locations of the name and the formula changed, more terms from Antti Karttunen, Oct 24 2017

A269223 Factorial of the sum of digits of n in base 3.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 2, 6, 24, 1, 2, 6, 2, 6, 24, 6, 24, 120, 2, 6, 24, 6, 24, 120, 24, 120, 720, 1, 2, 6, 2, 6, 24, 6, 24, 120, 2, 6, 24, 6, 24, 120, 24, 120, 720, 6, 24, 120, 24, 120, 720, 120, 720, 5040, 2, 6, 24, 6, 24, 120, 24, 120, 720, 6, 24, 120, 24, 120, 720, 120, 720

Offset: 0

M. F. Hasler, Mar 15 2016

Sequence A093659 is the binary (base 2) and sequence A269221 is the decimal (base 10) version.

Cf. A000142, A053735, A166350, A093659.

Table[Total[IntegerDigits[n, 3]]!, {n, 0, 70}] (* Michael De Vlieger, Mar 15 2016 *)

A269223(n)=sumdigits(n,3)! \\ sumdigits(.,3) requires version > 2.7.1; see A053735 for a substitute.

a(n) = vecsum(digits(n,3))!; \\ Michel Marcus, Mar 15 2016

a(n) = A000142(A053735(n)).

A136494 Number of permutation symmetries in the binary expansion of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 6, 6, 4, 4, 6, 4, 6, 6, 24, 24, 12, 12, 12, 12, 12, 12, 24, 12, 12, 12, 24, 12, 24, 24, 120, 120, 48, 48, 36, 48, 36, 36, 48, 48, 36, 36, 48, 36, 48, 48, 120, 48, 36, 36, 48, 36, 48, 48, 120, 36, 48, 48, 120, 48, 120, 120, 720, 720

Offset: 0

Max Sills, Apr 13 2008

			a(14) = 6 because there are 3! permutation symmetries of 1's * the 0! permutation symmetries of 0's.

Andrew Howroyd, Table of n, a(n) for n = 0..1023

Cf. A000120, A080791, A093659, A139329.

a[n_] := Times @@ (DigitCount[n, 2]!); Array[a, 65, 0] (* Amiram Eldar, Jul 29 2023 *)

a(n) = {if(n==0, 1, my(w=hammingweight(n)); w!*(1+logint(n,2)-w)!)} \\ Andrew Howroyd, Jan 12 2020

a(n) = A093659(n) * A139329(n).

a(n) = A000120(n)! * A080791(n)!.

Terms a(32) and beyond from Andrew Howroyd, Jan 12 2020

A188064 Partial sums of wt(n)! where wt(n) is the Hamming weight of n (A000120).

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 16, 17, 19, 21, 27, 29, 35, 41, 65, 66, 68, 70, 76, 78, 84, 90, 114, 116, 122, 128, 152, 158, 182, 206, 326, 327, 329, 331, 337, 339, 345, 351, 375, 377, 383, 389, 413, 419, 443, 467, 587, 589, 595, 601, 625, 631, 655, 679, 799, 805, 829, 853, 973, 997, 1117

Offset: 0

Joerg Arndt, Mar 20 2011

nonn

Partial sums of A093659, partial sums of the factorials of A000120.

A000522 is a subsequence: A000522(n)=a(2^n-1).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A093659, A000120, A000522.

FoldList[Plus, 0!, Table[(Plus @@ IntegerDigits[n, 2])!, {n, 1, 70}]] (* From Olivier Gérard, Mar 23 2011 *)
Accumulate[DigitCount[Range[0,70],2,1]!] (* Harvey P. Dale, Jun 26 2013 *)

bitcount(x)=
{ /* Return Hamming weight of x */
    local(p);  p = 0;
    while ( x, p+=bitand(x, 1); x>>=1; );
    return( p );
}
N=65; /* that many terms */
f=vector(N,n,bitcount(n-1)!); /* factorials of Hamming weights */
s=vector(N); s[1]=f[1]; /* for cumulative sums */
for (n=2,N,s[n]=s[n-1]+f[n]); /* sum up */
s /* show terms */ /* Joerg Arndt, Mar 20 2011 */

a(n)=sum(k=0,n,wt(k)!) where wt(k) is the Hamming weight of k.

A269224 Factorial of the sum of digits of n in base 4.

Original entry on oeis.org

1, 1, 2, 6, 1, 2, 6, 24, 2, 6, 24, 120, 6, 24, 120, 720, 1, 2, 6, 24, 2, 6, 24, 120, 6, 24, 120, 720, 24, 120, 720, 5040, 2, 6, 24, 120, 6, 24, 120, 720, 24, 120, 720, 5040, 120, 720, 5040, 40320, 6, 24, 120, 720, 24, 120, 720, 5040, 120, 720, 5040, 40320, 720, 5040, 40320

Offset: 0

M. F. Hasler, Mar 15 2016

See sequences A093659, A269223 and A269221 for the base 2, base 3 and base 10 analog.

Cf. A000142, A053737, A166350, A093659, A269223, A269221.

Table[Total[IntegerDigits[n, 4]]!, {n, 0, 62}] (* Michael De Vlieger, Mar 15 2016 *)

A269224(n)=sumdigits(n,4)! \\ sumdigits(.,4) requires version >= 2.7; see A053737 for a substitute.

a(n) = vecsum(digits(n,4))!; \\ Michel Marcus, Mar 15 2016

a(n) = A000142(A053737(n)).

A357534 Number of compositions (ordered partitions) of n into two or more powers of 2.

Original entry on oeis.org

0, 0, 1, 3, 5, 10, 18, 31, 55, 98, 174, 306, 542, 956, 1690, 2983, 5271, 9310, 16448, 29050, 51318, 90644, 160118, 282826, 499590, 882468, 1558798, 2753448, 4863696, 8591212, 15175514, 26805983, 47350055, 83639030, 147739848, 260967362, 460972286, 814260544, 1438308328

Offset: 0

Ilya Gutkovskiy, Oct 02 2022

nonn

Cf. A023359, A093659, A209229, A357476.

b:= proc(n) option remember;
      `if`(n=0, 1, add(b(n-2^i), i=0..ilog2(n)))
    end:
a:= n-> b(n)-`if`(2^ilog2(n)=n, 1, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 02 2022

b[n_] := b[n] = If[n == 0, 1, Sum[b[n - 2^i], {i, 0, Floor@ Log2[n]}]];
a[n_] :=  b[n] - If[2^Floor@Log2[n] == n, 1, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 26 2022, after Alois P. Heinz *)

a(n) = A023359(n) - A209229(n) for n > 0.

A384955 a(n) is the multinomial coefficient of the digits of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005

Offset: 0

Stefano Spezia, Jun 13 2025

			a(35) = (3+5)!/(3!*5!) = 40320/(6*120) = 56;
a(1512) = (1+5+1+2)!/(1!*5!*1!*2!) = 362880/(120*2) = 1512.

Stefano Spezia, Table of n, a(n) for n = 0..10000

Cf. A037124, A066459, A093659, A269221.

a:= n-> (l-> combinat[multinomial](add(i,i=l), l[]))(convert(n, base, 10)):
seq(a(n), n=0..69);  # Alois P. Heinz, Jun 15 2025

a[n_]:=Multinomial @@IntegerDigits[n]; Array[a,70,0]

from math import factorial, prod
def a(n): return factorial(sum(digits:=list(map(int, str(n))))) // prod(factorial(x) for x in digits)
print([a(n) for n in range(70)]) # David Radcliffe, Jun 15 2025

a(n) = A269221(n)/A066459(n).
a(n) = 1 iff n is equal to 0 or has only one nonzero digit (cf. A037124).
Conjecture: a(n) = n iff n = 1 or n = 1512.

cp's OEIS Frontend

A093677 Self-convolution of A093659, which is the first column of triangle A093658.

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A093658 Lower triangular matrix, read by rows, defined as the convergent of the concatenation of matrices using the iteration: M(n+1) = [[M(n),0*M(n)],[M(n)^2,M(n)]], with M(0) = [1].

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A269221 Factorial of the sum of decimal digits of n.

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A139329 a(n) = (factorial of the number of 0's in the binary expansion of n).

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A269223 Factorial of the sum of digits of n in base 3.

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A136494 Number of permutation symmetries in the binary expansion of n.

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A188064 Partial sums of wt(n)! where wt(n) is the Hamming weight of n (A000120).

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A269224 Factorial of the sum of digits of n in base 4.

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