A166350 -id:A166350 - cp's OEIS frontend

A047916 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk! if k|n else 0 (1<=k<=n).

1, 2, 2, 6, 0, 6, 8, 8, 0, 24, 20, 0, 0, 0, 120, 12, 36, 48, 0, 0, 720, 42, 0, 0, 0, 0, 0, 5040, 32, 64, 0, 384, 0, 0, 0, 40320, 54, 0, 324, 0, 0, 0, 0, 0, 362880, 40, 200, 0, 0, 3840, 0, 0, 0, 0, 3628800, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39916800, 48, 144

Offset: 1

N. J. A. Sloane

T(n,k) = A054523(n,k) * A010766(n,k)^A002260(n,k) * A166350(n,k). - Reinhard Zumkeller, Jan 20 2014

			1; 2,2; 6,0,6; 8,8,0,24; 20,0,0,0,120; 12,36,48,0,0,720; ...

J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
N. J. A. Sloane, Notes on A002618, A002619, etc.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]

A064649 gives the row sums.
Cf. A002618 (left edge), A000142 (right edge), A049820 (zeros per row), A000005 (nonzeros per row).
See also A247917, A047918, A047919.

import Data.List (zipWith4)
a047916 n k = a047916_tabl !! (n-1) !! (k-1)
a047916_row n = a047916_tabl !! (n-1)
a047916_tabl = zipWith4 (zipWith4 (\x u v w -> x * v ^ u * w))
               a054523_tabl a002260_tabl a010766_tabl a166350_tabl
-- Reinhard Zumkeller, Jan 20 2014

a[n_, k_] := If[Divisible[n, k], EulerPhi[n/k]*(n/k)^k*k!, 0]; Flatten[ Table[ a[n, k], {n, 1, 12}, {k, 1, n}]] (* Jean-François Alcover, May 04 2012 *)

a(n,k)=if(n%k, 0, eulerphi(n/k)*(n/k)^k*k!) \\ Charles R Greathouse IV, Feb 09 2017

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)).

A288777 Triangle read by rows in which column k lists the positive multiples of the factorial of k, with 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 4, 6, 4, 6, 12, 24, 5, 8, 18, 48, 120, 6, 10, 24, 72, 240, 720, 7, 12, 30, 96, 360, 1440, 5040, 8, 14, 36, 120, 480, 2160, 10080, 40320, 9, 16, 42, 144, 600, 2880, 15120, 80640, 362880, 10, 18, 48, 168, 720, 3600, 20160, 120960, 725760, 3628800, 11, 20, 54, 192, 840, 4320, 25200, 161280, 1088640

Offset: 1

Omar E. Pol, Jun 15 2017

T(n,k) is the number of k-digit numbers in base n+1 with distinct positive digits that form an integer interval when sorted.
T(9,k) is also the number of numbers with k digits in A288528.
The number of terms in A288528 is also A014145(9) = 462331, the same as the sum of the 9th row of this triangle.
Removing the left column of A137267 and of A137948 then this triangle appears in both cases.

			Triangle begins:
   1;
   2,  2;
   3,  4,  6;
   4,  6, 12,  24;
   5,  8, 18,  48, 120;
   6, 10, 24,  72, 240,  720;
   7, 12, 30,  96, 360, 1440,  5040;
   8, 14, 36, 120, 480, 2160, 10080,  40320;
   9, 16, 42, 144, 600, 2880, 15120,  80640,  362880;
  10, 18, 48, 168, 720, 3600, 20160, 120960,  725760, 3628800;
  11, 20, 54, 192, 840, 4320, 25200, 161280, 1088640, 7257600, 39916800;
  ...
For n = 9 and k = 2: T(9,2) is the number of numbers with two digits in A288528.
For n = 9 the row sum is 9 + 16 + 42 + 144 + 600 + 2880 + 15120 + 80640 + 362880 = 462331, the same as A014145(9) and also the same as the number of terms in A288528.

Right border gives A000142, n>=1.
Middle diagonal gives A001563, n>=1.
Row sums give A014145, n>=1.
Column 1..4: A000027, A005843, A008588, A008606.
Cf. A007489, A000142, A004736, A137267, A137948, A166350, A288528, A288778, A288780.

Table[(n - k + 1) k!, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Jun 15 2017 *)

T(n,k) = (n-k+1)*k! = (n-k+1)*A000142(k) = A004736(n,k)*A166350(n,k).
T(n,k) = Sum_{j=1..n} A166350(j,k).
T(n,k) =  A288778(n,k) + A000142(k-1).

A014454 Sum_{1<=k

Original entry on oeis.org

0, 1, 2, 5, 6, 21, 22, 73, 210, 1693, 1694, 2097, 2098, 12997, 21468, 174169, 174170, 1986237, 1986238, 10178833, 16875654, 246551437, 246551438, 2032266537, 3767596738, 45445808989, 260705796192, 2932954933753, 2932954933754, 5496591783573, 5496591783574

Offset: 1

Reinhard Zumkeller, Feb 12 2002

nonn

			a(5) = gcd(1!,5!)+gcd(2!,3*4*5)+gcd(3!,4*5)+gcd(4!,5)=1+2+2+1 = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..500

Cf. A000142.

Cf. A166350.

a014454 n = sum $ zipWith gcd kfs $ map (div nf) kfs
   where (nf:kfs) = reverse $ a166350_row n
-- Reinhard Zumkeller, Nov 11 2013

a(n) = sum(k=1, n-1, gcd(k!, n!/k!)); \\ Michel Marcus, Aug 04 2013

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)).

A288778 Triangle read by rows (1<=k<=n): T(n,k) = (n-k+1)*k! - (k-1)!

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 3, 5, 10, 18, 4, 7, 16, 42, 96, 5, 9, 22, 66, 216, 600, 6, 11, 28, 90, 336, 1320, 4320, 7, 13, 34, 114, 456, 2040, 9360, 35280, 8, 15, 40, 138, 576, 2760, 14400, 75600, 322560, 9, 17, 46, 162, 696, 3480, 19440, 115920, 685440, 3265920, 10, 19, 52, 186, 816, 4200, 24480, 156240, 1048320, 6894720, 36288000

Offset: 1

Omar E. Pol, Jun 15 2017

T(10,k) is also the number of positive integers with k digits in the sequence A215014. See Franklin T. Adams-Watters's comment in that entry. See also A288780.

			Triangle begins:
0;
1,   1;
2,   3,  4;
3,   5, 10,  18;
4,   7, 16,  42,  96;
5,   9, 22,  66, 216,  600;
6,  11, 28,  90, 336, 1320,  4320;
7,  13, 34, 114, 456, 2040,  9360,  35280;
8,  15, 40, 138, 576, 2760, 14400,  75600,  322560;
9,  17, 46, 162, 696, 3480, 19440, 115920,  685440, 3265920;
10, 19, 52, 186, 816, 4200, 24480, 156240, 1048320, 6894720, 36288000;
...
For n = 10 and k = 2; T(10,2) = 17 coincides with the number of positive terms with two digits in A215014 (see the first comment above).

Column 1 gives A001477.
Row sums give A288780.
Cf. A000142, A004736, A166350, A215014, A288777.

Table[(n - k + 1) k! - (k - 1)!, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Jun 16 2017 *)

T(n,k) = A288777(n,k) - A000142(k-1), n>=1.

A256589 Triangle read by rows: T(n,k) divided by (n-k+1)! is the expected value of number of possible subsets in a partition of a set of n elements with no subsets of cardinality smaller than k.

Original entry on oeis.org

1, 3, 1, 11, 2, 1, 50, 8, 2, 1, 274, 36, 6, 2, 1, 1764, 200, 30, 6, 2, 1, 13068, 1300, 168, 24, 6, 2, 1, 109584, 9720, 1080, 144, 24, 6, 2, 1, 1026576, 82180, 8100, 960, 120, 24, 6, 2, 1

Offset: 1

Martin Y. Champel, Apr 03 2015

This sequence can be seen as a generalization of A233744 which is the particular case where minimum subset cardinality is 2 (k=2).
T(n,k) / A004736(n,k)! = 1 + 1 / ( A002024(n,k) - A002260(n,k) + 1) * (Sum of (T(n,p) / A004736(n,p)!) for p starting at A002260(n,k) up to A002024(n) - A002260(n) if 2 * A002260(n) <= A002024(n)).
The triangle below but including the diagonal is A166350 because there is only one possible partition of subsets of cardinality >= k in any set whose cardinality is between k and 2*k-1.

			The triangle T(n, k) starts:
n\k  1   2   3  4  5  6  ...
1:   1
2:   3   1
3:  11   2   1
4:  50   8   2  1
5: 274  36   6  2  1
6:1764 200  36  6  2  1

Martin Y. Champel, Table of n, a(n) for n = 1..10000

Column 2 is A233744.

Cf. A002024, A002260, A004736, A026791, A166350, A135010.

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)).

A233543 Triangle read by rows: T(n,k) = k!.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Dec 12 2013

			Triangle begins:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 6;
  1, 1, 2, 6, 24;
  1, 1, 2, 6, 24, 120;
  ...

Cf. A000142, A166350.

Row sums give A003422(n+1).

t[, m] := m!; Table[Table[t[n, m], {m, 0, n}], {n, 0, 9}] // Flatten (* Jean-François Alcover, Dec 12 2013 *)

T(n,k) = A000142(k).

cp's OEIS Frontend

A047916 Triangular array read by rows: a(n,k) = phi(n/k)*(n/k)^k*k! if k|n else 0 (1<=k<=n).

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A269221 Factorial of the sum of decimal digits of n.

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A288777 Triangle read by rows in which column k lists the positive multiples of the factorial of k, with 1 <= k <= n.

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A014454 Sum_{1<=k

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

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A288778 Triangle read by rows (1<=k<=n): T(n,k) = (n-k+1)*k! - (k-1)!

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A256589 Triangle read by rows: T(n,k) divided by (n-k+1)! is the expected value of number of possible subsets in a partition of a set of n elements with no subsets of cardinality smaller than k.

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

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A047916 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk! if k|n else 0 (1<=k<=n).