A005649 -id:A005649 - cp's OEIS frontend

A386638 Number of integer partitions of n of inseparable type.

0, 0, 1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 19, 30, 30, 45, 45, 67, 67, 97, 97, 139, 139, 195, 195, 272, 272, 373, 373, 508, 508, 684, 684, 915, 915, 1212, 1212, 1597, 1597, 2087, 2087, 2714, 2714, 3506, 3506, 4508, 4508, 5763, 5763, 7338, 7338, 9296, 9296

Offset: 0

Gus Wiseman, Aug 14 2025

nonn

A multiset is inseparable iff it has no permutation without adjacent equal parts. It is of inseparable type iff any multiset with those multiplicities (type) is inseparable. For example, {1,1,2} is separable but {1,1,1,2} is not; hence (2,1) is of separable type but (3,1) is not.

Also the number of integer partitions of n whose greatest part is at least two more than the sum of all the other parts.

			The a(2) = 1 through a(10) = 12 partitions (A=10):
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)     (A)
            (31)  (41)  (42)   (52)   (53)    (63)    (64)
                        (51)   (61)   (62)    (72)    (73)
                        (411)  (511)  (71)    (81)    (82)
                                      (521)   (621)   (91)
                                      (611)   (711)   (622)
                                      (5111)  (6111)  (631)
                                                      (721)
                                                      (811)
                                                      (6211)
                                                      (7111)
                                                      (61111)

Reduplication of A000070 shifted right.
Same as A025065 shifted right twice.
The Heinz numbers of these partitions are A335126.
Row sums of A386586.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A239455 counts Look-and-Say partitions, inseparable case A386632.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
A336106 counts separable type partitions, ranks A335127, sums of A386585.
A386633 counts separable type set partitions, row sums of A386635.
A386634 counts inseparable type set partitions, row sums of A386636.
Cf. A000110, A008284, A047966, A072233, A106351, A124762, A217605, A239964, A279790, A335125, A351293, A386577.

Table[Length[Select[IntegerPartitions[n],2*Max@@#>1+n&]],{n,0,15}]

For n>1, a(n) = A025065(n-2).

a(n) = A000041(n) - A336106(n).

A084416 Triangle read by rows: T(n,k) = Sum_{i=k..n} i!*Stirling2(n,i), n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 3, 2, 13, 12, 6, 75, 74, 60, 24, 541, 540, 510, 360, 120, 4683, 4682, 4620, 4080, 2520, 720, 47293, 47292, 47166, 45360, 36960, 20160, 5040, 545835, 545834, 545580, 539784, 498960, 372960, 181440, 40320, 7087261, 7087260, 7086750, 7068600, 6882120, 6048000, 4142880, 1814400, 362880

Offset: 1

N. J. A. Sloane, Jun 24 2003

Interpolates between A000670 and factorials.
From Thomas Scheuerle, Apr 25 2022: (Start)
Number of preferential arrangements of n labeled elements when at least k ranks are required.
This sequence starts for k and n with offset 1. If it would start with k = 0, we would observe in column k = 0 an exact copy of column k = 1 with a preceding one at n = 0, k = 0. The difference between 0 ranks and one rank (all in the same rank) is only for n = 0 where k = 0 allows zero-filled ranks as an valid arrangement, too. (End)

			Triangle begins with T(n,k):
   k=   1,   2,   3,   4,   5
  n=1   1
  n=2   3,   2
  n=3  13,  12,   6
  n=4  75,  74,  60,  24
  n=5 541, 540, 510, 360, 120
...
From _Thomas Scheuerle_, Apr 25 2022: (Start)
If we would add n = 0, k = 0 to the data of this sequence:
   k=   0,   1,   2,
  n=0   1
  n=1   1,   1
  n=2   3,   3,   2
...
T(n, 3) with 3 preceding zeros is: 0,0,0,6,60,510,4620,...
This sequence has the e.g.f.: (e^x-1)^3/(2-e^x).
.
13 arrangements for n = 3 and k = 1 (one rank required):
1,2,3  1,2|3  2,3|1  1,3|2  1|2,3  2|1,3  3|1,2  1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
12 arrangements for n = 3 and k = 2 (two ranks required):
1,2|3  2,3|1  1,3|2  1|2,3  2|1,3  3|1,2  1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
6 arrangements for n = 3 and k = 3 (three ranks required):
1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
. (End)

Mirror image of array in A084417.
Cf. A005649, A069321 (row sums).
A000670(n) (column k = 1), A052875(n) (column k = 2), A102232(n) (column k = 3).
Cf. A000142, A008277.

T := (n,k)->sum(i!*Stirling2(n,i),i=k..n): seq(seq(T(n,k),k=1..n),n=1..10);

row(n) = vector(n, k, sum(i=k, n, i!*stirling(n, i, 2))); \\ Michel Marcus, Apr 20 2022

E.g.f. for m-th column: (exp(x)-1)^m/(2-exp(x)). - Vladeta Jovovic, Sep 14 2003
T(n, k) = Sum_{m = k..n} A090582(n + 1, m + 1).
From Thomas Scheuerle, Apr 25 2022: (Start)
Sum_{k = 0..n} T(n, k) = A005649(n). Column k = 0 is not part of data.
Sum_{k = 1..n} T(n, k) = A069321(n).
T(n, 0) = A000670(n). Column k = 0 is not part of data.
T(n, 1) = A000670(n), for n > 0.
T(n, 2) = A052875(n).
T(n, 3) = A102232(n).
T(n, n) = n! = A000142. (End)

More terms from Emeric Deutsch, May 11 2004

More terms from Michel Marcus, Apr 20 2022

A217388 Alternating sums of the ordered Bell numbers (number of preferential arrangements) A000670.

Original entry on oeis.org

1, 0, 3, 10, 65, 476, 4207, 43086, 502749, 6584512, 95663051, 1526969522, 26564598073, 500293750308, 10141049220135, 220142141757718, 5095512540223637, 125275254488912264, 3260259408767933059, 89541327910560478074, 2588146468333823725041

Offset: 0

Emanuele Munarini, Oct 02 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Mathematics Stack Exchange, Simple closed form for signed partial sums of Fubini numbers, Aug 8 2025.

Cf. A000670, A006957, A005649, A217389, A217391, A217392.

List([0..30],n->Sum([0..n],k->(-1)^(n-k)*Sum([0..k], j-> Factorial(j)*Stirling2(k,j)))); # Muniru A Asiru, Feb 07 2018

A000670:=func;
[&+[(-1)^(n-k)*A000670(k): k in [0..n]]: n in [0..20]]; // Bruno Berselli, Oct 03 2012

with(combinat):
seq(sum((-1)^(n-k)*sum(factorial(j)*stirling2(k,j), j=0..k), k=0..n), n=0..30); # Muniru A Asiru, Feb 07 2018

t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[(-1)^(n - k)t[k], {k, 0, n}], {n, 0, 100}]
(* second program: *)
Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1; a[n_] := Sum[(-1)^(n-k) Fubini[k, 1], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 31 2016 *)

t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum((-1)^(n-k)*t(k),k,0,n),n,0,40);

for(n=0,30, print1(sum(k=0,n, (-1)^(n-k)*sum(j=0,k, j!*stirling(k,j,2))), ", ")) \\ G. C. Greubel, Feb 07 2018

a(n) = sum(k=0, n, k!*stirling(n+2,k+2,2)*(2^(k+1)-1)*(-1)^(n-k)) \\ Mikhail Kurkov, Aug 08 2025

a(n) = sum((-1)^(n-k)*t(k), k=0..n), where t = A000670 (ordered Bell numbers).
E.g.f.: 1/(2-exp(x))-exp(-x)*log(1/(2-exp(x))). [Typo corrected by Vaclav Kotesovec, Oct 08 2013]
G.f.: 1/(1+x)/Q(0), where Q(k)= 1 - x*(k+1)/(1 - x*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
a(n) ~ n! /(2*(log(2))^(n+1)). - Vaclav Kotesovec, Oct 08 2013
a(n) = Sum_{k=0..n} k!*Stirling2(n+2,k+2)*(2^(k+1)-1)*(-1)^(n-k). - Mikhail Kurkov, Aug 08 2025

A226739 Row 4 of array in A226513.

Original entry on oeis.org

1, 5, 35, 305, 3155, 37625, 507035, 7608305, 125687555, 2265230825, 44210200235, 928594230305, 20880079975955, 500343586672025, 12726718227077435, 342425052939060305, 9715738272696568355, 289901469137229041225, 9074304882434034258635, 297297854264669632338305

Offset: 0

Vincenzo Librandi, Jun 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.

Cf. rows 0, 1, 2, 3 and 5 of A226513: A000670, A005649, A226515, A226738, A226740.

m:=4; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]];

Range[0,20]! CoefficientList[Series[(2 - Exp@x)^-5, {x, 0, 20}],x]

E.g.f.: 1/(2 - exp(x))^5 (see the Ahlbach et al. paper, Theorem 4).
a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(4+i,i), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3).
G.f.: 1/Q(0), where Q(k) = 1 - x*(3*k + 1 + m) - 2*x^2*(k + 1)*(k + 1 + m)/Q(k+1), m = 4 is row 4 of array in A226513; (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) ~ n! * n^4 / (768 * log(2)^(n+5)). - Vaclav Kotesovec, Oct 11 2022
Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 5*x/(1 - 2*x/(1 - 6*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - (n+4)*x/(1 - 2*n*x/(1 - ... ))))))))). - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (4*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 5*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)

A226740 Row 5 of array in A226513.

Original entry on oeis.org

1, 6, 48, 468, 5340, 69516, 1014348, 16372908, 289366860, 5553635436, 114964523148, 2552305112748, 60474398655180, 1522843616043756, 40605864407444748, 1142786353739186988, 33848016050071188300, 1052381222812017946476, 34266937867683980363148, 1166071764343727862515628

Offset: 0

Vincenzo Librandi, Jun 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.

Cf. rows 0, 1, 2, 3, 4 of A226513: A000670, A005649, A226515, A226738, A226739.

m:=5; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]];

Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-6, {x, 0, 20}], x]

E.g.f.: 1/(2 - exp(x))^6 (see the Ahlbach et al. paper, Theorem 4).
a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(5+i,i), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3).
a(n) ~ n! * n^5 / (7680 * log(2)^(n+6)). - Vaclav Kotesovec, Oct 11 2022
Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 6*x/(1 - 2*x/(1 - 7*x/(1 - 4*x/(1 - 8*x/(1 - 6*x/(1 - (n+5)*x/(1 - 2*n*x/(1 - ... ))))))))). - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (5*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 6*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)

A317280 Expansion of e.g.f. 1/(1 - log(1 + x))^2.

Original entry on oeis.org

1, 2, 4, 10, 30, 108, 444, 2112, 11040, 65712, 414816, 2992944, 21876816, 188936928, 1527813216, 15991733376, 133364903040, 1794144752640, 13329036288000, 270750383400960, 1167153128110080, 57074973648030720, -103080839984916480, 17319631144046423040, -171982551742151685120

Offset: 0

Ilya Gutkovskiy, Jul 25 2018

sign

Exponential self-convolution of A006252.

Seiichi Manyama, Table of n, a(n) for n = 0..451
N. J. A. Sloane, Transforms

Cf. A005444, A005649, A006252, A052801, A089064.

Cf. A006252, A354120, A354121.

a:=series(1/(1 - log(1 + x))^2, x=0, 25): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 26 2019

nmax = 24; CoefficientList[Series[1/(1 - Log[1 + x])^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] (k + 1)!, {k, 0, n}], {n, 0, 24}]

a(n) = Sum_{k=0..n} Stirling1(n,k)*(k + 1)!.
a(n) ~ n! * 2 * (-1)^(n+1) / (n * log(n)^3) * (1 - 3*(gamma+1) / log(n) + (6*gamma^2 + 12*gamma + 6 - Pi^2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 15 2022
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A332668 Number of strict integer partitions of n without three consecutive parts in arithmetic progression.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 11, 15, 20, 19, 26, 31, 34, 41, 50, 53, 67, 78, 84, 99, 120, 130, 154, 177, 193, 226, 262, 291, 332, 375, 419, 479, 543, 608, 676, 765, 859, 961, 1075, 1202, 1336, 1495, 1672, 1854, 2050, 2301, 2536, 2814, 3142, 3448, 3809

Offset: 0

Gus Wiseman, Mar 28 2020

nonn

Also the number of strict integer partitions of n whose first differences are an anti-run, meaning there are no adjacent equal differences.

			The a(1) = 1 through a(10) = 9 partitions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)
                        (41)  (51)  (52)   (62)   (63)   (73)
                                    (61)   (71)   (72)   (82)
                                    (421)  (431)  (81)   (91)
                                           (521)  (621)  (532)
                                                         (541)
                                                         (631)
                                                         (721)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..450
Wikipedia, Arithmetic progression

Anti-run compositions are counted by A003242.
Normal anti-runs of length n + 1 are counted by A005649.
Strict partitions with equal differences are A049980.
Partitions with equal differences are A049988.
The non-strict version is A238424.
The version for permutations is A295370.
Anti-run compositions are ranked by A333489.
Cf. A006560, A007862, A238423, A307824, A325328, A325852, A325874, A333195.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MatchQ[Differences[#],{_,x_,x_,_}]&]],{n,0,30}]

A386577 Irregular triangle read by rows where T(n,k) is the number of permutations of the multiset of prime factors of n with k adjacent equal terms.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 01 2025

Are the rows all unimodal?

Counts permutations of prime factors by "inseparability". For "separability" we have A374252.

			The prime indices of 12 are {1,1,2}, and we have:
- 1 permutation (1,2,1) with 0 adjacent equal parts
- 2 permutations (1,1,2), (2,1,1) with 1 adjacent equal part
- 0 permutations with 2 adjacent equal parts
so row 12 is (1,2,0).
Row 48 counts the following permutations:
  .  .  (1,1,1,2,1)  (1,1,1,1,2)  .
        (1,1,2,1,1)  (2,1,1,1,1)
        (1,2,1,1,1)
Row 144 counts the following permutations:
  .  (1,1,2,1,2,1)  (1,1,1,2,1,2)  (1,1,1,2,2,1)  (1,1,1,1,2,2)  .
     (1,2,1,1,2,1)  (1,1,2,1,1,2)  (1,1,2,2,1,1)  (2,2,1,1,1,1)
     (1,2,1,2,1,1)  (1,2,1,1,1,2)  (1,2,2,1,1,1)
                    (2,1,1,1,2,1)  (2,1,1,1,1,2)
                    (2,1,1,2,1,1)
                    (2,1,2,1,1,1)
Triangle begins:
   1:
   2: 1
   3: 1
   4: 0  1
   6: 1
   6: 2  0
   7: 1
   8: 0  0  1
   9: 0  1
  10: 2  0
  11: 1
  12: 1  2  0
  13: 1
  14: 2  0
  15: 2  0
  16: 0  0  0  1
  17: 1
  18: 1  2  0
  19: 1
  20: 1  2  0
  21: 2  0
  22: 2  0
  23: 1
  24: 0  2  2  0

Row lengths are A001222.
The minima of each row are A010051.
Sorted positions of first appearances appear to be A025487.
Column k = last is A069513.
Row sums are A168324 or A008480.
The number of trailing zeros in each row is A297155 = A001221-1.
Column k = 1 is A335452.
The number of leading zeros in each row is A374246.
For separability instead of inseparability we have A374252.
For a multiset with prescribed multiplicities we have A386578, separability A386579.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A124762 gives inseparability of standard compositions, separability A333382.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts partitions of separable type, ranks A335127, sums of A386585.
Cf. A001221, A051903, A051904, A106351, A238130, A336102, A373957, A382525, A386581, A386632.

Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],Function[q,Length[Select[Range[Length[q]-1],q[[#]]==q[[#+1]]&]]==k]]],{n,30},{k,0,PrimeOmega[n]-1}]

A386579 Number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent unequal parts.

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 0, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 6, 0, 2, 2, 2, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 6, 6, 1, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 2, 3, 4, 1, 0, 0, 0, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 6, 12, 12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 12, 2, 0, 2, 4, 6, 3, 0

Offset: 2

Gus Wiseman, Aug 04 2025

Row 1 is empty, so offset is 2.
Same as A386578 with rows reversed.
This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			Row n = 21 counts the following permutations:
  .  111122  111221  111212  112121  .
     221111  112211  112112  121121
             122111  121112  121211
             211112  211121
                     211211
                     212111
Triangle begins:
  .
  1
  1  0
  0  2
  1  0  0
  0  2  1
  1  0  0  0
  0  0  6
  0  2  2  2
  0  2  2  0
  1  0  0  0  0
  0  0  6  6
  1  0  0  0  0  0
  0  2  3  0  0
  0  2  3  4  1
  0  0  0 24
  1  0  0  0  0  0  0
  0  0  6 12 12
  1  0  0  0  0  0  0  0
  0  0  6 12  2
  0  2  4  6  3  0

Column k = 0 is A010051.
Row lengths are A056239.
Row sums are A318762.
Column k = last is A335125.
For prime indices we have A374252, reverse A386577.
Reversing all rows gives A386578.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A124762 gives inseparability of standard compositions, separability A333382.
A305936 is a multiset whose multiplicities are the prime indices of n.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts partitions of separable type, ranks A335127, sums of A386585.
Cf. A001221, A001222, A003963, A051903, A051904, A106351, A112798, A238130, A336102, A373957, A386581.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
ugt[c_,x_]:=Select[Permutations[c],Function[q,Length[Select[Range[Length[q]-1],q[[#]]!=q[[#+1]]&]]==x]];
Table[Table[Length[ugt[nrmptn[n],k]],{k,0,Length[nrmptn[n]]-1}],{n,30}]

A098348 Triangular array read by rows: a(n, k) = number of ordered factorizations of a "hook-type" number with n total prime factors and k distinct prime factors. "Hook-type" means that only one prime can have multiplicity > 1.

Original entry on oeis.org

1, 2, 3, 4, 8, 13, 8, 20, 44, 75, 16, 48, 132, 308, 541, 32, 112, 368, 1076, 2612, 4683, 64, 256, 976, 3408, 10404, 25988, 47293, 128, 576, 2496, 10096, 36848, 116180, 296564, 545835, 256, 1280, 6208, 28480, 120400, 454608, 1469892, 3816548

Offset: 1

Alford Arnold, Sep 04 2004

The first three columns are A000079, A001792 and A098385.
The first two diagonals are A000670 and A005649.
A070175 gives the smallest representative of each hook-type prime signature, so this sequence is a rearrangement of A074206(A070175).

			a(4, 2) = 20 because 24=2*2*2*3 has 20 ordered factorizations and so does any other number with the same prime signature.

Cf. A050324, A070175, A070826, A074206, A095705. A098349 gives the row sums. A098384.

a(n, k) = 1 + (Sum_{i=1..k-1} binomial(k-1, i)*a(i, i)) + (Sum_{j=1..k} Sum_{i=j..j+n-k-1} binomial(k-1, j-1)*a(i, j)) + (Sum_{j=1..k-1} binomial(k-1,j-1)*a(j+n-k, j)). - David Wasserman, Feb 21 2008
a(n, k) = A074206(2^(n+1-k)*A070826(k)). - David Wasserman, Feb 21 2008
The following conjectural formula for the triangle entries agrees with the values listed above: T(n,k) = Sum_{j = 0..n-k} 2^(n-k-j)*binomial(n-k,j)*a(k,j), where a(k,j) = 2^j*Sum_{i = j+1..k+1} binomial(i,j+1)*(i-1)!*Stirling2(k+1,i). See A098384 for related conjectures. - Peter Bala, Apr 20 2012

Edited and extended by David Wasserman, Feb 21 2008

cp's OEIS Frontend

A386638 Number of integer partitions of n of inseparable type.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A084416 Triangle read by rows: T(n,k) = Sum_{i=k..n} i!*Stirling2(n,i), n >= 1, 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A217388 Alternating sums of the ordered Bell numbers (number of preferential arrangements) A000670.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

Maxima

PARI

PARI

Formula

A226739 Row 4 of array in A226513.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A226740 Row 5 of array in A226513.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A317280 Expansion of e.g.f. 1/(1 - log(1 + x))^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A332668 Number of strict integer partitions of n without three consecutive parts in arithmetic progression.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A386577 Irregular triangle read by rows where T(n,k) is the number of permutations of the multiset of prime factors of n with k adjacent equal terms.