A108461 -id:A108461 - cp's OEIS frontend

A057567 Number of partitions of n where the product of parts divides n.

1, 2, 2, 4, 2, 5, 2, 7, 4, 5, 2, 11, 2, 5, 5, 12, 2, 11, 2, 11, 5, 5, 2, 21, 4, 5, 7, 11, 2, 15, 2, 19, 5, 5, 5, 26, 2, 5, 5, 21, 2, 15, 2, 11, 11, 5, 2, 38, 4, 11, 5, 11, 2, 21, 5, 21, 5, 5, 2, 36, 2, 5, 11, 30, 5, 15, 2, 11, 5, 15, 2, 52, 2, 5, 11, 11, 5, 15, 2, 38, 12, 5, 2, 36, 5, 5, 5, 21

Offset: 1

Leroy Quet, Oct 04 2000

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1). - Christian G. Bower, Jun 03 2005

			From _Gus Wiseman_, Jul 04 2019: (Start)
The a(1) = 1 through a(9) = 5 partitions are the following. The Heinz numbers of these partitions are given by A326155.
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (321)     (1111111)  (4211)
                    (211)            (3111)               (22211)
                    (1111)           (21111)              (41111)
                                     (111111)             (221111)
                                                          (2111111)
                                                          (11111111)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A000070, A000110, A001055 (Mobius transform), A002110, A025487, A057568, A108464, A113309, A131802.
Any prime numbered column of array A108461.
Cf. A028422, A096276, A114324, A318950, A319000, A319005, A326152, A326155.

Table[Function[m, Count[Map[Times @@ # &, IntegerPartitions[m]], P_ /; Divisible[m, P]] - Boole[n == 1]]@ Apply[Times, #] &@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]], {n, 88}] (* Michael De Vlieger, Aug 16 2017 *)

fcnt(n, m) = {local(s); s=0; if(n == 1, s=1, fordiv(n, d, if(d > 1 & d <= m, s=s+fcnt(n/d, d)))); s}
A001055(n) = fcnt(n, n) \\ This function from Michael B. Porter, Oct 29 2009
A057567(n) = sumdiv(n, d, A001055(d)); \\ After Jovovic's formula. Antti Karttunen, May 25 2017

from sympy import divisors, isprime
def T(n, m):
    if isprime(n): return 1 if n <= m else 0
    A = (d for d in divisors(n) if 1 < d < n and d <= m)
    s = sum(T(n // d, d) for d in A)
    return s + 1 if n <= m else s
def a001055(n): return T(n, n)
def a(n): return sum(a001055(d) for d in divisors(n))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 19 2017

a(n) = Sum_{d|n} A001055(d). - Vladeta Jovovic, Nov 19 2000
a(A025487(n)) = A108464(n).
a(p^k) = A000070(k).
a(A002110(n)) = A000110(n+1).
Dirichlet g.f.: zeta(s) * Product_{k>=2} 1/(1 - 1/k^s). - Ilya Gutkovskiy, Nov 03 2020

More terms from James Sellers, Oct 09 2000

More terms from Vladeta Jovovic, Nov 19 2000

A108459 Number of labeled partitions of (n,n) into pairs (i,j).

Original entry on oeis.org

1, 1, 5, 52, 855, 19921, 614866, 24040451, 1152972925, 66200911138, 4465023867757, 348383154017581, 31052765897026352, 3128792250765898965, 353179564583216567917, 44320731930172534543092, 6141797839043095806714667, 934330605640859569909566925

Offset: 0

Christian G. Bower, Jun 03 2005

nonn

Partitions of n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one white object.

a(n) is also the number of elements of the partition monoid P_n with domain {1,...,n}. Elements of P_n are set partitions of {1,1',...,n,n'}, and the domain of such a partition is the set of all points in {1,...,n} that belong to a block containing a dashed element. - James East, Apr 10 2018

Seiichi Manyama, Table of n, a(n) for n = 0..200

Main diagonal of A108458. Cf. A108461.
Cf. A048993 (Stirling2), A068424 (falling factorial).
Cf. A326600, A326270, A326271, A326288.
Bisection of A124421 (even part).

b:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
a:= n-> add(coeff(b(n), x, j)*j^n, j=0..n):
seq(a(n), n=0..21);  # Alois P. Heinz, Dec 02 2023

a[n_] := If[n == 0, 1, Sum[k^n*StirlingS2[n, k], {k, 0, n}]];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 10 2024 *)

{a(n)=polcoeff(sum(m=0, n, m^m*x^m/prod(k=1, m, 1-m*k*x +x*O(x^n))), n)} \\ Paul D. Hanna, Sep 17 2013

{a(n)=n!*polcoeff(sum(m=0, n, (exp(m*x+x*O(x^n))-1)^m/m!), n)} \\ Paul D. Hanna, Sep 17 2013

a(n) = Sum_{k=0..n} k^n*Stirling2(n,k). - Vladeta Jovovic, Aug 31 2006
E.g.f.: Sum_{n>=0} (exp(n*x)-1)^n / n!. - Vladeta Jovovic, Jul 12 2007
E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( -exp(n*x) ) / n!. - Paul D. Hanna, Jun 28 2019
O.g.f.: Sum_{n>=0} n^n * x^n / Product_{k=1..n} (1 - n*k*x). - Paul D. Hanna, Sep 17 2013
a(n) = Sum_{k=0..n} Stirling2(n,k) * Sum_{l=k..n} Stirling2(n,l)*T(l,k). Here T(l,k) are the falling factorials. - James East, Apr 10 2018

A051707 Number of factorizations of (n,n) into pairs (j,k).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 8, 3, 5, 1, 23, 1, 5, 5, 23, 1, 23, 1, 23, 5, 5, 1, 91, 3, 5, 8, 23, 1, 52, 1, 60, 5, 5, 5, 143, 1, 5, 5, 91, 1, 52, 1, 23, 23, 5, 1, 328, 3, 23, 5, 23, 1, 91, 5, 91, 5, 5, 1, 339, 1, 5, 23, 161, 5, 52, 1, 23, 5, 52, 1, 686, 1, 5, 23, 23, 5, 52, 1, 328, 23, 5, 1, 339, 5

Offset: 1

Yasutoshi Kohmoto

Pairs (j,k) must satisfy j>1, k>=1; (a,b)*(x,y)=(a*x,b*y); unit is (1,1).

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).

			(6,6)=(2,1)*(3,6)=(2,6)*(3,1)=(2,2)*(3,3)=(2,3)*(3,2), so a(6)=5.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A050354, A108461, A108455, A348161 (into at most two pairs).
a(A025487) = A108460.
a(p^k) = A108457(k).
a(A002110) = A108459.
Main diagonal of A108455.

Edited by Christian G. Bower, Jun 03 2005

A090806 Triangular array read by rows: T(n,k) (n >= 2, 1 <= k <= n) = number of partitions of k white balls and n-k black balls in which each part has at least one ball of each color. Also limit of the joint major-index / inversion polynomial for permutations of n elements, as n becomes infinite.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 4, 3, 1, 1, 3, 5, 5, 3, 1, 1, 4, 7, 9, 7, 4, 1, 1, 4, 9, 12, 12, 9, 4, 1, 1, 5, 11, 17, 20, 17, 11, 5, 1, 1, 5, 13, 22, 28, 28, 22, 13, 5, 1, 1, 6, 16, 29, 40, 45, 40, 29, 16, 6, 1, 1, 6, 18, 35, 53, 64, 64, 53, 35, 18, 6, 1, 1, 7, 21, 44, 70, 91, 100, 91

Offset: 2

Don Knuth, Feb 10 2004

Alternatively, square array read by antidiagonals: a(n,k) (n >= 1, k >= 1) = number of partitions of (n,k) into pairs (i,j) with i,j>0. The addition rule is (a,b)+(x,y)=(a+x,b+y). E.g., (4,3) = (3,2)+(1,1) = (3,1)+(1,2) = (2,2)+(2,1) = (2,1)+(1,1)+(1,1), so T(4,3)=5. - Christian G. Bower, Jun 03 2005

Permutations of n elements have a polynomial sum x^{ind pi}y^{inv pi} where ind denotes the major index and inv the number of inversions. For example when n=3 the polynomial is 1 + xy + xy^2 + x^2y + x^2y^2 + x^3y^3. The coefficient of x^i y^j when i+j <= n is given by this sequence; in other words, the polynomials approach 1 + xy + x^2y + xy^2 + x^3y + 2x^2y^2 + xy^3 + ... + 4x^3y^3 + ... as n grows. The reasons can be found in the Garsia-Gessel reference.

			Triangle T(n,k) begins
      1
     1 1
    1 2 1
   1 2 2 1
  1 3 4 3 1
The first row is for n=2. When n=6 and there are 3 balls of each color, the four partitions in question are bbbwww; bbww|bw; bw|bw|bw; bbw|bww.
Square array a(n,k) begins:
  1 1 1  1  1 ...
  1 2 2  3  3 ...
  1 2 4  5  7 ...
  1 3 5  9 12 ...
  1 3 7 12 20 ...

Alter, Ronald; Curtz, Thaddeus B.; Wang, Chung C. Permutations with fixed index and number of inversions. Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1974), pp. 209-228. Congressus Numerantium, No. X, Utilitas Math., Winnipeg, Man., 1974. From N. J. A. Sloane, Mar 20 2012
M. S. Cheema and T. S. Motzkin, "Multipartitions and multipermutations," Proc. Symp. Pure Math. 19 (1971), 39-70, eq. (3.1.3).

A. M. Garsia and I. Gessel, Permutation statistics and partitions, Advances in Mathematics, Volume 31, Issue 3, March 1979, Pages 288-305.
Günter Meinardus, Zur additiven Zahlentheorie in mehreren Dimensionen, Teil I, Math. Ann. 132 (1956), 333-346. [Gives asymptotic growth]
N. J. A. Sloane, Transforms

Cf. A108461. Main diagonal: A108469.

G.f. for T(n, k): 1/Product_{i>=1, j>=1} (1 - w^i * z^j).
Recurrence: m*T(m, n) = Sum_{L>0, j>0, k>=0} j*T(m-L*j, n-L*k). [Cheema and Motzkin]
Also, Euler transform of the table whose g.f. is xy/((1-x)*(1-y)). - Christian G. Bower, Jun 03 2005

More terms from Christian G. Bower, Jun 03 2005

Entry revised by N. J. A. Sloane, Jul 07 2005

A108455 Table read by antidiagonals: T(n,k) = number of factorizations of (n,k) into any number of pairs (i,j) with i > 0, j > 0 (and if i=1 then j=1).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 2, 0, 1, 1, 3, 1, 3, 1, 0, 1, 1, 2, 1, 3, 1, 3, 0, 1, 1, 3, 1, 4, 1, 4, 2, 0, 1, 1, 2, 1, 3, 1, 4, 2, 2, 0, 1, 1, 3, 1, 5, 1, 6, 2, 3, 1, 0, 1, 1, 3, 1, 3, 1, 4, 3, 3, 1, 4, 0, 1, 1, 3, 1, 5, 1, 7, 2

Offset: 1

Christian G. Bower, Jun 03 2005

(a,b)*(x,y) = (a*x,b*y); unit is (1,1).

			Table begins
  1 0 0 0 0 ...
  1 1 1 1 1 ...
  1 1 1 1 1 ...
  2 2 2 3 2 ...
  1 1 1 1 1 ...
  ...
(6,4) = (3,4)*(2,1) = (3,1)*(2,4) = (3,2)*(2,2), so a(6,4)=4.

Cf. A108461, A051707 (main diagonal), A348157. Column 1: A001055.

Dirichlet g.f.: A(s, t) = exp(B(s, t)/1 + B(2*s, 2*t)/2 + B(3*s, 3*t)/3 + ...) where B(s, t) = zeta(s)*(zeta(t)-1).

Definition clarified and original interpretation restored by Sean A. Irvine, Oct 03 2021

A108457 Number of partitions of (n,n) into pairs (i,j) with i>0, j>=0.

Original entry on oeis.org

1, 1, 3, 8, 23, 60, 161, 404, 1011, 2440, 5804, 13462, 30798, 69082, 152862, 333094, 716781, 1522534, 3198011, 6642274, 13657417, 27805917, 56097811, 112183092, 222496203, 437787036, 854936195, 1657526884, 3191478236, 6104379829, 11602006003, 21916355387

Offset: 0

Christian G. Bower, Jun 03 2005

nonn

(a,b)+(x,y)=(a+x,b+y); unit is (0,0).

			(2,2)=(1,2)+(1,0)=(1,1)+(1,1), so a(2)=3.

Alois P. Heinz, Table of n, a(n) for n = 0..99

Main diagonal of A108456.

Cf. A000041, A091438, A108461.

a(n) = A091438(2n,n). - Alois P. Heinz, Sep 24 2018

A108458 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the last block is the singleton {k}, 1<=k<=n; the blocks are ordered with increasing least elements.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 5, 10, 15, 0, 1, 9, 22, 37, 52, 0, 1, 17, 52, 99, 151, 203, 0, 1, 33, 130, 283, 471, 674, 877, 0, 1, 65, 340, 855, 1561, 2386, 3263, 4140, 0, 1, 129, 922, 2707, 5451, 8930, 12867, 17007, 21147, 0, 1, 257, 2572, 8919, 19921, 35098, 53411, 73681, 94828, 115975

Offset: 1

Christian G. Bower, Jun 03 2005; Emeric Deutsch, Nov 14 2006

Another way to obtain this sequence (with offset 0): Form the infinite array U(n,k) = number of labeled partitions of (n,k) into pairs (i,j), for n >= 0, k >= 0 and read it by antidiagonals. In other words, U(n,k) = number of partitions of n black objects labeled 1..n and k white objects labeled 1..k. Each block must have at least one white object.

Then T(n,k)=U(n+k,k+1). Thus the two versions are related like "multichoose" to "choose". - Augustine O. Munagi, Jul 16 2007

			Triangle T(n,k) starts:
  1;
  0,1;
  0,1,2;
  0,1,3,5;
  0,1,5,10,15;
T(5,3)=5 because we have 1245|3, 145|2|3, 14|25|3, 15|24|3 and 1|245|3.
The arrays U(n,k) starts:
   1  0  0   0   0 ...
   1  1  1   1   1 ...
   2  3  5   9  17 ...
   5 10 22  52 130 ...
  15 37 99 283 855 ...

Alois P. Heinz, Rows n = 1..141, flattened

Row sums of T(n, k) yield A124496(n, 1).
Cf. A108461.
Columns of U(n, k): A000110, A005493, A033452.
Rows of U(n, k): A000007, A000012, A000051.
Main diagonal: A108459.

T[n_, k_] := Sum[If[n == k, 1, i^(n-k)]*StirlingS2[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 10 2024, after Vladeta Jovovic *)

T(n,1)=0 for n>=2; T(n,2)=1 for n>=2; T(n,3)=1+2^(n-3) for n>=3; T(n,n)=B(n-1), T(n,n-1)=B(n-1)-B(n-2), where B(q) are the Bell numbers (A000110).
Double e.g.f.: exp(exp(x)*(exp(y)-1)).
U(n,k) = Sum_{i=0..k} i^(n-k)*Stirling2(k,i). - Vladeta Jovovic, Jul 12 2007

Edited by N. J. A. Sloane, May 22 2008, at the suggestion of Vladeta Jovovic. This entry is a composite of two entries submitted independently by Christian G. Bower and Emeric Deutsch, with additional comments from Augustine O. Munagi.

A108462 Number of factorizations of (n,n) into pairs (i,j) with i,j >= 1, not both 1.

Original entry on oeis.org

1, 2, 2, 9, 2, 15, 2, 31, 9, 15, 2, 92, 2, 15, 15, 109, 2, 92, 2, 92, 15, 15, 2, 444, 9, 15, 31, 92, 2, 203, 2, 339, 15, 15, 15, 712, 2, 15, 15, 444, 2, 203, 2, 92, 92, 15, 2, 1903, 9, 92, 15, 92, 2, 444, 15, 444, 15, 15, 2, 1663, 2, 15, 92, 1043, 15, 203, 2, 92, 15, 203, 2

Offset: 1

Christian G. Bower, Jun 03 2005

nonn

The rule of building products is (a,b)*(x,y) = (a*x,b*y).

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).

			From _Alois P. Heinz_ and _Antti Karttunen_, Nov 24 2017: (Start)
a(4) = 9 because for pair (4,4) there are nine factorizations:
  (4,4)
  (1,4)*(4,1)
  (1,2)*(4,2)
  (2,1)*(2,4)
  (2,2)*(2,2)
  (1,2)*(2,1)*(2,2)
  (1,4)*(2,1)*(2,1)
  (4,1)*(1,2)*(1,2)
  (1,2)*(1,2)*(2,1)*(2,1)
(End)
a(pq) = 15 for primes p<>q: (pq,pq); (p,1)(q,pq); (p,1)(q,1)(1,pq); (p,1)(q,1)(1,p)(1,q); (p,1)(q,q)(1,p); (p,1)(q,p)(1,q); (p,q)(q,p); (p,q)(q,1)(1,p); (p,p)(q,q) ; (p,p)(q,1)(1,q); (p,pq)(q,1); (pq,1)(1,pq); (pq,1)(1,p)(1,q); (pq,q)(1,p); (pq,p)(1,q). - _R. J. Mathar_, Nov 30 2017

Main diagonal of A108461.

Cf. A002774, A020557, A108463.

a(n) = if(n==1, return(1)); my(b, c, r, x, y, v=List([]), w=List([[n]])); while(#w>r, c++; for(k=r+1, r=#w, y=w[k]; if(!isprime(x=y[c]), fordiv(x, d, if(d!=1&&d!=x, listput(w, concat([y[1..c-1], d, x/d]))))))); for(i=1, #w, x=w[i]; r=#x; for(j=1, #w, y=w[j]; for(k=0, 2^r-1, b=concat(b=binary(k), vector(r-#b)); if(#y>=t=vecsum(b), c=0; listput(v, vecsort(vector(r+#y-t, m, if(m>r, [1, y[m-r+t]], if(b[m], [x[m], y[c++]], [x[m], 1]))))))))); #Set(v); \\ Jinyuan Wang, Jan 17 2022

a(A025487(n)) = A108463(n).
a(p^k) = A002774(k).
a(A002110(n)) = A020557(n).
a(n) = A108461(n,n).

A108466 Number of factorizations of (n,n) into pairs (i,j) with i,j>1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 11, 1, 3, 3, 9, 1, 11, 1, 11, 3, 3, 1, 35, 2, 3, 4, 11, 1, 25, 1, 20, 3, 3, 3, 55, 1, 3, 3, 35, 1, 25, 1, 11, 11, 3, 1, 106, 2, 11, 3, 11, 1, 35, 3, 35, 3, 3, 1, 132, 1, 3, 11, 45, 3, 25, 1, 11, 3, 25, 1, 222, 1, 3, 11, 11, 3, 25, 1, 106, 9, 3, 1, 132, 3, 3

Offset: 1

Christian G. Bower, Jun 03 2005

nonn

(a,b)*(x,y)=(a*x,b*y).

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).

Cf. A108461. Main diagonal of A108465. a(A025487)=A108467. a(p^k)=A108469(k). a(A002110)=A023997.

A108470 Table read by antidiagonals: T(n,k) = number of labeled partitions of (n,k) into pairs (i,j).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 25, 15, 1, 1, 31, 79, 79, 31, 1, 1, 63, 241, 339, 241, 63, 1, 1, 127, 727, 1351, 1351, 727, 127, 1, 1, 255, 2185, 5235, 6721, 5235, 2185, 255, 1, 1, 511, 6559, 20119, 31831, 31831, 20119, 6559, 511, 1, 1, 1023, 19681, 77379

Offset: 1

Christian G. Bower, Jun 03 2005

Partitions of n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one black object and at least one white object.

			1 1 1 1 1 ...
1 3 7 15 31 ...
1 7 25 79 241 ...
1 15 79 339 1351 ...
1 31 241 1351 6721 ...

Cf. A108461. Columns 1-3: A000012, A000225, A058481. Main diagonal: A023997.

T(n,k):=sum(m!*stirling2(k,m)*stirling2(n-k+1,m),m,1,min(k,n-k+1)); /* Vladimir Kruchinin, Apr 11 2015 */

antidiag(nn) = {for (n=1, nn, for (k=1, n, print1(sum(m=1, min(k, n-k+1), m!*stirling(k, m, 2)*stirling(n-k+1, m, 2)), ", "); ); print(););} \\ Michel Marcus, Apr 11 2015

tabl(nn) = {default(seriesprecision, nn); for (n=1, nn, for (k=1, nn, print1(k!*polcoeff(polcoeff(n!*exp((exp(x)-1)*(exp(y)-1))+O(x^(n+1)), n, x), k, y), ", "); ); print(););} \\ Michel Marcus, Apr 11 2015

Double e.g.f.: exp((exp(x)-1)*(exp(y)-1)).

T(n,k) = Sum{m=1..min(k,n-k+1)} m!*stirling2(k,m)*stirling2(n-k+1,m). - Vladimir Kruchinin, Apr 11 2015

cp's OEIS Frontend

A057567 Number of partitions of n where the product of parts divides n.

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A108459 Number of labeled partitions of (n,n) into pairs (i,j).

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A051707 Number of factorizations of (n,n) into pairs (j,k).

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A090806 Triangular array read by rows: T(n,k) (n >= 2, 1 <= k <= n) = number of partitions of k white balls and n-k black balls in which each part has at least one ball of each color. Also limit of the joint major-index / inversion polynomial for permutations of n elements, as n becomes infinite.

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A108455 Table read by antidiagonals: T(n,k) = number of factorizations of (n,k) into any number of pairs (i,j) with i > 0, j > 0 (and if i=1 then j=1).

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A108457 Number of partitions of (n,n) into pairs (i,j) with i>0, j>=0.

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A108458 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the last block is the singleton {k}, 1<=k<=n; the blocks are ordered with increasing least elements.

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A108462 Number of factorizations of (n,n) into pairs (i,j) with i,j >= 1, not both 1.

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