A134055 -id:A134055 - cp's OEIS frontend

A245059 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) * 2^(n-k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.

1, 1, 3, 17, 129, 1177, 12463, 149053, 1975473, 28628865, 449059179, 7562334793, 135837896769, 2588529249737, 52093016105575, 1102851978691749, 24480094135644513, 568066476383361793, 13745454515733689427, 346020796943921077057, 9043636093339718229697, 244954584886648170627641

Offset: 0

Paul D. Hanna, Jul 10 2014

nonn

			O.g.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 129*x^4 + 1177*x^5 + 12463*x^6 +...
where
A(x) = 1 + x/(1-2*x)*exp(-x/(1-2*x)) + 2^2*x^2/(1-4*x)^2*exp(-2*x/(1-4*x))/2! + 3^3*x^3/(1-6*x)^3*exp(-3*x/(1-6*x))/3! + 4^4*x^4/(1-8*x)^4*exp(-4*x/(1-8*x))/4! +...
simplifies to a power series in x with integer coefficients.
Illustrate the definition of the terms by:
a(2) = 1*1*2 + 1*1 = 3;
a(3) = 1*1*2^2 + 2*3*2 + 1*1 = 17;
a(4) = 1*1*2^3 + 3*7*2^2 + 3*6*2 + 1*1 = 129;
a(5) = 1*1*2^4 + 4*15*2^3 + 6*25*2^2 + 4*10*2 + 1*1 = 1177;
a(6) = 1*1*2^5 + 5*31*2^4 + 10*90*2^3 + 10*65*2^2 + 5*15*2 + 1*1 = 12463; ...

Cf. A134055, A245060, A218667, A218670.

{a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1)*polcoeff(1/prod(i=0, k, 1-i*x +x*O(x^(n-k))), n-k)*2^(n-k)))}
for(n=0, 25, print1(a(n), ", "))

{a(n)=polcoeff(sum(k=0, n+1, (k*x)^k/(1-2*k*x)^k*exp(-k*x/(1-2*k*x+x*O(x^n)))/k!), n)}
for(n=0, 25, print1(a(n), ", "))

O.g.f.: Sum_{n>=0} (n*x)^n/(1-2*n*x)^n * exp(-n*x/(1-2*n*x)) / n!.

A245060 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) * 3^(n-k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.

Original entry on oeis.org

1, 1, 4, 28, 271, 3172, 43174, 666577, 11445214, 215478712, 4401799930, 96757165012, 2273105615356, 56755763435503, 1499039156935948, 41714498328290992, 1218787798107634291, 37275555462806318512, 1190200470204107432854, 39581409916012393962280, 1368112674516484881342244

Offset: 0

Paul D. Hanna, Jul 10 2014

nonn

			O.g.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 271*x^4 + 3172*x^5 + 43174*x^6 +...
where
A(x) = 1 + x/(1-3*x)*exp(-x/(1-3*x)) + 2^2*x^2/(1-6*x)^2*exp(-2*x/(1-6*x))/2! + 3^3*x^3/(1-9*x)^3*exp(-3*x/(1-9*x))/3! + 4^4*x^4/(1-12*x)^4*exp(-4*x/(1-12*x))/4! +...
simplifies to a power series in x with integer coefficients.
Illustrate the definition of the terms by:
a(2) = 1*1*3 + 1*1 = 4;
a(3) = 1*1*3^2 + 2*3*3 + 1*1 = 28;
a(4) = 1*1*3^3 + 3*7*3^2 + 3*6*3 + 1*1 = 271;
a(5) = 1*1*3^4 + 4*15*3^3 + 6*25*3^2 + 4*10*3 + 1*1 = 3172;
a(6) = 1*1*3^5 + 5*31*3^4 + 10*90*3^3 + 10*65*3^2 + 5*15*3 + 1*1 = 43174; ...

Cf. A134055, A245059, A218667, A218670.

{a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1)*polcoeff(1/prod(i=0, k, 1-i*x +x*O(x^(n-k))), n-k)*3^(n-k)))}
for(n=0, 25, print1(a(n), ", "))

{a(n)=polcoeff(sum(k=0, n+1, (k*x)^k/(1-3*k*x)^k*exp(-k*x/(1-3*k*x+x*O(x^n)))/k!), n)}
for(n=0, 25, print1(a(n), ", "))

O.g.f.: Sum_{n>=0} (n*x)^n/(1-3*n*x)^n * exp(-n*x/(1-3*n*x)) / n!.

A218680 O.g.f.: A(x) = Sum_{n>=0} n^nx^n/(1-nx)^(2n)/n! exp(-nx/(1-nx)^2).

Original entry on oeis.org

1, 1, 3, 16, 111, 911, 8622, 91414, 1067579, 13564195, 185687381, 2718184470, 42288343176, 695667651368, 12049465530936, 218945489692574, 4160440403683643, 82448824370010887, 1699889286488298603, 36384381642357676480, 806926050321577391347, 18510872795071148287531

Offset: 0

Paul D. Hanna, Nov 06 2012

nonn

			O.g.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 111*x^4 + 911*x^5 + 8622*x^6 +...
where
A(x) = 1 + x/(1-x)^2*exp(-x/(1-x)^2) + 2^2*x^2/(1-2*x)^4/2!*exp(-2*x/(1-2*x)^2) + 3^3*x^3/(1-3*x)^6/3!*exp(-3*x/(1-3*x)^2) + 4^4*x^4/(1-4*x)^8/4!*exp(-4*x/(1-4*x)^2) + 5^5*x^5/(1-5*x)^10/5!*exp(-5*x/(1-5*x)^2) +...
simplifies to a power series in x with integer coefficients.

Cf. A134055.

{a(n)=local(A=1+x);A=sum(k=0,n,k^k/(1-k*x)^(2*k)*x^k/k!*exp(-k*x/(1-k*x)^2+x*O(x^n)));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A363849 Triangular array read by rows. T(n,k) is the number of Green's H-classes of rank k in the semigroup of partial transformations, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 18, 1, 1, 60, 150, 40, 1, 1, 155, 900, 650, 75, 1, 1, 378, 4515, 7000, 2100, 126, 1, 1, 889, 20286, 59535, 36750, 5586, 196, 1, 1, 2040, 84700, 435120, 486570, 148176, 12936, 288, 1, 1, 4599, 335880, 2864820, 5358150, 2876202, 493920, 27000, 405, 1

Offset: 0

Geoffrey Critzer, Jun 24 2023

Let H_f denote the H-class in the semigroup of partial transformations containing f. Then H_f contains an idempotent iff the image of f is a transversal for the kernel of f.

Let H_f ~ H_g iff the image of f is contained in the image of g and the kernel of f is more coarse than the kernel of g. Then ~ is a partial order on the H-classes, hence a preorder (quasi-order) on the semigroup. The poset is isomorphic to the Segre product of the Boolean lattice of rank n and the partition lattice of [n+1].

			Triangle begins:
 1;
 1,   1;
 1,   6,   1;
 1,  21,  18,   1;
 1,  60, 150,  40,  1;
 1, 155, 900, 650, 75, 1;
 ...

O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, 2009, Chapter 4.4 - 4.6.

Wikipedia, Green's relations
Wikipedia, Transversal (combinatorics)

Columns k=0-1 give: A000012, A066524.
Row sums give A134055(n+1).
T(n,n-1) gives A002411.
Cf. A000169, A007318, A008277, A090683, A101818.

T:= (n, k)-> binomial(n, k)*Stirling2(n+1, k+1):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 24 2023

Table[Table[Binomial[n, k] StirlingS2[n + 1, k + 1], {k, 0, n}], {n,0, 5}] // Grid

T(n,k) = A007318(n,k)*A008277(n+1,k+1).
Sum_{k=0..n} T(n,k)*k! = (n+1)^n = A000169(n+1).
T(n,1) = A101818(n,1) = A066524(n) = n*(2^n - 1).  (Every partial function of rank 1 is idempotent.)

cp's OEIS Frontend

A245059 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) * 2^(n-k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.

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A245060 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) * 3^(n-k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.

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A218680 O.g.f.: A(x) = Sum_{n>=0} n^n*x^n/(1-n*x)^(2*n)/n! * exp(-n*x/(1-n*x)^2).

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A363849 Triangular array read by rows. T(n,k) is the number of Green's H-classes of rank k in the semigroup of partial transformations, n >= 0, 0 <= k <= n.

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A218680 O.g.f.: A(x) = Sum_{n>=0} n^nx^n/(1-nx)^(2n)/n! exp(-nx/(1-nx)^2).