A092552 -id:A092552 - cp's OEIS frontend

A373872 a(n) = Sum_{k=1..n} (-1)^(n-k) * k! * k^(n-3) * Stirling2(n,k).

0, 1, 0, 1, 15, 391, 16275, 999391, 85314915, 9682617631, 1411532175075, 257220473522431, 57317980108103715, 15338554965273810271, 4855172557420679314275, 1794588990417909081447871, 766066194581899382513514915, 374061220058388896558805473311

Offset: 0

Seiichi Manyama, Jun 20 2024

nonn

Cf. A373869, A373870, A373871.

Cf. A092552, A220181.

a(n) = sum(k=1, n, (-1)^(n-k)*k!*k^(n-3)*stirling(n, k, 2));

E.g.f.: Sum_{k>=1} (1 - exp(-k*x))^k / k^3.

Sum_{k>=0} a(k+2) * x^k/k! = Sum_{k>=0} k * (1 - exp(-k*x))^k.

A382678 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+3,3) * Stirling2(n+1,k+1)^2.

Original entry on oeis.org

1, 5, 77, 2357, 118061, 8712245, 886143917, 118592620277, 20176999414061, 4249819031692085, 1084956766012858157, 329975948760472311797, 117851658189070970988461, 48830366210401091606537525, 23228207308210113849419226797, 12571433948267218576823401692917

Offset: 0

Seiichi Manyama, Apr 03 2025

nonn

Main diagonal of A382674.

Cf. A048163, A092552, A382676.

a(n) = sum(k=0, n, k!^2*binomial(k+3, 3)*stirling(n+1, k+1, 2)^2);

a(n) = (n!)^2 * [(x*y)^n] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^4.

A244437 E.g.f.: exp( Sum_{n>=1} (1 - exp(-n*x))^n / n ).

Original entry on oeis.org

1, 1, 4, 41, 845, 30012, 1650475, 130216865, 13944696526, 1945060435587, 342412144747677, 74216506678085290, 19414505134246518741, 6029823819095965829293, 2193174302711080501699684, 923346371767630311443639677, 445468655004100653462280596881, 244137607569262412209821327718964

Offset: 0

Paul D. Hanna, Aug 21 2014

nonn

Compare to: exp( Sum_{n>=1} (1 - exp(-x))^n/n ) = 1/(2-exp(x)), the e.g.f. of Fubini numbers (A000670).

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 41*x^3/3! + 845*x^4/4! + 30012*x^5/5! +...
where
log(A(x)) = (1-exp(-x)) + (1-exp(-2*x))^2/2 + (1-exp(-3*x))^3/3 + (1-exp(-4*x))^4/4 + (1-exp(-5*x))^5/5 + (1-exp(-6*x))^6/6 +...
Explicitly,
log(A(x)) = x + 3*x^2/2! + 31*x^3/3! + 675*x^4/4! + 25231*x^5/5! + 1441923*x^6/6! +...+ A092552(n)*x^n/n! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A092552, A243802.

max = 20; s = Exp[Sum[(1 - Exp[-n x])^n/n, {n, 1, max}]] + O[x]^max; CoefficientList[s, x] Range[0, max-1]! (* Jean-François Alcover, Mar 31 2016 *)

{a(n) = n!*polcoeff( exp( sum(m=1,n+1, (1 - exp(-m*x +x*O(x^n)))^m / m) ), n)}
for(n=0,20,print1(a(n),", "))

E.g.f.: exp( Sum_{n>=1} A092552(n)*x^n/n! ), where A092552(n) = Sum_{k=1..n} k!*(k-1)! * Stirling2(n, k)^2.

a(n) ~ (n!)^2 / (2 * sqrt(Pi) * sqrt(1-log(2)) * n^(3/2) * log(2)^(2*n)). - Vaclav Kotesovec, Aug 21 2014

A306267 Number of permutations of [n] within distance floor(n/2) of a fixed permutation.

Original entry on oeis.org

1, 1, 2, 3, 14, 31, 230, 675, 6902, 25231, 329462, 1441923, 22934774, 116914351, 2193664790, 12764590275, 276054834902, 1805409270031, 44222780245622, 321113303226243, 8787513806478134, 70146437009397871, 2121181056663291350, 18462286083671614275

Offset: 0

Alois P. Heinz, Feb 01 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..475

Cf. A306209.

Shifted bisections give: A048163 (even part), A092552 (odd part).

a(n) = A306209(n,floor(n/2)).

A243807 G.f.: exp( Integral Sum_{n>=1} n!n^(n-1)x^(n-1) / Product_{k=1..n} (1+knx) dx ).

Original entry on oeis.org

1, 1, 2, 12, 181, 5237, 245776, 16954562, 1612833457, 202233823341, 32315380158578, 6409484794915012, 1544967825490593319, 444799853104579872759, 150750913498484630903772, 59410000121654748323276898, 26938215605761889373324449091, 13925028099872858626544313312207

Offset: 0

Paul D. Hanna, Jun 11 2014

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 181*x^4 + 5237*x^5 + 245776*x^6 +...
where the logarithmic derivative is given by the series:
A'(x)/A(x) = 1/(1+x) + 2!*2^1*x/((1+1*2*x)*(1+2*2*x)) + 3!*3^2*x^2/((1+1*3*x)*(1+2*3*x)*(1+3*3*x)) + 4!*4^3*x^3/((1+1*4*x)*(1+2*4*x)*(1+3*4*x)*(1+4*4*x)) + 5!*5^4*x^4/((1+1*5*x)*(1+2*5*x)*(1+3*5*x)*(1+4*5*x)*(1+5*5*x)) +...
Explicitly,
A'(x)/A(x) = 1 + 3*x + 31*x^2 + 675*x^3 + 25231*x^4 + 1441923*x^5 + 116914351*x^6 +...+ A092552(n+1)*x^n +...
compare to:
G(x) = x + 3*x^2/2! + 31*x^3/3! + 675*x^4/4! + 25231*x^5/5! + 1441923*x^6/6! +...+ A092552(n)*x^n/n! +...
where G(x) = (1-exp(-x)) + (1-exp(-2*x))^2/2 + (1-exp(-3*x))^3/3 + (1-exp(-4*x))^4/4 +...

Cf. A092552, A243809.

{a(n)=local(A=1+x); A=exp(intformal(sum(m=1, n+1, m^(m-1)*m!*x^(m-1)/prod(k=1, m, 1+m*k*x +x*O(x^n))))); polcoeff(A,n)}
for(n=0, 20, print1(a(n), ", "))

/* From g.f. exp( Sum_{n>=1} A092552(n)*x^n/n ): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{A092552(n)=if(n<=0, 0, sum(k=1, n, k!*(k-1)! * Stirling2(n, k)^2))}
{a(n)=polcoeff(exp(sum(m=1,n,A092552(m)*x^m/m) +x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} A092552(n)*x^n/n ) where Sum_{n>=1} A092552(n)*x^n/n! = Sum_{n>=1} (1 - exp(-n*x))^n / n.

A255192 Triangle of number of connected subgraphs of K(n,n) with m edges.

Original entry on oeis.org

1, 4, 1, 81, 78, 36, 9, 1, 4096, 8424, 9552, 7464, 4272, 1812, 560, 120, 16, 1, 390625, 1359640, 2696200, 3880300, 4394600, 4059000, 3111140, 1994150, 1070150, 478800, 176900, 53120, 12650, 2300, 300, 25, 1, 60466176, 314452800, 939988800, 2075760000

Offset: 1

Thomas Dybdahl Ahle, Feb 16 2015

m ranges from 2n-1 to n^2.

First column is A068087.

			Triangle begins:
----|------------------------------------------------------------
n\m |  1 2 3 4  5  6    7    8    9   10   11   12  13  14 15 16
----|------------------------------------------------------------
1   |  1
2   |  - - 4 1
3   |  - - - - 81 78   36    9    1
4   |  - - - -  -  - 4096 8424 9552 7464 4272 1812 560 120 16  1

Cf. A005333 (row sums?).

from math import comb as binomial
def f(x, a, b, k):
    if b == k == 0:
        return 1
    if b == 0 or k == 0:
        return 0
    if x == a:
        return sum(binomial(a, n) * f(x, x, b - 1, k - n) for n in range(1, a + 1))
    return sum(binomial(b, n) * f(x, x, n, k2) * f(n, b, a - x, k - k2)
        for n in range(1, b + 1) for k2 in range(0, k + 1) )
def a(n, m):
    return f(1, n, n, m)
for n in range(1, 5):
    print([a(n, m) for m in range(1, n * n + 1)])

Sum(k>=0, T(n,k)*(-1)^k ) = A136126(2*n-1,n-1) = A092552(n+1), alternating row sums.

A382827 a(n) = Sum_{k=0..n} k! * (k+1)! * Stirling1(n+1,k+1)^2.

Original entry on oeis.org

1, 3, 34, 854, 37556, 2546852, 246113904, 32104625520, 5433891955968, 1157778241057152, 303197684900579712, 95717977509042032256, 35847800701044816248064, 15713483696924130220098816, 7969364997624587289470810112, 4630203661005094483980386924544

Offset: 0

Seiichi Manyama, Apr 06 2025

nonn

Main diagonal of A382824.

Cf. A092552, A382804.

a(n) = sum(k=0, n, k!*(k+1)!*stirling(n+1, k+1, 1)^2);

a(n) = (n!)^2 * [(x*y)^n] 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y))^2 ).

a(n) = (n!)^2 * [(x*y)^n] 1 / ( (1+x) * (1+y) * (1 - log(1+x) * log(1+y))^2 ).

A256286 Number of Hamiltonian cycles in a tournament on 3n vertices constructed by taking 3 copies of a transitive tournament on n vertices and placing each copy on a vertex of a directed 3-cycle, with all edges between the copies oriented in the direction of the cycle.

Original entry on oeis.org

1, 5, 181, 20381, 4940101, 2230319165, 1692864345061, 1997649164976701, 3461226344139932101, 8430034728440212411325, 27875832970537774479832741, 121651171242426267003975420221, 684351364639262056751911086836101, 4865203490721997132612204548628407485

Offset: 1

Hayato Ushijima-Mwesigwa, Jun 03 2015

nonn

N. J. Calkin, B. Novick and H. Ushijima-Mwesigwa, What Moser Could Have Asked: Counting Hamilton Cycles in Tournaments, arXiv:1506.00699 [math.CO], 2015.

Cf. A000629, A000670, A048993, A092552, A242280.

a(n)=sum(k=1,n, (stirling(n,k,2)*k!)^3/k) \\ Charles R Greathouse IV, Jun 03 2015

a(n) = Sum_{k=1..n} (S(n,k)*k!)^3/k, where S(n,k) is the Stirling number of the second kind (A048993, Stirling set numbers).

Offset changed to 1 by Georg Fischer, Jun 20 2022

cp's OEIS Frontend

A373872 a(n) = Sum_{k=1..n} (-1)^(n-k) * k! * k^(n-3) * Stirling2(n,k).

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A382678 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+3,3) * Stirling2(n+1,k+1)^2.

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A244437 E.g.f.: exp( Sum_{n>=1} (1 - exp(-n*x))^n / n ).

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Mathematica

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A306267 Number of permutations of [n] within distance floor(n/2) of a fixed permutation.

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A243807 G.f.: exp( Integral Sum_{n>=1} n!*n^(n-1)*x^(n-1) / Product_{k=1..n} (1+k*n*x) dx ).

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A255192 Triangle of number of connected subgraphs of K(n,n) with m edges.

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A382827 a(n) = Sum_{k=0..n} k! * (k+1)! * Stirling1(n+1,k+1)^2.

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A256286 Number of Hamiltonian cycles in a tournament on 3n vertices constructed by taking 3 copies of a transitive tournament on n vertices and placing each copy on a vertex of a directed 3-cycle, with all edges between the copies oriented in the direction of the cycle.

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A243807 G.f.: exp( Integral Sum_{n>=1} n!n^(n-1)x^(n-1) / Product_{k=1..n} (1+knx) dx ).