A220181 -id:A220181 - cp's OEIS frontend

A204066 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n + kx) / (1 + nx + k*x^2).

1, 1, 4, 16, 82, 502, 3574, 29002, 264166, 2668666, 29612014, 358025986, 4684916902, 65966957722, 994546450174, 15984888286642, 272845934899606, 4929166716321706, 93963635086523374, 1884915966747571906, 39691711412770983622, 875410001054417122042, 20180907494704416823774

Offset: 0

Paul D. Hanna, Jan 09 2013

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 82*x^4 + 502*x^5 + 3574*x^6 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(2+x)*(2+2*x)/((1+2*x+x^2)*(1+2*x+2*x^2)) + x^3*(3+x)*(3+2*x)*(3+3*x)/((1+3*x+x^2)*(1+3*x+2*x^2)*(1+3*x+3*x^2)) + x^4*(4+x)*(4+2*x)*(4+3*x)*(4+4*x)/((1+4*x+x^2)*(1+4*x+2*x^2)*(1+4*x+3*x^2)*(1+4*x+4*x^2)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A204064, A187741, A187742, A220181.

{a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (m+k*x)/(1+m*x+k*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

a(n) ~ exp(1/2) * n! * n/2. - Vaclav Kotesovec, Nov 02 2014

A249584 E.g.f.: Sum_{n>=0} (1 - exp(-n^n*x))^n.

Original entry on oeis.org

1, 1, 31, 117715, 103060086811, 35762522984596014091, 7426384178532990386136937008571, 1294894823429942167077615079432957932604708795, 253092741940931724342360814670783323840910439695820558059377771, 71352024076415778396125047299738711969981288880155877082517115026490261985701403851

Offset: 0

Paul D. Hanna, Nov 01 2014

nonn

			E.g.f.: A(x) = 1 + x + 31*x^2/2! + 117715*x^3/3! + 103060086811*x^4/4! +...
where
A(x) = 1 + (1-exp(-x)) + (1-exp(-2^2*x))^2 + (1-exp(-3^3*x))^3 + (1-exp(-4^4*x))^4 + (1-exp(-5^5*x))^5 + (1-exp(-6^6*x))^6 +...
Ordinary generating function.
O.g.f.: F(x) = 1 + x + 31*x^2 + 117715*x^3 + 103060086811*x^4 +...
where
F(x) = 1 + x/(1+x) + 2^4*2!*x^2/((1+2^2*1*x)*(1+2^2*2*x)) + 3^9*3!*x^3/((1+3^3*1*x)*(1+3^3*2*x)*(1+3^3*3*x)) + 4^16*4!*x^4/((1+4^4*1*x)*(1+4^4*2*x)*(1+4^4*3*x)*(1+4^4*4*x)) + 5^25*5!*x^5/((1+5^5*1*x)*(1+5^5*2*x)*(1+5^5*3*x)*(1+5^5*4*x)*(1+5^5*5*x)) +...

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A220181, A187755, A203798.

Flatten[{1, Table[Sum[(-1)^(n-k)*(k^k)^n*k!*StirlingS2[n, k], {k, 1, n}], {n, 1, 10}]}] (* Vaclav Kotesovec, Aug 24 2017 *)

{a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k^k*x +x*O(x^n)))^k), n)}
for(n=0, 15, print1(a(n), ", "))

{a(n)=polcoeff(sum(m=0, n, m^(m^2)*m!*x^m/prod(k=1, m, 1+m^m*k*x +x*O(x^n))), n)}
for(n=0, 15, print1(a(n), ", "))

{a(n) = sum(k=0, n, (-1)^(n-k)*k^(k*n)*k!*stirling(n, k, 2))}
for(n=0, 10, print1(a(n), ", "))

a(n) == 1 (mod 6) for n>=0.
O.g.f.: Sum_{n>=0} n^(n^2) * n! * x^n / Product_{k=1..n} (1 + n^n*k*x).
a(n) = Sum_{k=0..n} (-1)^(n-k) * (k^k)^n * k! * Stirling2(n,k).
a(n) ~ n! * n^(n^2). - Vaclav Kotesovec, Aug 24 2017

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

Original entry on oeis.org

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.

A373873 a(n) = Sum_{k=1..n} k! * k^(n-2) * Stirling2(n,k).

Original entry on oeis.org

0, 1, 3, 31, 765, 34651, 2502213, 263824891, 38248036725, 7298877611371, 1773652375115973, 534749297993098651, 195883403209280580885, 85687658454617655817291, 44120264185381411695106533, 26413555571018242181844978811

Offset: 0

Seiichi Manyama, Jun 20 2024

nonn

Cf. A122399, A244585, A373871.

Cf. A220181, A373874, A373875.

Table[Sum[k! k^(n-2) StirlingS2[n,k],{k,n}],{n,0,20}] (* Harvey P. Dale, Jul 13 2025 *)

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

E.g.f.: Sum_{k>=1} (exp(k*x) - 1)^k / k^2.

A373874 a(n) = Sum_{k=1..n} k! * k^(n-2) * Stirling1(n,k).

Original entry on oeis.org

0, 1, 1, 8, 142, 4534, 229658, 16951416, 1718394312, 229119947280, 38881745126112, 8183542269446928, 2092128552508587360, 638590833851037194256, 229398149222697428624688, 95801846241560025353728512, 46025711723325944648182502016

Offset: 0

Seiichi Manyama, Jun 20 2024

nonn

Cf. A320083, A373857, A373869.

Cf. A220181, A373873, A373875.

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

E.g.f.: Sum_{k>=1} log(1 + k*x)^k / k^2.

A373875 a(n) = Sum_{k=1..n} k! * k^(n-2) * |Stirling1(n,k)|.

Original entry on oeis.org

0, 1, 3, 32, 802, 36854, 2698598, 288450168, 42388536888, 8198703649296, 2019226648157472, 616991110153816848, 229048514514263311008, 101540936651344709359632, 52984383824921037875927760, 32145394332240602286960456192

Offset: 0

Seiichi Manyama, Jun 20 2024

nonn

Cf. A320096, A373855, A373870.

Cf. A220181, A373873, A373874.

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

E.g.f.: Sum_{k>=1} (-log(1 - k*x))^k / k^2.

A338044 E.g.f.: Sum_{j>=0} 2^j * (1 - exp(-j*x))^j.

Original entry on oeis.org

1, 2, 30, 1106, 75870, 8355602, 1349011230, 300225115346, 88096432294110, 32956583516814482, 15309575613991708830, 8646194423981547656786, 5834064910665307876000350, 4635347672272868599469126162, 4283458291212292843946379302430

Offset: 0

Vaclav Kotesovec, Oct 08 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..221

Cf. A195005, A220181.

nmax = 20; CoefficientList[Series[1 + Sum[2^j*(1 - Exp[-j*x])^j, {j, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[(-1)^(n-j) * 2^j * j^n * j! * StirlingS2[n, j], {j, 0, n}], {n, 1, 20}]

a(n) = sum(k=0, n, (-1)^(n-k)*2^k*k^n*k!*stirling(n, k, 2)); \\ Seiichi Manyama, Jan 31 2022

a(n) = Sum_{j=0..n} (-1)^(n-j) * 2^j * j^n * j! * Stirling2(n,j).

a(n) ~ c * d^n * n!^2 / sqrt(n), where d = 4.888902442941545347850916031937657541653741222401134656609725875258275714... and c = 0.4779849579705948535026794982366398948961135521828033628215401277586...

A187803 E.g.f.: Sum_{n>=0} Product_{k=1..n} (1 - exp(-nkx)).

Original entry on oeis.org

1, 1, 15, 829, 113487, 31137061, 15015039495, 11636995485949, 13584094722071007, 22735343245138118101, 52487807127760090483575, 162018777092079952134169069, 651747862300297714019151918927, 3344015911143306355676226376118341, 21488215819992049616143504500848490855

Offset: 0

Paul D. Hanna, Jan 06 2013

nonn

Compare to the e.g.f. of A079144, enumerating certain labeled interval orders:
Sum_{n>=0} Product_{k=1..n} (1 - exp(-k*x)).
Also compare to the e.g.f. of A220181: Sum_{n>=0} (1 - exp(-n*x))^n.

			E.g.f.: A(x) = 1 + x + 15*x^2/2! + 829*x^3/3! + 113487*x^4/4! +...
where
A(x) = 1 + (1-exp(-x)) + (1-exp(-2*1*x))*(1-exp(-2*2*x)) + (1-exp(-3*1*x))*(1-exp(-3*2*x))*(1-exp(-3*3*x)) + (1-exp(-4*1*x))*(1-exp(-4*2*x))*(1-exp(-4*3*x))*(1-exp(-4*4*x)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..140

Cf. A079144, A220181.

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

a(n) ~ c * d^n * (n!)^3 / sqrt(n), where d = 2.426663845780394275167988381..., c = 0.504146101604802096078745... . - Vaclav Kotesovec, Nov 03 2014

A227207 E.g.f.: Sum_{n>=0} n^n * x^n / (n! * Product_{k=0..n} (1 - nkx)).

Original entry on oeis.org

1, 1, 6, 105, 3568, 204745, 18028266, 2278860535, 394667414016, 90302033890953, 26525942216131330, 9775058594870836861, 4433256936788979640848, 2434899483389881601250937, 1597444746833206096334387802, 1237091666097626095124512681755, 1119205949224015886848972396596736

Offset: 0

Paul D. Hanna, Sep 18 2013

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 105*x^3/3! + 3568*x^4/4! + 204745*x^5/5! +...
where
A(x) = 1 + x/(1-x) + 2^2*x^2/(2!*(1-2*1*x)*(1-2*2*x)) + 3^3*x^3/(3!*(1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + 4^4*x^4/(4!*(1-4*1*x)*(1-4*2*x)*(1-4*3*x)*(1-4*4*x)) +...

Seiichi Manyama, Table of n, a(n) for n = 0..212

Cf. A119392, A229257, A229258, A229259, A229260, A229261.

Cf. A229233, A229234, A220181, A122399.

Flatten[{1,Table[n! * Sum[k^n * StirlingS2[n,k] / k!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

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

{a(n)=n!^2*polcoeff(sum(m=0, n, (exp(m*x+x*O(x^n))-1)^m/m!^2), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=n!*sum(k=0, n, k^n*stirling(n, k, 2)/k!)}
for(n=0, 20, print1(a(n), ", "))

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

a(n) = n! * Sum_{k=0..n} k^n * Stirling2(n,k) / k!.

A229045 O.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} (nk + x) / (1 + nk*x).

Original entry on oeis.org

1, 1, 8, 120, 3516, 166328, 11532458, 1100805591, 138367161579, 22150155031528, 4399449078063293, 1061636331152735214, 305921694251090263429, 103759120501255220937827, 40916086296549562038319264, 18562249570081891918836678474, 9599588267110033395243891612720

Offset: 0

Paul D. Hanna, Nov 02 2013

nonn

Compare to a g.f. of Fibonacci numbers (A000045):

Sum_{n>=0} x^n * Product_{k=1..n} (k + x)/(1 + k*x).

			G.f.: A(x) = 1 + x + 8*x^2 + 120*x^3 + 3516*x^4 + 166328*x^5 +...
where
A(x) = 1 + x*(1+x)/(1+x) + x^2*(2+x)*(4+x)/((1+2*x)*(1+4*x)) + x^3*(3+x)*(6+x)*(9+x)/((1+3*x)*(1+6*x)*(1+9*x)) + x^4*(4+x)*(8+x)*(12+x)*(16+x)/((1+4*x)*(1+8*x)*(1+12*x)*(1+16*x)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..230

Cf. A220181, A230682.

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

a(n) ~ sqrt(Pi) * n^(2*n+1/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Oct 28 2014

cp's OEIS Frontend

A204066 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n + k*x) / (1 + n*x + k*x^2).

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A249584 E.g.f.: Sum_{n>=0} (1 - exp(-n^n*x))^n.

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

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A373873 a(n) = Sum_{k=1..n} k! * k^(n-2) * Stirling2(n,k).

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A373874 a(n) = Sum_{k=1..n} k! * k^(n-2) * Stirling1(n,k).

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A373875 a(n) = Sum_{k=1..n} k! * k^(n-2) * |Stirling1(n,k)|.

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A338044 E.g.f.: Sum_{j>=0} 2^j * (1 - exp(-j*x))^j.

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Mathematica

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A187803 E.g.f.: Sum_{n>=0} Product_{k=1..n} (1 - exp(-n*k*x)).

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A227207 E.g.f.: Sum_{n>=0} n^n * x^n / (n! * Product_{k=0..n} (1 - n*k*x)).

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A204066 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n + kx) / (1 + nx + k*x^2).

A187803 E.g.f.: Sum_{n>=0} Product_{k=1..n} (1 - exp(-nkx)).

A227207 E.g.f.: Sum_{n>=0} n^n * x^n / (n! * Product_{k=0..n} (1 - nkx)).

A229045 O.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} (nk + x) / (1 + nk*x).