A122399 -id:A122399 - cp's OEIS frontend

A301583 G.f.: Sum_{n>=0} 4^n * ((1+x)^n - 1)^n.

1, 4, 64, 1792, 70736, 3600128, 224255040, 16521605376, 1405131880000, 135480346104896, 14602769310474240, 1739917222954854400, 227081534040721917952, 32217108743091290851328, 4936803887495636263284736, 812576030237749532251019264, 142976863303365903802301729024, 26781577193841845859144244087808, 5320767287406003709062843236972544, 1117525692987087894816123931091214336

Offset: 0

Paul D. Hanna, Mar 24 2018

nonn

In general, if k > 0 and g.f.: Sum_{j>=0} k^j * ((1+x)^j - 1)^j, then a(n) ~ c * (1 + k*exp(1/r))^n * r^(2*n) * n! / sqrt(n), where r is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/k and c is a constant (dependent only on k). - Vaclav Kotesovec, Oct 08 2020

			G.f.: A(x) = 1 + 4*x + 64*x^2 + 1792*x^3 + 70736*x^4 + 3600128*x^5 + 224255040*x^6 + 16521605376*x^7 + 1405131880000*x^8 + ...
such that
A(x) = 1 + 4*((1+x)-1) + 16*((1+x)^2-1)^2 + 64*((1+x)^3-1)^3 + 256*((1+x)^4-1)^4 + 1024*((1+x)^5-1)^5 + 4096*((1+x)^6-1)^6 + ...
Also,
A(x) = 1/5 + 4*(1+x)/(1 + 4*(1+x))^2 + 16*(1+x)^4/(1 + 4*(1+x)^2)^3 + 64*(1+x)^9/(1 + 4*(1+x)^3)^4 + 256*(1+x)^16/(1 + 4*(1+x)^4)^5 + 1024*(1+x)^25/(1 + 4*(1+x)^5)^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..318

Cf. A122400, A301581, A301582, A301463.

Cf. A122399, A195005, A195263, A338040.

nmax = 20; CoefficientList[Series[1 + Sum[4^j*((1 + x)^j - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2020 *)

{a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, 4^m * ((1+x +o)^m - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

G.f.: Sum_{n>=0} 4^n * (1+x)^(n^2) /(1 + 4*(1+x)^n)^(n+1).

a(n) ~ c * d^n * n! / sqrt(n), where d = (1 + 4*exp(1/r)) * r^2 = 11.35554580636894436474777793373210745006910386794268638744346793426715754570218..., where r = 0.95894043087329419322124137165060249611787608513866855417024... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/4 and c = 0.358692703763731594549618907599728117285634153... - Vaclav Kotesovec, Aug 09 2018, updated Oct 08 2020

A122418 a(n) = Sum_{k=0..n} (k-1)^nk!Stirling2(n,k).

Original entry on oeis.org

1, 0, 2, 54, 2534, 186030, 19794662, 2885980734, 552803552534, 134687987183790, 40686498089484422, 14925683377452413214, 6536580413039406774134, 3368723388994026165415950, 2018248855531992511720945382, 1390953089533285777007059354494, 1092714503596231472933813958469334

Offset: 0

Vladeta Jovovic, Sep 03 2006

G. C. Greubel, Table of n, a(n) for n = 0..229

Cf. A122419, A122420, A122399.

A122418 := proc(n) sum((k-1)^n*k!*combinat[stirling2](n,k),k=0..n) ; end; for n from 0 to 16 do print(A122418(n)) ; od ; # R. J. Mathar, Feb 10 2007

a[n_] := Sum[ (k-1)^n*k!*StirlingS2[n, k], {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 26 2013 *)

for(n=0,50, print1(sum(k=0,n, (k-1)^n*k!*stirling(n,k,2)), ", ")) \\ G. C. Greubel, Nov 15 2017

E.g.f.: Sum((exp((n-1)*x)-1)^n, n=0..infinity).

a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.10430562057820038909699083625848223918044424242153125547162600916636313858475... . - Vaclav Kotesovec, May 07 2014

More terms from R. J. Mathar, Feb 10 2007

A229257 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - n^2kx).

Original entry on oeis.org

1, 1, 2, 14, 168, 3147, 90563, 3561231, 185790622, 12599020184, 1071164190670, 111813313594259, 14140296360430353, 2132273568722682621, 378197030144360862958, 78127192632748956075174, 18627308660113953164384812, 5081218748742336002185874439, 1574128413278644602881499193579

Offset: 0

Paul D. Hanna, Sep 17 2013

nonn

			O.g.f.: A(x) = 1 + x + 2*x^2 + 14*x^3 + 168*x^4 + 3147*x^5 + 90563*x^6 +...
where
A(x) = 1 + x/(1-x) + x^2/((1-2^2*1*x)*(1-2^2*2*x)) + x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 2*x^2/2! + 14*x^3/3! + 168*x^4/4! + 3147*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/(2!*4^2) + (exp(9*x)-1)^3/(3!*9^3) + (exp(16*x)-1)^4/(4!*16^4) + (exp(25*x)-1)^5/(5!*25^5) +...

Seiichi Manyama, Table of n, a(n) for n = 0..242

Cf. A229258, A229259, A229260, A229261; A229233, A229234, A220181, A122399.

Flatten[{1,Table[Sum[(k^2)^(n-k) * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

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

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

{a(n)=sum(k=0, n, (k^2)^(n-k) * stirling(n, k, 2))}
for(n=0,30,print1(a(n),", "))

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

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

A242229 a(n) = Sum_{k=0..n} k^(3n) k! * StirlingS2(n,k).

Original entry on oeis.org

1, 1, 129, 121171, 421842405, 3921960731851, 80097035486409669, 3154805675402432477371, 218356776433458097793841045, 24765902586563160053438320367371, 4359137561016969073655241431827801509, 1139916274502953599866121961715757905518171

Offset: 0

Vaclav Kotesovec, May 08 2014

Seiichi Manyama, Table of n, a(n) for n = 0..125

Cf. A203798, A242228

Cf. A000670, A122399, A229260.

Table[Sum[k^(3*n) * k! * StirlingS2[n,k], {k,0,n}], {n,0,20}]

a(n) = sum(k=0, n, k!*k^(3*n)*stirling(n, k, 2)); \\ Seiichi Manyama, Feb 01 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (exp(k^3*x)-1)^k))) \\ Seiichi Manyama, Feb 01 2022

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k^3*x)^k/prod(j=1, k, 1-k^3*j*x))) \\ Seiichi Manyama, Feb 01 2022

a(n) ~ c * d^n * (n!)^4 / n^(3/2), where d = 20.5647332000203822461493845960846630764635... = r^4*(1+exp(3/r)), r = 0.97762267432285928683132021521727105447350... is the root of the equation (1+exp(-3/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.0600744446309702764688382302731840300640714536...
E.g.f.: Sum_{k>=0} (exp(k^3*x) - 1)^k. - Seiichi Manyama, Feb 01 2022
G.f.: Sum_{k>=0} k! * (k^3*x)^k/Product_{j=1..k} (1 - k^3*j*x). - Seiichi Manyama, Feb 01 2022

a(0)=1 prepended by Seiichi Manyama, Feb 01 2022

A350722 a(n) = Sum_{k=0..n} k! * k^(k+n) * Stirling2(n,k).

Original entry on oeis.org

1, 1, 33, 4567, 1652493, 1235777551, 1656820330173, 3619858882041487, 12034209740498292093, 57813156798714532953391, 385490564193781368103929213, 3454086424032897924417605526607, 40500898779980258599522326286912893

Offset: 0

Seiichi Manyama, Feb 03 2022

nonn

Cf. A122399, A195005, A195263, A338040.

Cf. A108459, A350721, A351117.

a[0] = 1; a[n_] := Sum[k! * k^(k+n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 13, 0] (* Amiram Eldar, Feb 03 2022 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(k*x)-1))^k)))

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

a(n) ~ exp(exp(-2)/2) * n! * n^(2*n). - Vaclav Kotesovec, Feb 04 2022

A373871 a(n) = Sum_{k=1..n} k! * k^(n-3) * Stirling2(n,k).

Original entry on oeis.org

0, 1, 2, 13, 233, 8311, 495437, 44495263, 5619239453, 949995402271, 207228784973597, 56681221280785663, 19000392210559326173, 7661410911700580500831, 3658694812581483750630557, 2042247041839449013948374463, 1317554928647608644852032652893

Offset: 0

Seiichi Manyama, Jun 20 2024

nonn

Cf. A373869, A373870, A373872.

Cf. A122399, A244585, A373873.

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

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

A351117 a(n) = Sum_{k=0..n} k! * k^(kn) Stirling2(n,k).

Original entry on oeis.org

1, 1, 33, 118483, 103098350565, 35763050750177408011, 7426387531294259002278007386693, 1294894837982331421844458945612619053737859003, 253092742000650212461957357208907985560332648454746968725711765

Offset: 0

Seiichi Manyama, Feb 01 2022

nonn

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

Cf. A000670, A122399, A229260, A242229.

Cf. A249584.

a:= n-> add(k!*k^(k*n)*Stirling2(n,k), k=0..n):
seq(a(n), n=0..10);  # Alois P. Heinz, Feb 01 2022

a[0] = 1; a[n_] := Sum[k! * k^(k*n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 9, 0] (* Amiram Eldar, Feb 02 2022 *)

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

my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (exp(k^k*x)-1)^k)))

my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k^k*x)^k/prod(j=1, k, 1-k^k*j*x)))

E.g.f.: Sum_{k>=0} (exp(k^k*x) - 1)^k.
G.f.: Sum_{k>=0} k! * (k^k*x)^k/Product_{j=1..k} (1 - k^k*j*x).
a(n) ~ n! * n^(n^2). - Vaclav Kotesovec, Feb 04 2022

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.

A248656 E.g.f.: Sum_{n>=0} exp(n(n+1)/2x) / (1 + exp(nx))^(n+1) = Sum_{n>=0} a(n) x^(2n) / (2n)!.

Original entry on oeis.org

1, -4, 1172, -2394604, 17925470132, -356711164156204, 15557257046545589492, -1306859934761006954164204, 192757826813283097789632563252, -46564510721452609888686654192978604, 17449940281041871638688960825766828695412, -9712709908164237387647891995373875626734039404

Offset: 0

Paul D. Hanna, Oct 26 2014

sign

Compare to an e.g.f. of A122399: Sum_{n>=0} exp(n^2*x)/(1 + exp(n*x))^(n+1).

			E.g.f.: A(x) = 1 - 4*x^2/2! + 1172*x^4/4! - 2394604*x^6/6! + 17925470132*x^8/8! -+...
where
A(x) = 1/2 + exp(x)/(1+exp(x))^2 + exp(3*x)/(1+exp(2*x))^3 + exp(6*x)/(1+exp(3*x))^4 + exp(10*x)/(1+exp(4*x))^5 + exp(15*x)/(1+exp(5*x))^6 + exp(21*x)/(1+exp(6*x))^7 +...

Cf. A122399, A248657.

\p100 \\ set precision
{A=Vec(serlaplace(sum(n=0,800,1.*exp((n^2+n)/2*x +O(x^31))/(1 + exp(n*x +O(x^31)))^(n+1)) ))}
for(n=1,#A\2,print1(round(A[2*n-1]),", "))

A248798 E.g.f.: Sum_{n>=0} (exp((n+1)*x) - 1)^n.

Original entry on oeis.org

1, 2, 22, 554, 25366, 1844042, 195320182, 28410656234, 5435279204566, 1323405341744522, 399637856402514742, 146583029519189084714, 64192080498935634774166, 33083140834428825424557002, 19821855421651521815140243702, 13662206025133299916665629413994

Offset: 0

Paul D. Hanna, Oct 30 2014

nonn

			E.g.f.: E(x) = 1 + 2*x + 22*x^2/2! + 554*x^3/3! + 25366*x^4/4! + 1844042*x^5/5! +...
such that
E(x) = 1 + (exp(2*x)-1) + (exp(3*x)-1)^2 + (exp(4*x)-1)^3 + (exp(5*x)-1)^4 +...
The e.g.f. is also given by the infinite series:
E(x) = 1/2 + exp(2*x)/(1+exp(x))^2 + exp(6*x)/(1+exp(2*x))^3 + exp(12*x)/(1+exp(3*x))^4 + exp(20*x)/(1+exp(4*x))^5 + exp(30*x)/(1+exp(5*x))^6 +...
or, equivalently,
E(x) = 1/2 + 1/(1+exp(-x))^2 + 1/(1+exp(-2*x))^3 + 1/(1+exp(-3*x))^4 + 1/(1+exp(-4*x))^5 + 1/(1+exp(-5*x))^6 + 1/(1+exp(-6*x))^7 +...
ORDINARY GENERATING FUNCTION.
O.g.f.: A(x) = 1 + 2*x + 22*x^2 + 554*x^3 + 25366*x^4 + 1844042*x^5 +...
where
A(x) = 1 + 2*x/(1-2*x) + 3^2*2!*x^2/((1-3*1*x)*(1-3*2*x)) + 4^3*3!*x^3/((1-4*1*x)*(1-4*2*x)*(1-4*3*x)) + 5^4*4!*x^4/((1-5*1*x)*(1-5*2*x)*(1-5*3*x)*(1-5*4*x)) + 6^5*5!*x^5/((1-6*1*x)*(1-6*2*x)*(1-6*3*x)*(1-6*4*x)*(1-6*5*x)) +...

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

Cf. A248656, A122399.

Table[Sum[(k+1)^n * k! * StirlingS2[n,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Nov 02 2014 *)

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

/* From e.g.f. infinite series: */
\p100 \\ set precision
{A=Vec(serlaplace(sum(n=0, 500, 1.*exp(n*(n+1)*x +O(x^26))/(1 + exp(n*x +O(x^26)))^(n+1)) ))}
for(n=0, #A-1, print1(round(A[n+1]), ", "))

/* From o.g.f.: */
{a(n)=polcoeff(sum(m=0, n, (m+1)^m*m!*x^m/prod(k=1, m, 1-(m+1)*k*x+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=sum(k=0, n, (k+1)^n * k! * Stirling2(n,k))}
for(n=0, 25, print1(a(n), ", "))

E.g.f.: Sum_{n>=0} exp(n*(n+1)*x) / (1 + exp(n*x))^(n+1).
E.g.f.: Sum_{n>=0} 1 / (1 + exp(-n*x))^(n+1).
O.g.f.: Sum_{n>=0} (n+1)^n * n! * x^n / Product_{k=1..n} (1 - (n+1)*k*x).
a(n) = Sum_{k=0..n} (k+1)^n * k! * Stirling2(n,k).
a(n) ~ c * r^(2*n) * n^(2*n+1/2) * (1+exp(1/r))^n / exp(2*n), where r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation (1+exp(-1/r)) * LambertW(-exp(-1/r)/r) = -1/r, and c = 6.4659886138749084757432110017013709735979825189027... . - Vaclav Kotesovec, Nov 07 2014

cp's OEIS Frontend

A301583 G.f.: Sum_{n>=0} 4^n * ((1+x)^n - 1)^n.

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A122418 a(n) = Sum_{k=0..n} (k-1)^n*k!*Stirling2(n,k).

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A229257 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - n^2*k*x).

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A242229 a(n) = Sum_{k=0..n} k^(3*n) * k! * StirlingS2(n,k).

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A350722 a(n) = Sum_{k=0..n} k! * k^(k+n) * Stirling2(n,k).

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A373871 a(n) = Sum_{k=1..n} k! * k^(n-3) * Stirling2(n,k).

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A351117 a(n) = Sum_{k=0..n} k! * k^(k*n) * Stirling2(n,k).

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A122418 a(n) = Sum_{k=0..n} (k-1)^nk!Stirling2(n,k).

A229257 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - n^2kx).

A242229 a(n) = Sum_{k=0..n} k^(3n) k! * StirlingS2(n,k).

A351117 a(n) = Sum_{k=0..n} k! * k^(kn) Stirling2(n,k).

A248656 E.g.f.: Sum_{n>=0} exp(n(n+1)/2x) / (1 + exp(nx))^(n+1) = Sum_{n>=0} a(n) x^(2n) / (2n)!.