A226750 -id:A226750 - cp's OEIS frontend

A226775 Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.

4, 0, 6, 3, 7, 5, 7, 3, 9, 9, 5, 9, 9, 5, 9, 9, 0, 7, 6, 7, 6, 9, 5, 8, 1, 2, 4, 1, 2, 4, 8, 3, 9, 7, 5, 8, 2, 1, 0, 9, 9, 7, 5, 7, 5, 1, 8, 1, 1, 4, 0, 6, 3, 5, 0, 0, 0, 4, 9, 5, 4, 8, 8, 3, 0, 3, 9, 1, 5, 0, 1, 5, 1, 8, 3, 8, 1, 2, 0, 4, 9, 7, 6, 7, 2, 5, 0, 0, 7, 2, 3, 3, 8, 1, 5, 5, 9, 2, 8, 5, 8, 2, 9, 3, 8

Offset: 0

José María Grau Ribas, Jun 17 2013

			-0.4063757399599599076769581241248397582109975751811406350004954883....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A106533, A187655, A187657, A217899, A217900, A226750, A243227, A245109, A247238, A258467, A337458.

RealDigits[N[ProductLog[-2/E^2], 105]][[1]] (* corrected by Vaclav Kotesovec, Feb 21 2014 *)

solve(x=-1, x=0, x*exp(x) + 2*exp(-2)) \\ G. C. Greubel, Nov 15 2017

Equals -2*A106533.

Equals LambertW(-2*exp(-2)).

A217913 O.g.f.: Sum_{n>=0} (n^3)^n * exp(-n^3x) x^n / n!.

Original entry on oeis.org

1, 1, 31, 3025, 611501, 210766920, 110687251039, 82310957214948, 82318282158320505, 106563273280541795575, 173373343599189364594756, 346289681454731077633095526, 833091176987705031151553054843, 2376102520162485084539597049185710

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

			O.g.f.: A(x) = 1 + x + 31*x^2 + 3025*x^3 + 611501*x^4 + ... + Stirling2(3*n, n)*x^n + ...
where
A(x) = 1 + 1^3*x*exp(-1^3*x) + 2^6*exp(-2^3*x)*x^2/2! + 3^9*exp(-3^3*x)*x^3/3! + 4^12*exp(-4^3*x)*x^4/4! + 5^15*exp(-5^3*x)*x^5/5! + ...
simplifies to a power series in x with integer coefficients.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A219228, A007820, A217914, A217915, A217900, A008277.

Table[StirlingS2[3*n,n],{n,0,20}] (* Vaclav Kotesovec, Feb 28 2013 *)

makelist(stirling2(3*n, n), n, 0, 13); /* Martin Ettl, Oct 15 2012 */

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

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^3)^k*x^k/(1+k^3*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), 2*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(3*n, n)}
for(n=0,20,print1(a(n),", "))

a(n) = Stirling2(3*n, n).
a(n) = [x^(3*n)] (3*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(2*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^3)^k*x^k / (1 + k^3*x)^(k+1).
a(n) ~ 9^n*exp(n*(c+1))*n^(2*n)/((c+3)^(2*n)*sqrt(2*Pi*(c+1)*n)), where c = -0.1785606278779211... = LambertW(-3/exp(3)) = A226750. - Vaclav Kotesovec, Feb 28 2013

A305140 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x)^2 dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 1, 33, 3422, 710395, 245288190, 127281447538, 92967363233586, 91202509214139831, 115939599286159123295, 185623891076803234259504, 365706330842590993758556662, 869715472542563657980211015186, 2456766458611829222907737567821138, 8131203421875726862447708824758000364

Offset: 0

Paul D. Hanna, May 31 2018

nonn

It is remarkable that this sequence should consist entirely of integers.

Note: 0 = [x^n] exp( n * Integral F(x)^2 dx ) / F(x) holds for n > 0 when F(x) = 1 + x*F(x)^3 is a g.f. of A001764.

			O.g.f.: A(x) = 1 + x + 33*x^2 + 3422*x^3 + 710395*x^4 + 245288190*x^5 + 127281447538*x^6 + 92967363233586*x^7 + 91202509214139831*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^3*Integral A(x)^2 dx)/A(x) begins:
n=0: [1, -1, -32, -3357, -702560, -243654950, ...];
n=1: [1, 0, -63/2, -3367, -5633901/8, -2440775421/10, ...];
n=2: [1, 7, 0, -3325, -715316, -1235155194/5, ...];
n=3: [1, 26, 665/2, 0, -5720533/8, -5095053859/20, ...];
n=4: [1, 63, 2016, 41699, 0, -1290302622/5, ...];
n=5: [1, 124, 15561/2, 328643, 80013395/8, 0, ...];
n=6: [1, 215, 23296, 1697283, 93264388, 19574613422/5, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^3*Integral A(x)^2 dx)/A(x), for n > 0.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 67*x^2 + 6910*x^3 + 1428723*x^4 + 492223022*x^5 + 255112067610*x^6 + 186210340326168*x^7 + 182601537143712727*x^8 + ...
exp( Integral A(x)^2 dx) = 1 + x + 5*x^2/2! + 415*x^3/3! + 167521*x^4/4! + 172296341*x^5/5! + 355443416701*x^6/6! + 1288266047868955*x^7/7! + 7518341623369166465*x^8/8! + ...
A'(x)/A(x) = 1 + 65*x + 10168*x^2 + 2825845*x^3 + 1222346736*x^4 + 762046826846*x^5 + 649809039848130*x^6 + 728835192043655757*x^7 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..250

Cf. A305137, A305138, A305139, A305141, A305142, A305143.

{a(n) = my(A=[1],m); for(i=1,n+1, m=#A; A=concat(A,0); A[m+1] = Vec( exp(m^3*intformal(Ser(A)^2)) / Ser(A) )[m+1] );A[n+1]}
for(n=0,20,print1(a(n),", "))

a(n) ~ 3^(3*n - 1) * n^(2*n - 1/2) / (sqrt(2*Pi*(1-c)) * c^n * (3-c)^(2*n - 1) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = -A226750. - Vaclav Kotesovec, Oct 19 2020

A304323 O.g.f. A(x) satisfies: [x^n] exp( n^3 * x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 1, 25, 2317, 466241, 162016980, 85975473871, 64545532370208, 65062315637060121, 84756897268784533255, 138581022247955235150982, 277878562828788369685779910, 670574499099019193091230751539, 1917288315895234006935990419270242, 6409780596355519454337664637246378856, 24774712941456386970945752104780461007848, 109632095120643795798521114315908854415860345

Offset: 0

Paul D. Hanna, May 11 2018

nonn

It is conjectured that the coefficients of o.g.f. A(x) consist entirely of integers.
Equals row 3 of table A304320.
O.g.f. A(x) = 1/(1 - x*B(x)), where B(x) is the o.g.f. of A107675.
Logarithmic derivative of o.g.f. A(x), A'(x)/A(x), equals o.g.f. of A304312.
Conjecture: given o.g.f. A(x), the coefficient of x^n in A'(x)/A(x) is the number of connected n-state finite automata with 3 inputs (A006692).

			O.g.f.: A(x) = 1 + x + 25*x^2 + 2317*x^3 + 466241*x^4 + 162016980*x^5 + 85975473871*x^6 + 64545532370208*x^7 + 65062315637060121*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^3*x) / A(x) begins:
n=0: [1, -1, -48, -13608, -11065344, -19317285000, -61649646030720, ...];
n=1: [1, 0, -49, -13754, -11120067, -19372748284, -61765715993765, ...];
n=2: [1, 7, 0, -14440, -11517184, -19768841352, -62587640670464, ...];
n=3: [1, 26, 627, 0, -12292251, -20908064898, -64905483973113, ...];
n=4: [1, 63, 3920, 227032, 0, -22551552136, -69768485886848, ...];
n=5: [1, 124, 15327, 1874642, 213958781, 0, -75806801733845, ...];
n=6: [1, 215, 46176, 9893016, 2100211968, 416846973816, 0, ...];
n=7: [1, 342, 116915, 39937660, 13616254341, 4604681316698, 1458047845980391, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^3*x ) / A(x) = 0 for n>=0.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304313:
A'(x)/A(x) = 1 + 49*x + 6877*x^2 + 1854545*x^3 + 807478656*x^4 + 514798204147*x^5 + 451182323794896*x^6 + 519961864703259753*x^7 + ... + A304313(n)*x^n +...
INVERT TRANSFORM.
1/A(x) = 1 - x*B(x), where B(x) is the o.g.f. of A107675:
B(x) = 1 + 24*x + 2268*x^2 + 461056*x^3 + 160977375*x^4 + 85624508376*x^5 + 64363893844726*x^6 + ... + A107675(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A304320, A304313, A304321, A304322, A304324, A304325.

Cf. A006691, A107675.

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^3 +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0,25, print1( a(n),", "))

a(n) ~ sqrt(1-c) * 3^(3*n) * n^(2*n - 1/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n) * exp(2*n)), where c = -A226750 = -LambertW(-3*exp(-3)). - Vaclav Kotesovec, Aug 31 2020

A317346 O.g.f. A(x) satisfies: [x^n] exp( n^3x - n^2A(x) ) = 0 for n >= 1.

Original entry on oeis.org

1, 2, 72, 8096, 1839000, 695334816, 392764566208, 309340607492096, 323795915817507936, 434750954619876448000, 728547799352068864173632, 1490865523016798790557180928, 3659466509860384349989504297344, 10614823215131644149237135937187328, 35927108634064565449228268842108588800, 140351379904337650357154561973550135705600

Offset: 1

Paul D. Hanna, Jul 26 2018

nonn

It is remarkable that this sequence should consist entirely of integers.

			O.g.f.: A(x) = x + 2*x^2 + 72*x^3 + 8096*x^4 + 1839000*x^5 + 695334816*x^6 + 392764566208*x^7 + 309340607492096*x^8 + ...
such that [x^n] exp( n^3*x - n^2*A(x) ) = 0  for n >= 1.
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^3*x - n^2*A(x) ) begins:
n=1: [1, 0, -4, -432, -194256, -220662720, -500627544000, ...];
n=2: [1, 4, 0, -1856, -805376, -898258176, -2023715201024, ...];
n=3: [1, 18, 288, 0, -1989792, -2154563712, -4727980751616, ...];
n=4: [1, 48, 2240, 94464, 0, -4244861952, -9137589559296, ...];
n=5: [1, 100, 9900, 959200, 84852400, 0, -15901448888000, ...];
n=6: [1, 180, 32256, 5738688, 1003636224, 161358324480, 0, ...];
n=7: [1, 294, 86240, 25218144, 7335234144, 2103824749824, 557359956846336, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 5*x^2/2! + 445*x^3/3! + 196105*x^4/4! + 221673401*x^5/5! + 501981700621*x^6/6! + 1983064113021685*x^7/7!  + ... + A317345(n)*x^n/n! + ...

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A317344, A317345, A317347, A319941, A319942, A319943, A319944.

{a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^3*x +x*O(x^#A)) / Ser(A)^(m^2) )[m+1]/m^2 ); polcoeff( log(Ser(A)),n)}
for(n=1,20,print1(a(n),", "))

a(n) ~ sqrt(1 - c) * 3^(3*n - 7/3) * n^(2*n - 5/2) / (sqrt(2*Pi) * exp(2*n) * c^(n - 1/3) * (3 - c)^(2*n - 2)), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968086697... = -A226750. - Vaclav Kotesovec, Aug 07 2018

A296172 E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^3) = [x^n] A(x)^(n^3) for n>=1.

Original entry on oeis.org

1, 1, -5, -197, -65111, -62390159, -125012786669, -447082993406405, -2583111044504384687, -22511408975342644804991, -281350305428215911326408789, -4850582201056517165575319399909, -111834955668396093904661955538037255, -3361788412998032560821833199260880942287, -128987969989211586699135087535153035663946301, -6203990036027464835833031041177436339788197962789

Offset: 0

Paul D. Hanna, Dec 07 2017

sign

Compare e.g.f. to: [x^(n-1)] exp(x)^n = [x^n] exp(x)^n for n>=1.

			E.g.f.: A(x) = 1 + x - 5*x^2/2! - 197*x^3/3! - 65111*x^4/4! - 62390159*x^5/5! - 125012786669*x^6/6! - 447082993406405*x^7/7! - 2583111044504384687*x^8/8! - 22511408975342644804991*x^9/9! - 281350305428215911326408789*x^10/10! - 4850582201056517165575319399909*x^11/11! - 111834955668396093904661955538037255*x^12/12! +...
To illustrate [x^(n-1)] A(x)^(n^3) = [x^n] A(x)^(n^3), form a table of coefficients of x^k in A(x)^(n^3) that begins as
n=1: [(1), (1), -5/2, -197/6, -65111/24, -62390159/120, -125012786669/720, ...];
n=2: [1, (8), (8), -1040/3, -71152/3, -64676744/15, -63817770776/45, ...];
n=3: [1, 27, (567/2), (567/2), -787941/8, -648507951/40, -405807483249/80, ...];
n=4: [1, 64, 1856, (88448/3), (88448/3), -689015872/15, -611019817664/45, ...];
n=5: [1, 125, 14875/2, 1649375/6, (156207625/24), (156207625/24), ...];
n=6: [1, 216, 22680, 1533168, 73812816, (12455715384/5), (12455715384/5), ...];
n=7: [1, 343, 115591/2, 38174185/6, 12294445009/24, 3808296195823/120, (1051338418817239/720), (1051338418817239/720), ...];
...
in which the diagonals indicated by parenthesis are equal.
Dividing the coefficients of x^(n-1)/(n-1)! in A(x)^(n^3) by n^3, we obtain the following sequence:
[1, 1, 21, 2764, 1249661, 1383968376, 3065126585473, 11913154589356672, 74286423963211939641, 696469981042645688972800, ...].
LOGARITHMIC PROPERTY.
Amazingly, the logarithm of the e.g.f. A(x) is an integer series:
log(A(x)) = x - 3*x^2 - 30*x^3 - 2686*x^4 - 517311*x^5 - 173118807*x^6 - 88535206152*x^7 - 63977172334344*x^8 - 61971659588102940*x^9 - 77470793599569049440*x^10 - 121439997599825393413344*x^11 - 233353875172602479932391040*x^12 - 539638027429765922735002220880*x^13 - 1479049138515818646669055218090480*x^14 - 4742815067612592169849894663392228480*x^15 +...

Paul D. Hanna, Table of n, a(n) for n = 0..180

Cf. A296173, A296170, A296174, A296176.

{a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^3)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^3 ); n!*A[n+1]}
for(n=0,20,print1(a(n),", "))

The logarithm of the e.g.f. A(x) is an integer series:
 log(A(x)) = Sum{n>=1} A296173(n) * x^n.
E.g.f. A(x) satisfies:
_ 1/n! * d^n/dx^n A(x)^(n^3) = 1/(n-1)! * d^(n-1)/dx^(n-1) A(x)^(n^3) for n>=1, when evaluated at x = 0.
a(n) ~ -sqrt(1-c) * 3^(3*n - 3) * n^(3*n - 3) / (c^n * (3-c)^(2*n - 3) * exp(3*n)), where c = -LambertW(-3*exp(-3)) = -A226750. - Vaclav Kotesovec, Oct 13 2020

A296173 G.f. equals the logarithm of the e.g.f. of A296172.

Original entry on oeis.org

1, -3, -30, -2686, -517311, -173118807, -88535206152, -63977172334344, -61971659588102940, -77470793599569049440, -121439997599825393413344, -233353875172602479932391040, -539638027429765922735002220880, -1479049138515818646669055218090480, -4742815067612592169849894663392228480, -17597031102801426396121130730318359114880, -74817150772352720408567833273371047298417408

Offset: 1

Paul D. Hanna, Dec 07 2017

sign

E.g.f. G(x) of A296172 satisfies: [x^(n-1)] G(x)^(n^3) = [x^n] G(x)^(n^3) for n>=1.

			G.f. A(x) = x - 3*x^2 - 30*x^3 - 2686*x^4 - 517311*x^5 - 173118807*x^6 - 88535206152*x^7 - 63977172334344*x^8 - 61971659588102940*x^9 - 77470793599569049440*x^10 - 121439997599825393413344*x^11 - 233353875172602479932391040*x^12 - 539638027429765922735002220880*x^13 - 1479049138515818646669055218090480*x^14 - 4742815067612592169849894663392228480*x^15 +...
such that
G(x) = exp(A(x)) = 1 + x - 5*x^2/2! - 197*x^3/3! - 65111*x^4/4! - 62390159*x^5/5! - 125012786669*x^6/6! - 447082993406405*x^7/7! - 2583111044504384687*x^8/8! - 22511408975342644804991*x^9/9! - 281350305428215911326408789*x^10/10! - 4850582201056517165575319399909*x^11/11! - 111834955668396093904661955538037255*x^12/12! +...
satisfies [x^(n-1)] G(x)^(n^3) = [x^n] G(x)^(n^3) for n>=1.
Series_Reversion(A(x)) = x + 3*x^2 + 48*x^3 + 3271*x^4 + 575163*x^5 + 185377116*x^6 + 93039467356*x^7 + 66505075585875*x^8 + 63970743282062646*x^9 + 79580632411431634441*x^10 + 124299284968805234137968*x^11 + 238188439678208173206500760*x^12 +...+ A295813(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..180

Cf. A296172, A295813, A296171, A296175, A296177.

{a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^3)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^3 ); polcoeff(log(Ser(A)),n)}
for(n=1,30,print1(a(n),", "))

a(n) ~ -sqrt(1-c) * 3^(3*n - 3) * n^(2*n - 7/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n - 3) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = -A226750. - Vaclav Kotesovec, Oct 13 2020

A305145 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral 1/A(x) dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 1, 21, 1886, 381735, 134584434, 72514796422, 55192152857400, 56287911330435339, 74043167807482274450, 122040226074154110294114, 246341047594913378800486668, 597752265070243363135031803950, 1716967839431601765698468898047292, 5762431350664488199395983555754160140, 22346478647255335081326815815314403748524

Offset: 0

Paul D. Hanna, May 31 2018

nonn

Note: 0 = [x^n] exp( n * Integral 1/F(x) dx ) / F(x) holds for n > 0 when F(x) = 1 + x.

It is remarkable that this sequence should consist entirely of integers.

			O.g.f.: A(x) = 1 + x + 21*x^2 + 1886*x^3 + 381735*x^4 + 134584434*x^5 + 72514796422*x^6 + 55192152857400*x^7 + 56287911330435339*x^8 + 74043167807482274450*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^3*Integral 1/A(x) dx)/A(x) begins:
n=0: [1, -1, -20, -1845, -377584, -133748650, -72227419704, -55040493806445, ...];
n=1: [1, 0, -21, -1872, -379890, -134201604, -72383437035, -55123034324112, ...];
n=2: [1, 7, 0, -2033, -396970, -137452068, -73490534208, -55705843833995, ...];
n=3: [1, 26, 304, 0, -437155, -147006370, -76635381186, -57333497856168, ...];
n=4: [1, 63, 1932, 36075, 0, -163035066, -83375170872, -60709861617885, ...];
n=5: [1, 124, 7605, 304780, 8444291, 0, -92858506104, -66905102463320, ...];
n=6: [1, 215, 22984, 1625463, 84879650, 3287781224, 0, -74725745263095, ...];
n=7: [1, 342, 58290, 6597132, 556856100, 37129859844, 1920530286186, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^3 * Integral 1/A(x) dx)/A(x), for n > 0.
RELATED SERIES.
1/A(x) = 1 - x - 20*x^2 - 1845*x^3 - 377584*x^4 - 133748650*x^5 - 72227419704*x^6 - 55040493806445*x^7 - 56174066916766400*x^8 - 73928074251625193826*x^9 + ...
exp( Integral 1/A(x) dx) = 1 + x - 7*x^3 - 468*x^4 - 75978*x^5 - 22366934*x^6 - 10340491005*x^7 - 6890379290514*x^8 - 6248442860989378*x^9 - 7399048902607246248*x^10 + ..., which is an integer series.
A'(x)/A(x) = 1 + 41*x + 5596*x^2 + 1518597*x^3 + 670826996*x^4 + 434225271374*x^5 + 385813724342292*x^6 + 449847594913097949*x^7 + 665870324595294969196*x^8 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..198

Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.

Cf. A305144, A305146, A305147, A304861, A304862.

{a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^3*intformal(1/Ser(A))) / Ser(A) )[m+1] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ sqrt(1-c) * 3^(3*n - 1) * n^(2*n - 1/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n - 1) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = -A226750. - Vaclav Kotesovec, Oct 19 2020

A317347 O.g.f. A(x) satisfies: [x^n] exp( n^3x - nA(x) ) = 0 for n >= 1.

Original entry on oeis.org

1, 9, 552, 85842, 24653700, 11219022936, 7393496092416, 6649411839351120, 7822998961379912592, 11662362974001268456560, 21487905123054927319268352, 47958258768575173308988367040, 127523196462392124262710980808384, 398397752352904475778061859746030080, 1445051361690004153927005867189533921280

Offset: 1

Paul D. Hanna, Jul 29 2018

nonn

It is remarkable that this sequence should consist entirely of integers.

			O.g.f.: A(x) = x + 9*x^2 + 552*x^3 + 85842*x^4 + 24653700*x^5 + 11219022936*x^6 + 7393496092416*x^7 + 6649411839351120*x^8 + ...
such that [x^n] exp( n^3*x - n*A(x) ) = 0  for n >= 1.
ILLUSTRATION OF DEFINITION.
The table of coefficients in  begins:
n=1: [1, 0, -18, -3312, -2059236, -2957847840, -8077030651800, ...];
n=2: [1, 6, 0, -7056, -4281984, -6040453824, -16367904244224, ...];
n=3: [1, 24, 522, 0, -6980580, -9667325376, -25560523291464, ...];
n=4: [1, 60, 3528, 189792, 0, -14146669440, -37025599219200, ...];
n=5: [1, 120, 14310, 1679040, 181358460, 0, -51097553724600, ...];
n=6: [1, 210, 43992, 9173088, 1887214464, 358972896960, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 19*x^2/2! + 3367*x^3/3! + 2074537*x^4/4! + 2969379361*x^5/5! + 8096147776171*x^6/6! + 37321188279552199*x^7/7! + ...

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A317344, A317346.

{a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^3*x +x*O(x^#A)) / Ser(A)^m )[m+1]/m ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))

a(n) ~ sqrt(1-c) * 3^(3*n-1) * n^(2*n - 3/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n-1) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968086697... = -A226750. - Vaclav Kotesovec, Aug 06 2018

A221214 O.g.f.: Sum_{n>=0} (3n+1)^(3n+1) * exp(-(3n+1)^3x) * x^n / n!.

Original entry on oeis.org

1, 255, 395388, 1525953330, 10977340509135, 126827739333023274, 2148335345336441463090, 50163717301669569182864400, 1544377393328765493716910877185, 60615459491155396034172113103266025, 2954227738557038665136475801709196246304

Offset: 0

Paul D. Hanna, Feb 27 2013

nonn

			O.g.f.: A(x) = 1 + 255*x + 395388*x^2 + 1525953330*x^3 + 10977340509135*x^4 +...
where A(x) = exp(-x) + 4^4*x*exp(-4^3*x) + 7^7*exp(-7^3*x)*x^2/2! + 10^10*exp(-10^3*x)*x^3/3! + 13^13*exp(-13^3*x)*x^4/4! + 16^16*exp(-16^3*x)*x^5/5! +... is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..159

Cf. A220955, A213193, A217900, A007820, A242449.

Table[1/n!*Sum[(-1)^(n-k)*Binomial[n,k]*(3*k+1)^(3*n+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[Binomial[3*n+1,n+k]*3^(n+k)*StirlingS2[n+k,n],{k,0,2*n+1}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)

{a(n)=polcoeff(sum(k=0, n, (3*k+1)^(3*k+1)*exp(-(3*k+1)^3*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=(1/n!)*polcoeff(sum(k=0, n, (3*k+1)^(3*k+1)*x^k/(1+(3*k+1)^3*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(3*k+1)^(3*n+1))}
for(n=0, 20, print1(a(n), ", "))

a(n) = 1/n! * [x^n] Sum_{k>=0} (3*k+1)^(3*k+1) * x^k / (1 + (3*k+1)^3*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (3*k+1)^(3*n+1).
a(n) ~ n^(2*n+1/2) * 3^(6*n+7/3) / (sqrt(2*Pi*(1-r)) * exp(2*n) * r^(n+1/3) * (3-r)^(2*n+1)), where r = -LambertW(-3*exp(-3)) = 0.1785606278779211... (see A226750 = -r) . - Vaclav Kotesovec, May 13 2014

cp's OEIS Frontend

A226775 Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.

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A217913 O.g.f.: Sum_{n>=0} (n^3)^n * exp(-n^3*x) * x^n / n!.

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A305140 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x)^2 dx ) / A(x), for n > 0.

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A304323 O.g.f. A(x) satisfies: [x^n] exp( n^3 * x ) / A(x) = 0 for n>0.

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A317346 O.g.f. A(x) satisfies: [x^n] exp( n^3*x - n^2*A(x) ) = 0 for n >= 1.

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A296172 E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^3) = [x^n] A(x)^(n^3) for n>=1.

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A296173 G.f. equals the logarithm of the e.g.f. of A296172.

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A217913 O.g.f.: Sum_{n>=0} (n^3)^n * exp(-n^3x) x^n / n!.

A317346 O.g.f. A(x) satisfies: [x^n] exp( n^3x - n^2A(x) ) = 0 for n >= 1.

A317347 O.g.f. A(x) satisfies: [x^n] exp( n^3x - nA(x) ) = 0 for n >= 1.

A221214 O.g.f.: Sum_{n>=0} (3n+1)^(3n+1) * exp(-(3n+1)^3x) * x^n / n!.