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Search: a317855 -id:a317855
Displaying 1-10 of 19 results found.
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A122399
a(n) = Sum_{k=0..n} k^n * k! * Stirling2(n,k).
+10
36
1, 1, 9, 211, 9285, 658171, 68504709, 9837380491, 1863598406805, 450247033371451, 135111441590583909, 49300373690091496171, 21495577955682021043125, 11037123350952586270549531, 6591700149366720366704735109
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 2, 1, 3, 3, 0, 1, 2, 1, 3, 3, 0, ...], with an apparent period of 6. Cf. A338040. - Peter Bala, May 31 2022
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
E.g.f.: Sum_{n >= 0} (exp(n*x) - 1)^n. - Vladeta Jovovic, Sep 03 2006
E.g.f.: Sum_{n>=0} exp(n^2*x) / (1 + exp(n*x))^(n+1). - Paul D. Hanna, Oct 26 2014
E.g.f.: Sum_{n>=0} exp(-n*x) / (1 + exp(-n*x))^(n+1). - Paul D. Hanna, Oct 30 2014
O.g.f.: Sum_{n>=0} n^n * n! * x^n / Product_{k=1..n} (1 - n*k*x). - Paul D. Hanna, Jan 05 2013
Limit n->infinity (a(n)/n!)^(1/n)/n = ((1+exp(1/r))*r^2)/exp(1) = A317855/exp(1) = 1.162899527477400818845..., where r = 0.87370243323966833... is the root of the equation 1/(1+exp(-1/r)) = -r*LambertW(-exp(-1/r)/r). - Vaclav Kotesovec, Jun 21 2013
a(n) ~ c * A317855^n * (n!)^2 / sqrt(n), where c = 0.327628285569869481442286492410507030710253054522608... - Vaclav Kotesovec, Aug 09 2018
Let A(x) = 1 + x + 9*x^2/2! + 211*x^3/3! + ... denote the e.g.f. of the sequence. Let F(x) denote the series reversion of A(x) - 1 = x - 9*x^2/2 + 16*x^3/3 - 205*x^4/4 - 2714*x^5/5 - .... Then both dF/dx = 1 - 9*x + 16*x^2 - 205*x^3 - 2714*x^4 - ... and exp(F(x)) = 1 + x - 4*x^2 + x^3 - 38*x^4 - 606*x^5 - ... have integer coefficients. Note that 1 + series reversion(exp(F(x)) - 1) is the o.g.f. for A122400. - Peter Bala, Aug 09 2022
EXAMPLE
E.g.f.: A(x) = 1 + x + 9*x^2/2! + 211*x^3/3! + 9285*x^4/4! + 658171*x^5/5! + ...
such that
A(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2 + (exp(3*x)-1)^3 + (exp(4*x)-1)^4 + ...
The e.g.f. is also given by the series:
A(x) = 1/2 + exp(x)/(1+exp(x))^2 + exp(4*x)/(1+exp(2*x))^3 + exp(9*x)/(1+exp(3*x))^4 + exp(16*x)/(1+exp(4*x))^5 + exp(25*x)/(1+exp(5*x))^6 + ...
or, equivalently,
A(x) = 1/2 + exp(-x)/(1+exp(-x))^2 + exp(-2*x)/(1+exp(-2*x))^3 + exp(-3*x)/(1+exp(-3*x))^4 + exp(-4*x)/(1+exp(-4*x))^5 + exp(-5*x)/(1+exp(-5*x))^6 + ...
MAPLE
a := n -> add(k^n*k!*combinat[stirling2](n, k), k=0..n); # Max Alekseyev, Feb 01 2007
MATHEMATICA
Flatten[{1, Table[Sum[k^n*k!*StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jun 21 2013 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, m^m*m!*x^m/prod(k=1, m, 1-m*k*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 05 2013
(PARI) {a(n)=n!*polcoeff(sum(k=0, n, (exp(k*x +x*O(x^n)) - 1)^k), n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 26 2014
(PARI) /* From e.g.f. infinite series: */
\p100 \\ set precision
{A=Vec(serlaplace(sum(n=0, 500, 1.*exp(n^2*x +O(x^26))/(1 + exp(n*x +O(x^26)))^(n+1)) ))}
for(n=0, #A-1, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 30 2014
CROSSREFS
Cf. A195415, A122400, A220181, A224899, A245322, A320083, A338040.
KEYWORD
nonn,easy
AUTHOR
Vladeta Jovovic, Aug 31 2006
EXTENSIONS
More terms from Max Alekseyev, Feb 01 2007
STATUS
approved
A122400
Number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.
+10
25
1, 1, 4, 31, 338, 4769, 82467, 1687989, 39905269, 1069863695, 32071995198, 1062991989013, 38596477083550, 1523554760656205, 64961391010251904, 2975343608212835855, 145687881987604377815, 7594435556630244257213
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..375
FORMULA
a(n) = (1/n!)* Sum_{k=0..n} Stirling1(n,k)*A122399(k).
G.f.: Sum_{n>=0} ((1+x)^n - 1)^n. - Vladeta Jovovic, Sep 03 2006
G.f.: Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n)^(n+1). - Paul D. Hanna, Mar 23 2018
a(n) ~ c * d^n * n! / 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 = 0.2796968489586733500739737080739303725411427162653658... . - Vaclav Kotesovec, May 07 2014
MAPLE
A122399 := proc(n) option remember ; add( combinat[stirling2](n, k)*k^n*k!, k=0..n) ; end: A122400 := proc(n) option remember ; add( combinat[stirling1](n, k)*A122399(k), k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ", A122400(n)) ; od ; # R. J. Mathar, May 18 2007
MATHEMATICA
max = 17; CoefficientList[ Series[ 1 + Sum[ ((1 + x)^n - 1)^n, {n, 1, max}], {x, 0, max}], x] (* Jean-François Alcover, Mar 26 2013, after Vladeta Jovovic *)
CROSSREFS
Cf. A104602, A220353, A301581, A301582, A301583, A301584.
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Aug 31 2006
EXTENSIONS
More terms from R. J. Mathar, May 18 2007
STATUS
approved
A303056
G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - A(x))^n.
+10
18
1, 1, 1, 8, 89, 1326, 24247, 521764, 12867985, 357229785, 11017306489, 373675921093, 13825260663882, 554216064798423, 23934356706763264, 1108017262467214486, 54747529760516714323, 2876096694574711401525, 160092696678371426933342, 9413031424290635395882462, 583000844360279565483710624
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
for any fixed integer k; here, k = 1 with r = 1, p = -A(x), q = (1+x). - Paul D. Hanna, Jun 22 2019
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^n - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n*A(x))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.1610886538654... and c = 0.11739505492506... - Vaclav Kotesovec, Sep 26 2020
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 8*x^3 + 89*x^4 + 1326*x^5 + 24247*x^6 + 521764*x^7 + 12867985*x^8 + 357229785*x^9 + 11017306489*x^10 + ...
such that
1 = 1 + ((1+x) - A(x)) + ((1+x)^2 - A(x))^2 + ((1+x)^3 - A(x))^3 + ((1+x)^4 - A(x))^4 + ((1+x)^5 - A(x))^5 + ((1+x)^6 - A(x))^6 + ((1+x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1+x)/(1 + (1+x)*A(x))^2 + (1+x)^4/(1 + (1+x)^2*A(x))^3 + (1+x)^9/(1 + (1+x)^3*A(x))^4 + (1+x)^16/(1 + (1+x)^4*A(x))^5 + (1+x)^25/(1 + (1+x)^5*A(x))^6 + (1+x)^36/(1 + (1+x)^6*A(x))^7 + ...
RELATED SERIES.
log(A(x)) = x + x^2/2 + 22*x^3/3 + 325*x^4/4 + 6186*x^5/5 + 137380*x^6/6 + 3478651*x^7/7 + 98674253*x^8/8 + 3096911434*x^9/9 + ...
PARTICULAR VALUES.
Although the power series A(x) diverges at x = -1/2, it may be evaluated formally.
Let t = A(-1/2) = 0.545218973635949431234950245034944106957612798888179456724264...
then t satisfies
(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.
(2) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).
Also,
A(r) = 1/2 at r = -0.54683649902292991492196620520872286547799291909992048564578...
where
(1) 1 = Sum_{n>=0} ( (1+r)^n - 1/2 )^n.
(2) 1 = Sum_{n>=0} (1+r)^(-n) / ( 1/(1+r)^n + 1/2 )^(n+1).
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^m - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A304642, A304639, A303926.
Cf. A321602, A321603, A321604, A321605.
Cf. A326282, A326283, A326284.
Cf. A337755, A337756, A337757.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 19 2018
STATUS
approved
A244585
E.g.f.: Sum_{n>=1} (exp(n*x) - 1)^n / n.
+10
8
1, 5, 79, 2621, 149071, 12954365, 1596620719, 264914218301, 56934521042191, 15385666763366525, 5106110041462786159, 2041611328770984737981, 967972254733121945653711, 536962084044317668770841085, 344546100916295014902350596399
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Compare to: Sum_{n>=1} (1 - exp(-n*x))^n / n, the e.g.f. of A092552.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..200
FORMULA
O.g.f.: Sum_{n>=1} n^(n-1) * n! * x^n / Product_{k=1..n} (1 - n*k*x).
a(n) ~ c * d^n * (n!)^2 / n^(3/2), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491..., r = 0.873702433239668330496568304720719298... is the root of the equation exp(1/r)/r + (1+exp(1/r)) * LambertW(-exp(-1/r)/r) = 0, and c = 0.37498840921734807101035131780130551... . - Vaclav Kotesovec, Aug 21 2014
EXAMPLE
E.g.f.: A(x) = x + 5*x^2/2! + 79*x^3/3! + 2621*x^4/4! + 149071*x^5/5! +...
where
A(x) = (exp(x)-1) + (exp(2*x)-1)^2/2 + (exp(3*x)-1)^3/3 + (exp(4*x)-1)^4/4 + (exp(5*x)-1)^5/5 + (exp(6*x)-1)^6/6 + (exp(7*x)-1)^7/7 +...
Exponentiation yields:
exp(A(x)) = 1 + x + 6*x^2/2! + 95*x^3/3! + 3043*x^4/4! + 167342*x^5/5! +...+ A243802(n)*x^n/n! +...
The O.G.F. begins:
F(x) = x + 5*x^2 + 79*x^3 + 2621*x^4 + 149071*x^5 + 12954365*x^6 +...
where
F(x) = x/(1-x) + 2*2!*x^2/((1-2*x)*(1-4*x)) + 3^2*3!*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 4^3*4!*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 5^4*5!*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) +...
PROG
(PARI) {a(n) = n!*polcoeff( sum(m=1, n+1, (exp(m*x +x*O(x^n)) - 1)^m / m), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=if(n<1, 0, polcoeff(sum(m=1, n, m^(m-1) * m! * x^m / prod(k=1, m, 1-m*k*x +x*O(x^n))), n))}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
Cf. A092552, A243802.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 21 2014
STATUS
approved
A304639
G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n.
+10
8
1, 1, 2, 11, 117, 1735, 31853, 689043, 17079221, 476238926, 14742680162, 501584454703, 18605089712174, 747393133162471, 32332767332220442, 1498961537925543920, 74153115616699819304, 3899494667155151052688, 217246028175467702590241, 12783023090792392539557926, 792236994094236725330142276, 51585659784100723438219893047, 3520987513029712770759434038820
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n.
(2) 1 = Sum_{n>=0} ( 1 - (1-x)^n*A(x) )^n / (1-x)^(n^2).
(3) 1 = Sum_{n>=0} (1-x)^n / ( (1-x)^n + A(x) )^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.16107844724485... - Vaclav Kotesovec, Oct 14 2020
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 117*x^4 + 1735*x^5 + 31853*x^6 + 689043*x^7 + 17079221*x^8 + 476238926*x^9 + 14742680162*x^10 + 501584454703*x^11 + ...
is such that
1 = 1 + (1/(1-x) - A(x)) + (1/(1-x)^2 - A(x))^2 + (1/(1-x)^3 - A(x))^3 + (1/(1-x)^4 - A(x))^4 + (1/(1-x)^5 - A(x))^5 + (1/(1-x)^6 - A(x))^6 + (1/(1-x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1-x)/((1-x) + A(x))^2 + (1-x)^2/((1-x)^2 + A(x))^3 + (1-x)^3/((1-x)^3 + A(x))^4 + (1-x)^4/((1-x)^4 + A(x))^5 + (1-x)^5/((1-x)^5 + A(x))^6 + (1-x)^6/((1-x)^6 + A(x))^7 + ...
PARTICULAR VALUES.
Although the power series A(x) diverges at x = -1, it may be evaluated formally.
Let t = A(-1) = 0.5452189736359494312349502450349441069576127988881794567242641...
then t satisfies
(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.
(2) 1 = Sum_{n>=0} ( 1 - 2^n*t )^n / 2^(n^2).
(3) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (1/(1-x +x^2*O(x^n))^m - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A326262, A326263, A326264, A326265.
Cf. A303056.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 16 2018
STATUS
approved
A122418
a(n) = Sum_{k=0..n} (k-1)^n*k!*Stirling2(n,k).
+10
6
1, 0, 2, 54, 2534, 186030, 19794662, 2885980734, 552803552534, 134687987183790, 40686498089484422, 14925683377452413214, 6536580413039406774134, 3368723388994026165415950, 2018248855531992511720945382, 1390953089533285777007059354494, 1092714503596231472933813958469334
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..229
FORMULA
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
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) for(n=0, 50, print1(sum(k=0, n, (k-1)^n*k!*stirling(n, k, 2)), ", ")) \\ G. C. Greubel, Nov 15 2017
CROSSREFS
Cf. A122419, A122420, A122399.
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Sep 03 2006
EXTENSIONS
More terms from R. J. Mathar, Feb 10 2007
STATUS
approved
A301584
G.f.: Sum_{n>=0} ((1+x)^(2*n) - 1)^n.
+10
5
1, 2, 17, 264, 5784, 163610, 5667551, 232280480, 10991951114, 589780778314, 35379149504709, 2346218124687516, 170439977706143335, 13459938431949414118, 1148107512505151099653, 105194122765096703619248, 10303686044959088279454117, 1074408525677705370497704526, 118828297870115694372235974855, 13893778686151373846512389392672, 1712370237144948501135060958863978
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..340
FORMULA
G.f.: Sum_{n>=0} (1+x)^(2*n^2) /(1 + (1+x)^(2*n))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = 2*A317855 = 6.3221773077308576276603444051762649834527655483771126832545564150753941184386... and c = 0.302715376391132275494451399946850989516917... - Vaclav Kotesovec, Aug 09 2018
EXAMPLE
G.f.: A(x) = 1 + 2*x + 17*x^2 + 264*x^3 + 5784*x^4 + 163610*x^5 + 5667551*x^6 + 232280480*x^7 + 10991951114*x^8 + 589780778314*x^9 + ...
such that
A(x) = 1 + ((1+x)^2-1) + ((1+x)^4-1)^2 + ((1+x)^6-1)^3 + ((1+x)^8-1)^4 + ((1+x)^10-1)^5 + ((1+x)^12-1)^6 + ((1+x)^14-1)^7 + ...
Also,
A(x) = 1/2 + (1+x)^2/(1 + (1+x)^2)^2 + (1+x)^8/(1 + (1+x)^4)^3 + (1+x)^18/(1 + (1+x)^6)^4 + (1+x)^32/(1 + (1+x)^8)^5 + (1+x)^50/(1 + (1+x)^10)^6 + ...
PROG
(PARI) {a(n) = my(A, o=x*O(x^n)); A = sum(m=0, n, ((1+x +o)^(2*m) - 1)^m ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A122400, A301585, A301586.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 24 2018
STATUS
approved
A122419
Number of labeled digraphs with n arcs and with no vertex of indegree 0.
+10
4
1, 0, 1, 8, 93, 1354, 23900, 496244, 11855700, 320428318, 9667220397, 322072882348, 11744421711587, 465270864839688, 19899234175413257, 913836170567749048, 44849438199960187278, 2342666125012348876152
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Table of n, a(n) for n=0..17.
FORMULA
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A122418(k).
G.f.: Sum_{n>=0} ((1+x)^(n-1) - 1)^n.
a(n) ~ c * d^n * n! / 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 = 0.08904589343883135100956914504938... . - Vaclav Kotesovec, May 07 2014
MAPLE
A122418 := proc(n) option remember ; add( combinat[stirling2](n, k)*(k-1)^n*k!, k=0..n) ; end: A122419 := proc(n) option remember ; add( combinat[stirling1](n, k)*A122418(k), k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ", A122419(n)) ; od ; # R. J. Mathar, May 18 2007
MATHEMATICA
nmax=20; CoefficientList[Series[Sum[((1+x)^(n-1)-1)^n, {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 06 2014 *)
CROSSREFS
Cf. A122420, A122400.
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Sep 03 2006
EXTENSIONS
More terms from R. J. Mathar, May 18 2007
STATUS
approved
A122420
Number of labeled directed multigraphs with n arcs and with no vertex of indegree 0.
+10
4
1, 0, 1, 10, 120, 1778, 31685, 661940, 15882128, 430607370, 13022755068, 434697574538, 15875944361864, 629756003982336, 26963278837704185, 1239382820431888898, 60875147436141987437, 3181961834442383306068
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Table of n, a(n) for n=0..17.
FORMULA
a(n) = (1/n!)*Sum_{k=0..n} |Stirling1(n,k)|*A122418(k). G.f.: A(x/(1-x)) where A(x) is g.f. for A122419.
a(n) ~ c * d^n * n! / 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 = 0.1221803955695846906452721220983425... . - Vaclav Kotesovec, May 07 2014
MAPLE
A122418 := proc(n) option remember ; add( combinat[stirling2](n, k)*(k-1)^n*k!, k=0..n) ; end: A122420 := proc(n) option remember ; add( abs(combinat[stirling1](n, k))*A122418(k), k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ", A122420(n)) ; od ; # R. J. Mathar, May 18 2007
MATHEMATICA
Table[1/n!*Sum[Abs[StirlingS1[n, k]]*Sum[(m-1)^k*m!*StirlingS2[k, m], {m, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 07 2014 *)
CROSSREFS
Cf. A104209.
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Sep 03 2006
EXTENSIONS
More terms from R. J. Mathar, May 18 2007
STATUS
approved
A243802
E.g.f.: exp( Sum_{n>=1} (exp(n*x) - 1)^n / n ).
+10
4
1, 1, 6, 95, 3043, 167342, 14175447, 1715544861, 280986929888, 59828264507385, 16056622678756319, 5300955907062294008, 2110872493413444115109, 997542435957462115205773, 551887323312314977683048334, 353334615697796170374209624907, 259179558930246734075836153918127
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Compare to: exp( Sum_{n>=1} (exp(x) - 1)^n/n ) = 1/(2-exp(x)), the e.g.f. of Fubini numbers (A000670).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..200
FORMULA
a(n) ~ c * d^n * (n!)^2 / n^(3/2), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491..., r = 0.873702433239668330496568304720719298... is the root of the equation exp(1/r)/r + (1+exp(1/r)) * LambertW(-exp(-1/r)/r) = 0, and c = 0.37498840921734807101035131780130551... . - Vaclav Kotesovec, Aug 21 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 95*x^3/3! + 3043*x^4/4! + 167342*x^5/5! +...
PROG
(PARI) {a(n) = n!*polcoeff( exp( sum(m=1, n+1, (exp(m*x +x*O(x^n)) - 1)^m / m) ), n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
Cf. A244585, A244437.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 21 2014
STATUS
approved
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