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Decimal expansion of a constant related to the asymptotics of A122400.
+10
19
3, 1, 6, 1, 0, 8, 8, 6, 5, 3, 8, 6, 5, 4, 2, 8, 8, 1, 3, 8, 3, 0, 1, 7, 2, 2, 0, 2, 5, 8, 8, 1, 3, 2, 4, 9, 1, 7, 2, 6, 3, 8, 2, 7, 7, 4, 1, 8, 8, 5, 5, 6, 3, 4, 1, 6, 2, 7, 2, 7, 8, 2, 0, 7, 5, 3, 7, 6, 9, 7, 0, 5, 9, 2, 1, 9, 3, 0, 4, 6, 1, 1, 2, 1, 9, 7, 5, 7, 4, 6, 8, 5, 4, 9, 7, 8, 4, 5, 9, 3, 2, 4, 2, 2, 7
FORMULA
Equals (1+exp(1/r))*r^2, where r = 0.873702433239668330496568304720719298213992... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0.
EXAMPLE
3.161088653865428813830172202588132491726382774188556341627278...
MATHEMATICA
r = r /. FindRoot[E^(1/r)/r + (1 + E^(1/r)) * ProductLog[-E^(-1/r)/r] == 0, {r, 3/4}, WorkingPrecision -> 120]; RealDigits[(1 + Exp[1/r])*r^2][[1]]
PROG
(PARI) r=solve(r=.8, 1, exp(1/r)/r + (1+exp(1/r))*lambertw(-exp(-1/r)/r))
CROSSREFS
Cf. A121886, A122399, A122400, A122418, A122419, A122420, A227619, A232192, A243802, A244585, A248798, A317340, A326010.
a(n) = Sum_{k=1..n} k^(2*n-1) * k! * Stirling2(n,k).
+10
5
1, 17, 1651, 473741, 300257371, 355743405917, 706872713310331, 2182548723605418941, 9894910566488309801851, 63052832687428562206049117, 545439670961897317869306191611, 6226501736967631584015448186252541, 91619831483112536750163352484302283131
FORMULA
a(n) ~ c * d^n * (n!)^3 / n^2, where d = r^3*(1+exp(2/r)) = 7.8512435106631367719817991716164612615296980032514..., r = 0.94520217245242431308104743874492469552738... is the root of the equation (1+exp(-2/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.15095210978787998524366903417512193343948127919...
MATHEMATICA
Table[Sum[k^(2*n-1) * k! * StirlingS2[n, k], {k, 1, n}], {n, 1, 20}]
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
COMMENTS
Compare to: exp( Sum_{n>=1} (exp(x) - 1)^n/n ) = 1/(2-exp(x)), the e.g.f. of Fubini numbers ( A000670).
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), ", "))
a(n) = Sum_{k=1..n} k! * k^(n-1) * |Stirling1(n,k)|.
+10
4
0, 1, 5, 80, 2690, 155074, 13658386, 1706098008, 286888266696, 62485391828448, 17112247116585744, 5755236604915060944, 2331975856351260982848, 1120439648590390138640304, 629855675998212293917375344, 409557081242059531918330384896
FORMULA
E.g.f.: Sum_{k>=1} (-log(1 - k*x))^k / k.
MATHEMATICA
nmax=15; Range[0, nmax]!CoefficientList[Series[Sum[(-Log[1 - k*x])^k / k, {k, nmax}], {x, 0, nmax}], x] (* Stefano Spezia, Jun 19 2024 *)
PROG
(PARI) a(n) = sum(k=1, n, k!*k^(n-1)*abs(stirling(n, k, 1)));
a(n) = Sum_{k=1..n} k! * k^(n-3) * Stirling2(n,k).
+10
4
0, 1, 2, 13, 233, 8311, 495437, 44495263, 5619239453, 949995402271, 207228784973597, 56681221280785663, 19000392210559326173, 7661410911700580500831, 3658694812581483750630557, 2042247041839449013948374463, 1317554928647608644852032652893
FORMULA
E.g.f.: Sum_{k>=1} (exp(k*x) - 1)^k / k^3.
PROG
(PARI) a(n) = sum(k=1, n, k!*k^(n-3)*stirling(n, k, 2));
a(n) = Sum_{k=1..n} k! * k^(n-1) * Stirling1(n,k).
+10
3
0, 1, 3, 32, 734, 28994, 1752046, 150262104, 17356844088, 2597710341600, 488957612319984, 113044488306692304, 31490845086661001664, 10403092187976909854640, 4021236906890850070201488, 1798052050351216209712206336, 920859156623446912386646303104
FORMULA
E.g.f.: Sum_{k>=1} log(1 + k*x)^k / k.
MATHEMATICA
nmax=16; Range[0, nmax]!CoefficientList[Series[Sum[(Log[1 + k*x])^k / k, {k, nmax}], {x, 0, nmax}], x] (* Stefano Spezia, Jun 19 2024 *)
PROG
(PARI) a(n) = sum(k=1, n, k!*k^(n-1)*stirling(n, k, 1));
a(n) = Sum_{k=1..n} k! * k^(n-2) * Stirling2(n,k).
+10
3
0, 1, 3, 31, 765, 34651, 2502213, 263824891, 38248036725, 7298877611371, 1773652375115973, 534749297993098651, 195883403209280580885, 85687658454617655817291, 44120264185381411695106533, 26413555571018242181844978811
FORMULA
E.g.f.: Sum_{k>=1} (exp(k*x) - 1)^k / k^2.
PROG
(PARI) a(n) = sum(k=1, n, k!*k^(n-2)*stirling(n, k, 2));
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