A349561
E.g.f. satisfies: A(x)^A(x) = 1/(1 - x).
Original entry on oeis.org
1, 1, 0, 3, -8, 100, -834, 11438, -159928, 2762352, -52322160, 1124320032, -26509832040, 686751503568, -19306448087640, 586539826169880, -19131996548499264, 667157522614934016, -24762890955027112128, 974824890777753840576, -40566428716555791936000
Offset: 0
A(x) - 1 = x + 3*x^3/6 - 8*x^4/24 + ... = x + x^3/2 - x^4/3 + ... .
A(x)^A(x) = (1 + (A(x) - 1))^(1 + (A(x) - 1)) = Sum_{k>=0} A005727(k) * (A(x) - 1)^k / k! = 1 + 1 * (x + x^3/2 - x^4/3 + ... )/1! + 2 * (x + x^3/2 - x^4/3 + ... )^2/2! + 3 * (x + x^3/2 - x^4/3 + ... )^3/3! + ... = 1 + x + x^2 + x^3 + ... = 1/(1 - x).
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Join[{1}, Table[(n-1)! - (-1)^n*Sum[(k-1)^(k-1)*StirlingS1[n, k], {k, 2, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 22 2021 *)
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a(n) = (-1)^(n-1)*sum(k=0, n, (k-1)^(k-1)*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (k-1)^(k-1)*log(1-x)^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1-x)/lambertw(-log(1-x))))
A305981
Expansion of e.g.f. 1/(1 + LambertW(log(1 - x))).
Original entry on oeis.org
1, 1, 5, 41, 468, 6854, 122582, 2589978, 63129392, 1743732192, 53827681152, 1836453542472, 68620052332752, 2786929842106344, 122241516227220504, 5758920745460806824, 290017142065771138560, 15547326972257789803200, 883974436758296523437760, 53131928820278417749940544, 3366145488853852112016117504
Offset: 0
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a:=series(1/(1+LambertW(log(1-x))),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019
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nmax = 20; CoefficientList[Series[1/(1 + LambertW[Log[1 - x]]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] k^k, {k, n}], {n, 20}]]
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a(n) = sum(k=0, n, (-1)^(n-k)*k^k*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 05 2022
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my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(log(1-x))))) \\ Seiichi Manyama, Feb 05 2022
A351180
a(n) = Sum_{k=0..n} k^(k+n) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 15, 635, 53112, 7367444, 1529130770, 443685287576, 171495189203456, 85174828026304824, 52856314387144232184, 40077340463437963801752, 36457068309928364981668848, 39186634107857517367884040632
Offset: 0
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a[0] = 1; a[n_] := Sum[k^(k + n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 04 2022 *)
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a(n) = sum(k=0, n, k^(k+n)*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*log(1+k*x))^k/k!)))
A351182
a(n) = Sum_{k=0..n} k^(2*k) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 15, 683, 61332, 9135004, 2035708760, 634172615600, 263166948202080, 140322186951905736, 93484350581344936344, 76095870609142447018152, 74311960997497053384537408, 85748280952260853814490688656
Offset: 0
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a(n) = sum(k=0, n, k^(2*k)*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^2*log(1+x))^k/k!)))
A351280
a(n) = Sum_{k=0..n} k! * k^k * Stirling1(n,k).
Original entry on oeis.org
1, 1, 7, 140, 5254, 318854, 28455182, 3506576856, 570360248856, 118356589567440, 30512901324706608, 9566812017770347152, 3584662956711860108352, 1581905384865801328253712, 812047187127758913474118032, 479763784808095613489811245568
Offset: 0
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a[0] = 1; a[n_] := Sum[k! * k^k * StirlingS1[n, k], {k, 1, n}]; Array[a, 16, 0] (* Amiram Eldar, Feb 06 2022 *)
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a(n) = sum(k=0, n, k!*k^k*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*log(1+x))^k)))
A351274
a(0) = 1; thereafter a(n) = Sum_{k=1..n} (2*k)^k * Stirling1(n,k).
Original entry on oeis.org
1, 2, 14, 172, 2964, 65848, 1789688, 57521280, 2133964352, 89744964288, 4219022123328, 219246630903936, 12479659844383104, 772174659456713472, 51603153976362554112, 3704166182571098222592, 284239227254465994240000, 23218955083323248158556160
Offset: 0
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Join[{1},Table[Sum[(2k)^k StirlingS1[n,k],{k,n}],{n,20}]] (* Harvey P. Dale, Dec 31 2023 *)
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a(n) = sum(k=0, n, (2*k)^k*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-2*log(1+x)))))
A350726
a(n) = Sum_{k=0..n} k^(n-k) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 0, -3, 21, -100, -525, 33026, -860503, 16304464, -100885935, -12798492630, 1037135603845, -55556702499792, 2207903148318777, -31916679640973750, -6164889702150516015, 983802138243128355456, -100629406324320358067423
Offset: 0
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a[0] = 1; a[n_] := Sum[k^(n - k) * StirlingS1[n, k], {k, 1, n}]; Array[a, 19, 0] (* Amiram Eldar, Feb 03 2022 *)
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a(n) = sum(k=0, n, k^(n-k)*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, log(1+k*x)^k/(k!*k^k))))
A323619
Expansion of e.g.f. 1 - LambertW(-log(1+x))*(2 + LambertW(-log(1+x)))/2.
Original entry on oeis.org
1, 1, 0, 2, 3, 44, 260, 3534, 40796, 658440, 11066184, 220005840, 4750650432, 114430365048, 2993377996440, 85208541290040, 2611784941760640, 85941161628865344, 3018822193183216320, 112805065528683216192, 4467115744449046110720, 186900232401341222964480, 8237944325702047624948224
Offset: 0
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[n le 0 select 1 else (&+[StirlingFirst(n,k)*k^(k-2): k in [1..n]]): n in [0..25]]; // G. C. Greubel, Feb 07 2019
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seq(n!*coeff(series(1-LambertW(-log(1+x))*(2+LambertW(-log(1+x)))/2, x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 28 2019
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nmax = 22; CoefficientList[Series[1 - LambertW[-Log[1 + x]] (2 + LambertW[-Log[1 + x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k^(k - 2), {k, n}], {n, 22}]]
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{a(n) = if(n==0,1, sum(k=1,n, stirling(n,k,1)*k^(k-2)))};
vector(25, n, n--; a(n)) \\ G. C. Greubel, Feb 07 2019
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[1] + [sum((-1)^(k+n)*stirling_number1(n,k)*k^(k-2) for k in (1..n)) for n in (1..25)] # G. C. Greubel, Feb 07 2019
Showing 1-8 of 8 results.