A301419
a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 - n*j*x).
Original entry on oeis.org
1, 1, 3, 19, 201, 3176, 69823, 2026249, 74565473, 3376695763, 183991725451, 11854772145800, 890415496931689, 77023751991841669, 7592990698770559111, 845240026276785888451, 105409073489605774592897, 14625467507717709778793020, 2244123413703647502288608467, 378751257186051653931253015229
Offset: 0
Cf.
A000110,
A004211,
A004212,
A004213,
A005011,
A005012,
A008277,
A075506,
A075507,
A075508,
A075509,
A242817,
A292914,
A318183.
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List([0..20],n->Sum([0..n],k->n^(n-k)*Stirling2(n,k))); # Muniru A Asiru, Mar 20 2018
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Table[SeriesCoefficient[Sum[x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! SeriesCoefficient[Exp[(Exp[n x] - 1)/n], {x, 0, n}], {n, 19}]]
Join[{1}, Table[Sum[n^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 19}]]
(* Or: *)
A301419[n_] := If[n == 0, 1, n^n BellB[n, 1/n]];
Table[A301419[n], {n, 0, 19}] (* Peter Luschny, Dec 22 2021 *)
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a(n) = sum(k=0, n, n^(n-k)*stirling(n, k, 2)); \\ Michel Marcus, Mar 23 2018
A309386
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} (-k)^(n-j)*Stirling2(n,j).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 1, 1, 1, 1, -3, 1, 9, 2, 1, 1, 1, -4, 5, 19, -23, -9, 1, 1, 1, -5, 11, 25, -128, -25, 9, 1, 1, 1, -6, 19, 21, -343, 379, 583, 50, 1, 1, 1, -7, 29, 1, -674, 2133, 1549, -3087, -267, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 0, -1, -2, -3, -4, -5, ...
1, -1, -1, 1, 5, 11, 19, ...
1, 1, 9, 19, 25, 21, 1, ...
1, 2, -23, -128, -343, -674, -1103, ...
1, -9, -25, 379, 2133, 6551, 15211, ...
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T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-j)] * StirlingS2[n, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 07 2021 *)
A350261
Triangle read by rows. T(n, k) = k^n * BellPolynomial(n, -1/k) for k > 0, if k = 0 then T(n, k) = k^n.
Original entry on oeis.org
1, 0, -1, 0, 0, -1, 0, 1, 1, -1, 0, 1, 9, 19, 25, 0, -2, 23, 128, 343, 674, 0, -9, -25, 379, 2133, 6551, 15211, 0, -9, -583, -1549, 3603, 33479, 123821, 331827, 0, 50, -3087, -32600, -112975, -174114, 120865, 1619108, 5987745
Offset: 0
Triangle starts:
[0] 1
[1] 0, -1
[2] 0, 0, -1
[3] 0, 1, 1, -1
[4] 0, 1, 9, 19, 25
[5] 0, -2, 23, 128, 343, 674
[6] 0, -9, -25, 379, 2133, 6551, 15211
[7] 0, -9, -583, -1549, 3603, 33479, 123821, 331827
[8] 0, 50, -3087, -32600, -112975, -174114, 120865, 1619108, 5987745
-
A350261 := (n, k) -> ifelse(k = 0, k^n, k^n * BellB(n, -1/k)):
seq(seq(A350261(n, k), k = 0..n), n = 0..8);
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T[n_, k_] := If[k == 0, k^n, k^n BellB[n, -1/k]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
A349731
a(n) = -(-n)^n * FallingFactorial(1/n, n) for n >= 1 and a(0) = -1.
Original entry on oeis.org
-1, 1, 1, 10, 231, 9576, 623645, 58715280, 7547514975, 1270453824640, 271252029133449, 71635824470246400, 22929813173612997575, 8747686347650933760000, 3921812703436118765113125, 2041590849971133677650610176, 1221367737152989777782325269375, 832163138229382457228044554240000
Offset: 0
The main diagonal of
A349971 for n >= 1.
The Stirling set counterpart is
A318183.
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[-1,1] cat [Round(n^(n-1)*Gamma((n^2-1)/n)/Gamma((n-1)/n)): n in [2..30]]; // G. C. Greubel, Feb 22 2022
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A349731 := n -> -add((-1)^(n-k)*Stirling1(n, n-k)*(-n)^k, k = 0..n):
seq(A349731(n), n = 0..17);
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a[0] = -1; a[n_] := -(-n)^n * FactorialPower[1/n, n]; Array[a, 18, 0] (* Amiram Eldar, Dec 21 2021 *)
-
from sympy import ff
from fractions import Fraction
def A349731(n): return -1 if n == 0 else -(-n)**n*ff(Fraction(1,n),n) # Chai Wah Wu, Dec 21 2021
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def a(n): return -(-n)^n*falling_factorial(1/n, n) if n > 0 else -1
print([a(n) for n in (1..17)])
A334193
a(0) = 1; thereafter a(n) = exp(1/n) * Sum_{k>=0} (n*k + 1)^n / ((-n)^k * k!).
Original entry on oeis.org
1, 0, -2, -9, -16, 625, 21384, 571438, 13471744, 188661555, -9794500000, -1476328587789, -134710712340480, -10664210861777200, -744650964057237888, -37832162051689453125, 831929248561267474432, 725944099523076464203157, 167435684777981700601449984
Offset: 0
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Table[SeriesCoefficient[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - n j x/(1 - x)), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n! SeriesCoefficient[Exp[x + (1 - Exp[n x])/n], {x, 0, n}], {n, 1, 18}]]
Join[{1}, Table[Sum[Binomial[n, k]*n^k*BellB[k, -1/n], {k, 0, n}], {n, 1, 18}]] (* Vaclav Kotesovec, Apr 18 2020 *)
A355373
a(n) = Sum_{k=0..n} k! * (-1)^k * n^(n-k) * Stirling2(n,k).
Original entry on oeis.org
1, -1, 0, 3, 40, 455, 2016, -177373, -11564160, -497664081, -12796467200, 536297904659, 132025634657280, 14907422733429239, 1181852660381503488, 34684559693802943875, -11771644802057621110784, -3553614228958108389522721, -656899368126170250221715456
Offset: 0
-
a[n_] := Sum[k! * (-1)^k * n^(n - k) * StirlingS2[n, k], {k, 0, n}]; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Jun 30 2022 *)
-
a(n) = sum(k=0, n, k!*(-1)^k*n^(n-k)*stirling(n, k, 2));
A355375
a(n) = Sum_{k=0..n} (-k)^(n-k) * Stirling2(n,k).
Original entry on oeis.org
1, 1, 0, -4, 10, 67, -969, 3341, 86976, -1988704, 14144108, 405611857, -17544321563, 287677263837, 3595470378748, -421298868094940, 14476946230894114, -112253861285434961, -18711849695261432065, 1354595712379990848137, -44436925726445545236496
Offset: 0
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a[n_] := Sum[(-k)^(n - k) * StirlingS2[n, k], {k, 0, n}]; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Jun 30 2022 *)
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a(n) = sum(k=0, n, (-k)^(n-k)*stirling(n, k, 2));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (1-exp(-k*x))^k/(k^k*k!))))
Showing 1-7 of 7 results.