A334190
a(n) = exp(1/2) * Sum_{k>=0} (2*k + 1)^n / ((-2)^k * k!).
Original entry on oeis.org
1, 0, -2, -4, 4, 64, 248, 48, -6512, -51200, -171296, 830400, 17870400, 144684032, 441316224, -5976726784, -119879356160, -1123892297728, -3962230563328, 70410917051392, 1686366492509184, 19578100126072832, 101728414306826240, -1258662784047370240, -42727186269262737408
Offset: 0
-
nmax = 24; CoefficientList[Series[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - 2 j x/(1 - x)), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[x + (1 - Exp[2 x])/2], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] * 2^k * BellB[k, -1/2], {k, 0, n}], {n, 0, 24}] (* Vaclav Kotesovec, Apr 18 2020 *)
A308645
Expansion of e.g.f. exp(1 + x - exp(2*x)).
Original entry on oeis.org
1, -1, -3, 3, 41, 87, -571, -5701, -14575, 156655, 2094925, 9148851, -63364423, -1474212665, -11494853995, 10945362411, 1520718442785, 20719421344991, 100137575499165, -1638818071763869, -45333849658449847, -512404024891840969, -577060092568365467, 99142586163648127771
Offset: 0
-
nmax = 23; CoefficientList[Series[Exp[1 + x - Exp[2 x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Exp[1] Sum[(-1)^k (2 k + 1)^n/k!, {k, 0, Infinity}], {n, 0, 23}]
Table[Sum[Binomial[n, k] 2^k BellB[k, -1], {k, 0, n}], {n, 0, 23}]
A367743
Expansion of e.g.f. exp(1 - x - exp(2*x)).
Original entry on oeis.org
1, -3, 5, 1, -7, -75, -99, 1241, 10161, 18989, -332299, -3857551, -14440151, 141168997, 2807256333, 20182451657, -42073176479, -2999363709091, -38439478980891, -161835672017439, 3439471815545177, 87228227501354517, 937579822282327421, 216540362854403513, -198501712690150659055
Offset: 0
-
nmax = 24; CoefficientList[Series[Exp[1 - x - Exp[2 x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[Binomial[n - 1, k - 1] 2^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k BellB[k, -1], {k, 0, n}], {n, 0, 24}]
Showing 1-3 of 3 results.