A368766
a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+1,2) / k!).
Original entry on oeis.org
1, 0, 3, 3, 22, 95, 591, 4109, 32908, 296127, 2961325, 32574509, 390894186, 5081624327, 71142740683, 1067141110125, 17074257762136, 290262381956159, 5224722875211033, 99269734629009437, 1985394692580188950, 41693288544183967719, 917252347972047290071
Offset: 0
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nxt[{n_,a_}]:={n+1,a(n+1)+(-1)^(n+1) Binomial[n+2,2]}; NestList[nxt,{0,1},30][[;;,2]] (* Harvey P. Dale, Mar 26 2025 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x*sum(k=0, 1, binomial(1, k)*(-x)^k/(k+1)!)*exp(-x))/(1-x)))
A368767
a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+2,3) / k!).
Original entry on oeis.org
1, 0, 4, 2, 28, 105, 686, 4718, 37864, 340611, 3406330, 37469344, 449632492, 5845221941, 81833107734, 1227496615330, 19639945846096, 333879079382663, 6009823428889074, 114186645148891076, 2283732902977823060, 47958390962534282489, 1055084601175754216782
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x*sum(k=0, 2, binomial(2, k)*(-x)^k/(k+1)!)*exp(-x))/(1-x)))
A368764
a(n) = n! * (1 + Sum_{k=0..n} binomial(k+3,4) / k!).
Original entry on oeis.org
1, 2, 9, 42, 203, 1085, 6636, 46662, 373626, 3363129, 33632005, 369953056, 4439438037, 57712696301, 807977750594, 12119666261970, 193914660195396, 3296549223326577, 59337886019884371, 1127419834377810364, 22548396687556216135, 473516330438680549461
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x*sum(k=0, 3, binomial(3, k)*x^k/(k+1)!)*exp(x))/(1-x)))
A368765
a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * k / k!).
Original entry on oeis.org
1, 0, 2, 3, 16, 75, 456, 3185, 25488, 229383, 2293840, 25232229, 302786760, 3936227867, 55107190152, 826607852265, 13225725636256, 224837335816335, 4047072044694048, 76894368849186893, 1537887376983737880, 32295634916658495459, 710503968166486900120, 16341591267829198702737
Offset: 0
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Table[n!(1+Sum[(-1)^k k/k!,{k,0,n}]),{n,0,30}] (* Harvey P. Dale, Mar 26 2025 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x*exp(-x))/(1-x)))
Showing 1-4 of 4 results.