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)))
A368768
a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+3,4) / k!).
Original entry on oeis.org
1, 0, 5, 0, 35, 105, 756, 5082, 40986, 368379, 3684505, 40528554, 486344013, 6322470349, 88514587266, 1327718805930, 21243500898756, 361139515274007, 6500511274938111, 123509714223816794, 2470194284476344735, 51874079974003228809, 1141229759428071046448
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)))
A368763
a(n) = n! * (1 + Sum_{k=0..n} binomial(k+2,3) / k!).
Original entry on oeis.org
1, 2, 8, 34, 156, 815, 4946, 34706, 277768, 2500077, 25000990, 275011176, 3300134476, 42901748643, 600624481562, 9009367224110, 144149875586576, 2450547884972761, 44109861929510838, 838087376660707252, 16761747533214146580, 351996698197497079951
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)))
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.