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)))
A368585
a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(k+2,3) / k!.
Original entry on oeis.org
0, 1, 2, 4, 4, 15, -34, 322, -2456, 22269, -222470, 2447456, -29369108, 381798859, -5345183466, 80177752670, -1282844041904, 21808348713337, -392550276838926, 7458455259940924, -149169105198816940, 3132551209175157511, -68916126601853463218
Offset: 0
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my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(x*sum(k=0, 2, binomial(2, k)*x^k/(k+1)!)*exp(x)/(1+x))))
A368586
a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(k+3,4) / k!.
Original entry on oeis.org
0, 1, 3, 6, 11, 15, 36, -42, 666, -5499, 55705, -611754, 7342413, -95449549, 1336296066, -20044437930, 320711010756, -5452087178007, 98137569210111, -1864613814984794, 37292276299704735, -783137802293788809, 17229031650463366448, -396267727960657413354
Offset: 0
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my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(x*sum(k=0, 3, binomial(3, k)*x^k/(k+1)!)*exp(x)/(1+x))))
A368587
a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(k+4,5) / k!.
Original entry on oeis.org
0, 1, 4, 9, 20, 26, 96, -210, 2472, -20961, 211612, -2324729, 27901116, -362708320, 5077925048, -76168864092, 1218701840976, -20717931276243, 372922762998708, -7085532496941803, 141710649938878564, -2975923648716396714, 65470320271760793488
Offset: 0
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my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(x*sum(k=0, 4, binomial(4, k)*x^k/(k+1)!)*exp(x)/(1+x))))
Showing 1-4 of 4 results.