A330260
a(n) = n! * Sum_{k=0..n} binomial(n,k) * n^(n - k) / k!.
Original entry on oeis.org
1, 2, 17, 352, 13505, 830126, 74717857, 9263893892, 1513712421377, 315230799073690, 81499084718806001, 25612081645835777192, 9615370149488574778177, 4250194195208050117007942, 2184834047906975645398282625, 1292386053018890618812398220876
Offset: 0
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[Factorial(n)*&+[Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Dec 18 2019
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Join[{1}, Table[n! Sum[Binomial[n, k] n^(n - k)/k!, {k, 0, n}], {n, 1, 15}]]
Join[{1}, Table[n^n n! LaguerreL[n, -1/n], {n, 1, 15}]]
Table[n! SeriesCoefficient[Exp[x/(1 - n x)]/(1 - n x), {x, 0, n}], {n, 0, 15}]
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a(n) = n! * sum(k=0, n, binomial(n,k) * n^(n-k)/k!); \\ Michel Marcus, Dec 18 2019
A318224
a(n) = n! * [x^n] exp(x/(1 + n*x)).
Original entry on oeis.org
1, 1, -3, 37, -1007, 47901, -3514499, 367671697, -51952729023, 9529552851193, -2201241933756899, 625136460673954461, -214066473170125310063, 86976878219664125966677, -41368038169392401671082787, 22767783580493235411255966601, -14356419990032448099044028030719
Offset: 0
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Table[n! SeriesCoefficient[Exp[x/(1 + n x)], {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[(-n)^(n - k) Binomial[n - 1, k - 1] n!/k!, {k, n}], {n, 16}]]
Join[{1}, Table[(-1)^(n + 1) n^n (n - 1)! Hypergeometric1F1[1 - n, 2, 1/n], {n, 16}]]
Flatten[{1, Table[-(-1)^n * n^(n-1) * (n-1)! * LaguerreL[n-1, 1, 1/n], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 21 2018 *)
A341033
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-k*x)).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 37, 73, 1, 1, 1, 9, 73, 361, 501, 1, 1, 1, 11, 121, 1009, 4361, 4051, 1, 1, 1, 13, 181, 2161, 17341, 62701, 37633, 1, 1, 1, 15, 253, 3961, 48081, 355951, 1044205, 394353, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 3, 5, 7, 9, 11, ...
1, 13, 37, 73, 121, 181, ...
1, 73, 361, 1009, 2161, 3961, ...
1, 501, 4361, 17341, 48081, 108101, ...
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T[0, k_] = 1; T[n_, k_] := n!*Sum[If[k == n - j == 0, 1, k^(n - j)]*Binomial[n - 1, j - 1]/j!, {j, 1, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 03 2021 *)
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{T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^(n-j)*binomial(n-1, j-1)/j!))}
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{T(n, k) = if(n<2, 1, (2*k*n-2*k+1)*T(n-1, k)-k^2*(n-1)*(n-2)*T(n-2, k))}
A341056
a(n) = n! * [x^n] exp(x/(1 - n*x)) / (1 - x).
Original entry on oeis.org
1, 2, 9, 106, 2801, 132426, 9705577, 1015001954, 143392421601, 26298332570386, 6074043257989001, 1724846814877790682, 590605908915568818769, 239956225437223244619866, 114123836188192016600789481, 62808518765936960824453590226, 39603421893790601518269204039617
Offset: 0
a(3) = 3! * (1 + 1/1! + 7/2! + 73/3!) = 106.
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Table[n!*(1 + Sum[Sum[n^(j-k)*Binomial[j-1, k-1]/k!, {k, 1, j}], {j, 1, n}]), {n, 0, 20}] (* Vaclav Kotesovec, Feb 14 2021 *)
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{a(n) = n!*(1+sum(j=1, n, sum(k=1, j, n^(j-k)*binomial(j-1, k-1)/k!)))}
Showing 1-4 of 4 results.