A317277
a(n) = Sum_{k=0..n} binomial(n-1,k-1)*k^n*n!/k!; a(0) = 1.
Original entry on oeis.org
1, 1, 6, 81, 1828, 60565, 2734926, 160109005, 11724156648, 1045312448841, 111114793839610, 13845807451708441, 1994597720747571468, 328351264019737949341, 61162428777982281583302, 12782305566531823350524805, 2975150384583838798131401296, 766253903501365584725344992529
Offset: 0
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[1]cat[(&+[Binomial(n-1,j-1)*Binomial(n,j)*Factorial(n-j)*j^n: j in [0..n]]): n in [1..30]]; // G. C. Greubel, Mar 09 2021
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A317277:= n-> `if`(n=0,1, add(binomial(n-1,j-1)*binomial(n,j)*(n-j)!*j^n, j=0..n)); seq(A317277(n), n=0..30); # G. C. Greubel, Mar 09 2021
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Join[{1}, Table[Sum[Binomial[n - 1, k - 1] k^n n!/k!, {k, n}], {n, 17}]]
Join[{1}, Table[n! SeriesCoefficient[Sum[k^n (x/(1 - x))^k/k!, {k, n}], {x, 0, n}], {n, 17}]]
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a(n) = if (n==0, 1, sum(k=0, n, binomial(n-1, k-1)*k^n*n!/k!)); \\ Michel Marcus, Mar 10 2021; corrected Jun 15 2022
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[1]+[sum(binomial(n-1,j-1)*binomial(n,j)*factorial(n-j)*j^n for j in (0..n)) for n in (1..30)] # G. C. Greubel, Mar 09 2021
A317278
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*k^n*n!/k!.
Original entry on oeis.org
1, 1, 2, -15, -164, 4245, 46386, -4901939, 39141656, 11707820361, -671114863610, -29398709945319, 7385525824325364, -307076643365636963, -73748845974115224262, 14299745046516639280005, -237996466462017367478864, -377740669670216316717155055, 75515477307532501838072029326
Offset: 0
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[1]cat[(&+[(-1)^(n+j)*Binomial(n-1,j-1)*Binomial(n,j)*Factorial(n-j)*j^n: j in [0..n]]): n in [1..30]]; // G. C. Greubel, Mar 09 2021
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A317278:= n-> `if`(n=0,1, add((-1)^(n+j)*binomial(n-1,j-1)*binomial(n,j)*(n-j)!*j^n, j=0..n));
seq(A317278(n), n=0..30); # G. C. Greubel, Mar 09 2021
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Join[{1}, Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] k^n n!/k!, {k, n}], {n, 18}]]
Join[{1}, Table[n! SeriesCoefficient[Sum[k^n (x/(1 + x))^k/k!, {k, n}], {x, 0, n}], {n, 18}]]
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a(n) = if (n==0, 1, sum(k=0, n, (-1)^(n-k)*binomial(n-1, k-1)*k^n*n!/k!)); \\ Michel Marcus, Mar 10 2021; corrected Jun 13 2022
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[1]+[sum((-1)^(n+j)*binomial(n-1,j-1)*binomial(n,j)*factorial(n-j)*j^n for j in (0..n)) for n in (1..30)] # G. C. Greubel, Mar 09 2021
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 *)
Showing 1-3 of 3 results.
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