A341014
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * j! * binomial(n,j)^2.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 17, 34, 1, 1, 5, 31, 139, 209, 1, 1, 6, 49, 352, 1473, 1546, 1, 1, 7, 71, 709, 5233, 19091, 13327, 1, 1, 8, 97, 1246, 13505, 95836, 291793, 130922, 1, 1, 9, 127, 1999, 28881, 318181, 2080999, 5129307, 1441729, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
1, 7, 17, 31, 49, 71, ...
1, 34, 139, 352, 709, 1246, ...
1, 209, 1473, 5233, 13505, 28881, ...
1, 1546, 19091, 95836, 318181, 830126, ...
-
T[n_, k_] := Sum[If[j == k == 0, 1, k^j]*j!*Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 02 2021 *)
-
{T(n,k) = sum(j=0, n, k^j*j!*binomial(n, j)^2)}
A330497
a(n) = n! * Sum_{k=0..n} (-1)^k * binomial(n,k) * n^(n - k) / k!.
Original entry on oeis.org
1, 0, 1, 26, 1089, 70124, 6495985, 821315214, 136115947009, 28651724077976, 7470040450004001, 2363470644596843330, 892244303052345224641, 396227360441775922668036, 204487588996059177697597969, 121370399839482643287189048374
Offset: 0
-
[Factorial(n)*&+[(-1)^k*Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Dec 18 2019
-
Join[{1}, Table[n! Sum[(-1)^k 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}]
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.
-
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 *)
-
{a(n) = n!*(1+sum(j=1, n, sum(k=1, j, n^(j-k)*binomial(j-1, k-1)/k!)))}
Showing 1-3 of 3 results.