A348312
a(n) = n! * Sum_{k=0..n-1} 3^k / k!.
Original entry on oeis.org
0, 1, 8, 51, 312, 1965, 13248, 97839, 800208, 7260921, 72806040, 801515979, 9620317512, 125071036389, 1751016829968, 26265324194055, 420245416687392, 7144172815479921, 128595113003161512, 2443307154421058019, 48866143111666389720, 1026189005418216656541, 22576158119430894214368
Offset: 0
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Table[n! Sum[3^k/k!, {k, 0, n - 1}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[x Exp[3 x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
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a(n) = n!*sum(k=0, n-1, 3^k/k!); \\ Michel Marcus, Oct 11 2021
A134558
Array read by antidiagonals, a(n,k) = gamma(n+1,k)*e^k, where gamma(n,k) is the upper incomplete gamma function and e is the exponential constant 2.71828...
Original entry on oeis.org
1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 24, 16, 10, 4, 1, 120, 65, 38, 17, 5, 1, 720, 326, 168, 78, 26, 6, 1, 5040, 1957, 872, 393, 142, 37, 7, 1, 40320, 13700, 5296, 2208, 824, 236, 50, 8, 1, 362880, 109601, 37200, 13977, 5144, 1569, 366, 65, 9, 1, 3628800, 986410, 297856
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
2, 5, 10, 17, 26, 37, 50, ...
6, 16, 38, 78, 142, 236, 366, ...
24, 65, 168, 393, 824, 1569, 2760, ...
120, 326, 872, 2208, 5144, 10970, 21576, ...
720, 1957, 5296, 13977, 34960, 81445, 176112, ...
Cf. a(n, 0) =
A000142(n); a(n, 1) =
A000522(n); a(n, 2) =
A010842(n); a(n, 3) =
A053486(n); a(n, 4) =
A053487(n); a(n, 5) =
A080954(n); a(n, 6) =
A108869(n); a(1, k) =
A000027(k+1); a(2, k) =
A002522(k+1); a(n, n) =
A063170(n); a(n, n+1) =
A001865(n+1); a(n, n+2) =
A001863(n+2).
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T[n_,k_] := Gamma[n+1, k]*E^k; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] //Flatten (* Amiram Eldar, Jun 27 2020 *)
A295519
a(n) = e^3 * Sum_{k=0..n-1} Gamma(k + 1, 3).
Original entry on oeis.org
0, 1, 5, 22, 100, 493, 2701, 16678, 116704, 923473, 8204077, 81069166, 882762292, 10503611245, 135576241957, 1886597854894, 28151936397856, 448397396131969, 7592570340752053, 136187683731334054, 2579494839314653540, 51445637954467827661
Offset: 0
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a := proc(n) option remember; if n=0 then 0 elif n=1 then 1 elif n=2 then 5 else
(3*n-6)*a(n-3)+(4-4*n)*a(n-2)+(3+n)*a(n-1) fi end: seq(a(n), n=0..21);
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a[n_] := E^3 Sum[Gamma[k + 1, 3], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
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a=vector(1000); a[1]=1;a[2]=5;a[3]=22;for(n=4, #a, a[n] = (n+3)*a[n-1]+(4-4*n)*a[n-2]+(3*n-6)*a[n-3]); va=concat(0, vector(1000, n, a[n])) \\ Altug Alkan, Dec 17 2017