A368765
a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * k / k!).
Original entry on oeis.org
1, 0, 2, 3, 16, 75, 456, 3185, 25488, 229383, 2293840, 25232229, 302786760, 3936227867, 55107190152, 826607852265, 13225725636256, 224837335816335, 4047072044694048, 76894368849186893, 1537887376983737880, 32295634916658495459, 710503968166486900120, 16341591267829198702737
Offset: 0
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Table[n!(1+Sum[(-1)^k k/k!,{k,0,n}]),{n,0,30}] (* Harvey P. Dale, Mar 26 2025 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x*exp(-x))/(1-x)))
A165813
a(n) = n*(a(n-1)+3), a(0)=1.
Original entry on oeis.org
1, 4, 14, 51, 216, 1095, 6588, 46137, 369120, 3322107, 33221100, 365432133, 4385185632, 57007413255, 798103785612, 11971556784225, 191544908547648, 3256263445310067, 58612742015581260, 1113642098296043997
Offset: 0
A166677
a(n)= n*(a(n-1)+4), a(0)=1.
Original entry on oeis.org
1, 5, 18, 66, 280, 1420, 8544, 59836, 478720, 4308516, 43085200, 473937244, 5687246976, 73934210740, 1035078950416, 15526184256300, 248418948100864, 4223122117714756, 76016198118865680, 1444307764258447996
Offset: 0
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f:= gfun:-rectoproc({a(n)=n*(a(n-1)+4),a(0)=1},a(n),remember):
map(f, [$0..30]); # Robert Israel, May 22 2016
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FoldList[#1*#2 + 4 #2 &, 1, Range[19]] (* Robert G. Wilson v, Jul 07 2012 *)
A368759
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * (1 + Sum_{j=0..n} j^k/j!).
Original entry on oeis.org
2, 1, 3, 1, 2, 7, 1, 2, 6, 22, 1, 2, 8, 21, 89, 1, 2, 12, 33, 88, 446, 1, 2, 20, 63, 148, 445, 2677, 1, 2, 36, 141, 316, 765, 2676, 18740, 1, 2, 68, 351, 820, 1705, 4626, 18739, 149921, 1, 2, 132, 933, 2428, 4725, 10446, 32431, 149920, 1349290, 1, 2, 260, 2583, 7828, 15265, 29646, 73465, 259512, 1349289, 13492901
Offset: 0
Square array begins:
2, 1, 1, 1, 1, 1, 1, ...
3, 2, 2, 2, 2, 2, 2, ...
7, 6, 8, 12, 20, 36, 68, ...
22, 21, 33, 63, 141, 351, 933, ...
89, 88, 148, 316, 820, 2428, 7828, ...
446, 445, 765, 1705, 4725, 15265, 54765, ...
2677, 2676, 4626, 10446, 29646, 99366, 375246, ...