A334316
E.g.f. A(x) satisfies: A(x) = x * exp(A(x)) * (1 - A(x)).
Original entry on oeis.org
1, 0, -3, -8, 45, 576, 385, -54144, -499527, 4787200, 160740261, 558627840, -45943496027, -854266871808, 8403892043625, 590895130771456, 4982009666876145, -320936968832679936, -10133752613818727987, 75595253378088960000, 11587542472638176520861
Offset: 1
-
nmax = 21; CoefficientList[InverseSeries[Series[x Exp[-x]/(1 - x), {x, 0, nmax}], x], x] Range[0, nmax]! // Rest
Table[(n - 1)! Sum[(-1)^k Binomial[n, k] n^(n - k - 1)/(n - k - 1)!, {k, 0, n - 1}], {n, 1, 21}]
Table[HypergeometricU[1 - n, 2, n], {n, 1, 21}]
A373966
Triangle read by rows: T(n,k) = (-1)^(n+1) * A000166(n) + (-1)^(k) * A000166(k) for n >= 2 and 1 <= k <= n-1.
Original entry on oeis.org
-1, 2, 3, -9, -8, -11, 44, 45, 42, 53, -265, -264, -267, -256, -309, 1854, 1855, 1852, 1863, 1810, 2119, -14833, -14832, -14835, -14824, -14877, -14568, -16687, 133496, 133497, 133494, 133505, 133452, 133761, 131642, 148329, -1334961, -1334960, -1334963, -1334952, -1335005, -1334696, -1336815, -1320128, -1468457
Offset: 2
Triangle begins:
-1;
2, 3;
-9, -8, -11;
44, 45, 42, 53;
-265, -264, -267, -256, -309;
1854, 1855, 1852, 1863, 1810, 2119;
...
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T[n_,k_]:= (-1)^(n+1)*Subfactorial[n] + (-1)^k*Subfactorial[k]; Table[T[n,k],{n,2,10},{k,n-1}]// Flatten (* Stefano Spezia, Jun 24 2024 *)
Original entry on oeis.org
6, 24, 180, 1320, 11130, 103824, 1067976, 12014640, 146845710, 1938363240, 27489515196, 416924313624, 6734931220290, 115455963776160, 2093601476474640, 40040128237577184, 805513168073611926
Offset: 3
-
a:=n->sum(n!*sum((-1)^k/k!, j=0..n), k=0..n): seq(a(n)*n, n=2..17); # Zerinvary Lajos, Dec 18 2007
A144090
Triangle read by rows: T(n,k) is the number of partial bijections (or subpermutations) of an n-element set of height k (height(alpha) = |Im(alpha)|) and with exactly 1 fixed point.
Original entry on oeis.org
1, 2, 0, 3, 6, 3, 4, 24, 36, 8, 5, 60, 210, 220, 45, 6, 120, 780, 1920, 1590, 264, 7, 210, 2205, 9940, 19005, 12978, 1855, 8, 336, 5208, 37520, 130200, 203952, 118664, 14832, 9, 504, 10836, 114408, 630630, 1783656, 2369556, 1201464, 133497
Offset: 1
T(3,2) = 6 because there are exactly 6 partial bijections (on a 3-element set) with exactly 1 fixed point and of height 2, namely: (1,2)->(1,3), (1,2)->(3,2), (1,3)->(1,2), (1,3)->(2,3), (2,3)->(2,1), (2,3)->(1,3)- the mappings are coordinate-wise.
First six rows:
1
2 0
3 6 3
4 24 36 8
5 60 210 220 45
6 120 780 1920 1590 264
-
Table[(n!/(n - k)!) Sum[ ((-1)^m/m!) Binomial[n - 1 - m, k - 1 - m], {m, 0, k - 1}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Apr 27 2016 *)
-
T(n,k) = (n!/(n-k)!)*sum(m=0,k-1,((-1)^m/m!)*binomial(n-1-m,k-1-m));
for (n=1, 10, for (k=1, n, print1(T(n,k), ", "))) \\ Michel Marcus, Apr 27 2016
A271706
Triangle read by rows: T(n, k) = Sum_{j=0..n} C(-j-1, -n-1)*L(j, k), L the unsigned Lah numbers A271703, for n >= 0 and 0 <= k <= n.
Original entry on oeis.org
1, -1, 1, 1, 0, 1, -1, 3, 3, 1, 1, 8, 18, 8, 1, -1, 45, 110, 70, 15, 1, 1, 264, 795, 640, 195, 24, 1, -1, 1855, 6489, 6335, 2485, 441, 35, 1, 1, 14832, 59332, 67984, 32550, 7504, 868, 48, 1, -1, 133497, 600732, 789852, 445914, 126126, 19068, 1548, 63, 1
Offset: 0
Triangle starts:
[ 1]
[-1, 1]
[ 1, 0, 1]
[-1, 3, 3, 1]
[ 1, 8, 18, 8, 1]
[-1, 45, 110, 70, 15, 1]
[ 1, 264, 795, 640, 195, 24, 1]
[-1, 1855, 6489, 6335, 2485, 441, 35, 1]
-
L := (n, k) -> `if`(k<0 or k>n, 0, (n-k)!*binomial(n, n-k)*binomial(n-1, n-k)):
T := (n, k) -> add(L(j, k)*binomial(-j-1,-n-1), j=0..n):
seq(seq(T(n, k), k=0..n), n=0..9);
# Or:
T := (n, k) -> (-1)^(n-k)*binomial(n, k)*hypergeom([k-n, k], [], 1):
for n from 0 to 8 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, Jun 25 2025
A348311
a(n) = n! * Sum_{k=1..n} (-1)^k * (k-2) / (k-1)!.
Original entry on oeis.org
0, 1, 2, 3, 20, 85, 534, 3703, 29672, 266985, 2669930, 29369131, 352429692, 4581585853, 64142202110, 962133031455, 15394128503504, 261700184559313, 4710603322067922, 89501463119290195, 1790029262385804260, 37590614510101889061, 826993519222241559782
Offset: 0
-
Table[n! Sum[(-1)^k (k - 2)/(k - 1)!, {k, 1, n}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[x (1 + x) Exp[-x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
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a(n) = n!*sum(k=1, n, (-1)^k * (k-2) / (k-1)!); \\ Michel Marcus, Oct 20 2021
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