A122104
Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and such that the sum of the bottom levels of all columns is k (n>=1, k>=0; informally, the number of the "missing" cells in the right bottom corner of the polyomino). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
Original entry on oeis.org
1, 2, 5, 1, 16, 5, 3, 65, 23, 20, 10, 2, 326, 119, 115, 84, 57, 11, 8, 1957, 719, 714, 582, 526, 310, 137, 55, 34, 6, 13700, 5039, 5033, 4222, 4173, 3291, 2506, 972, 748, 348, 220, 38, 30, 109601, 40319, 40312, 34026, 34454, 29792, 28055, 18723, 10613, 6745
Offset: 1
Triangle starts:
1;
2;
5,1;
16,5,3;
65,23,20,10,2;
326,119,115,84,57,11,8;
- E. Barcucci, S. Brunetti and F. Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.
- E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
-
Q[1]:=x: for n from 2 to 10 do Q[n]:=simplify(subs(x=t*x,Q[n-1])/t+(n-1)*x*Q[n-1]) od: for n from 1 to 10 do P[n]:=sort(subs(x=1,Q[n])) od: for n from 1 to 10 do seq(coeff(P[n],t,j),j=0..floor((n-1)^2/4)) od; # yields sequence in triangular form
A196739
a(n) = n! - n^10.
Original entry on oeis.org
1, 0, -1022, -59043, -1048552, -9765505, -60465456, -282470209, -1073701504, -3486421521, -9996371200, -25897507801, -61438362624, -131631471049, -202076363776, 731023977375, 19823278260224, 353671434195551, 6398803238501376, 121638969342574199, 2432891768176640000
Offset: 0
-
[Factorial(n)-n^10: n in [0..20]];
-
Table[n!-n^10,{n,0,20}] (* Harvey P. Dale, May 19 2012 *)
A262376
a(n) = Sum_{k=0..n} (k! - k).
Original entry on oeis.org
1, 1, 1, 4, 24, 139, 853, 5886, 46198, 409069, 4037859, 43954648, 522956236, 6749977023, 93928268209, 1401602636194, 22324392524178, 378011820620161, 6780385526348143, 128425485935180124, 2561327494111820104, 53652269665821260083, 1177652997443428940061
Offset: 0
a(3) = 4, which is the following sum: (0!-0) + (1!-1) + (2!-2) + (3!-3).
-
Table[Sum[k! - k, {k, 0, n}], {n, 0, 22}] (* Michael De Vlieger, Sep 21 2015 *)
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a(n)=sum(i=0,n,i!-i) \\ Anders Hellström, Sep 20 2015
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var sum = 0;
range(0, 10).each { |n|
sum += (n! - n);
say(n, "\t", sum);
};
A141693
Triangle read by rows: T(n,k) = (2*k - n)*A008292(n,k) with T(n,n) = n, 0 <= k <= n, where A008292 is the triangle of Eulerian numbers.
Original entry on oeis.org
0, -1, 1, -2, 0, 2, -3, -4, 1, 3, -4, -22, 0, 2, 4, -5, -78, -66, 26, 3, 5, -6, -228, -604, 0, 114, 4, 6, -7, -600, -3573, -2416, 1191, 360, 5, 7, -8, -1482, -17172, -31238, 0, 8586, 988, 6, 8, -9, -3514, -73040, -264702, -156190, 88234, 43824, 2510, 7, 9, -10
Offset: 0
Triangle begins:
0;
-1, 1;
-2, 0, 2;
-3, -4, 1, 3;
-4, -22, 0, 2, 4;
-5, -78, -66, 26, 3, 5;
-6, -228, -604, 0, 114, 4, 6;
-7, -600, -3573, -2416, 1191, 360, 5, 7;
-8, -1482, -17172, -31238, 0, 8586, 988, 6, 8;
-9, -3514, -73040, -264702, -156190, 88234, 43824, 2510, 7, 9;
...
-
T:= proc(n,k) `if`(n=k,n,(2*k-n)*add((-1)^j*(k-j+1)^n*binomial(n+1,j),j=0..k)); end proc: seq(seq(T(n,k),k=0..n),n=0..10); # Muniru A Asiru, Oct 06 2018
T := (n, k) -> `if`(n = k, n, (2*k - n)*combinat:-eulerian1(n,k)):
seq(seq(T(n,k), k=0..n), n=0..9); # Peter Luschny, Oct 06 2018
-
T[n_, k_] = If[n == k, n, (2*k - n)*Sum[(-1)^j*(k - j + 1)^n*Binomial[n + 1, j], {j, 0, k}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}]//Flatten
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T(n, k) := if n = k then n else (2*k - n)*sum((-1)^j*(k - j + 1)^n*binomial(n + 1, j), j, 0, k)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* Franck Maminirina Ramaharo, Oct 05 2018 */
A267897
a(n) = prime(n)! - prime(n).
Original entry on oeis.org
0, 3, 115, 5033, 39916789, 6227020787, 355687428095983, 121645100408831981, 25852016738884976639977, 8841761993739701954543615999971, 8222838654177922817725562879999969, 13763753091226345046315979581580902399999963
Offset: 1
-
[Factorial(p)-p: p in PrimesUpTo(40)];
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Array[Prime[#]! - Prime[#] &, 20]
#!-#&/@Prime[Range[20]] (* Harvey P. Dale, Oct 23 2024 *)
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lista(nn) = forprime(p=2, nn, print1(p! - p, ", ")); \\ Altug Alkan, Jan 23 2016
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