A140709 -id:A140709 - cp's OEIS frontend

A132371 a(n) = n! - Sum_{j=1..n-1} j!.

1, 1, 3, 15, 87, 567, 4167, 34407, 316647, 3219687, 35878887, 435046887, 5704064487, 80428314087, 1213746099687, 19521187251687, 333363035571687, 6024361885107687, 114864714882483687, 2304476522241459687, 48529614677597619687, 1070348458111786419687

Offset: 1

Ben Branman, Nov 09 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..440

Cf. A000142, A003422, A007489, A140709.

A132371:= func< n | n eq 1 select 1 else  Factorial(n) - (&+[Factorial(j): j in [1..n-1]]) >;
[A132371(n): n in [1..30]]; // G. C. Greubel, May 02 2021

seq(factorial(n)-(sum(factorial(j),j=1..n-1)),n=1..22); # Emeric Deutsch, May 26 2008

With[{fctrls=Range[30]!},Table[fctrls[[n]]-Total[Take[fctrls, n-1]], {n,30}]] (* Harvey P. Dale, Feb 27 2012 *)
Table[n! -Sum[j!, {j, n-1}], {n, 30}] (* G. C. Greubel, May 02 2021 *)

[factorial(n) - sum(factorial(j) for j in (1..n-1)) for n in (1..30)] # G. C. Greubel, May 02 2021

From G. C. Greubel, May 02 2021: (Start)
a(n) = A000142(n) - A007489(n-1).
a(n) = n! - A003422(n) + 1. (End)

Corrected and extended by N. J. A. Sloane, Nov 11 2007

Better definition from Emeric Deutsch, May 26 2008

A140710 Number of maximal initial consecutive columns ending at the same level, summed over all deco polyominoes of height n.

Original entry on oeis.org

1, 3, 10, 38, 172, 944, 6208, 47696, 417952, 4101824, 44491648, 528068096, 6804155392, 94559581184, 1409615239168, 22434345998336, 379633330204672, 6805952938041344, 128854632579186688, 2568966172926181376

Offset: 1

Emeric Deutsch, Jun 03 2008

nonn

A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

			a(3)=10 because the 6 deco polyominoes of height 3 have columns ending at levels 3, 22, 12, 111, 22, 122, respectively and 1+2+1+3+2+1=10.

G. C. Greubel, Table of n, a(n) for n = 1..440
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Cf. A000142, A011782, A140709.

Row sums of A227550/2.

[2^(n-1)*(&+[j*Factorial(j)/2^j: j in [1..n-1]]): n in [1..30]]; // G. C. Greubel, May 02 2021

a:=proc(n) options operator, arrow: 2^(n-1)*(1+sum(j^2*factorial(j-1)/2^j, j= 1..n-1)) end proc: seq(a(n),n=1..20);

Table[2^(n-1)*(1 + Sum[j*j!/2^j, {j,n-1}]), {n,30}] (* G. C. Greubel, May 02 2021 *)

[2^(n-1)*sum(j*factorial(j)/2^j for j in (1..n-1)) for n in (1..30)] # G. C. Greubel, May 02 2021

a(n) = 2^(n-1) * (1 + Sum_{j=1..n-1} j*j!/2^j ).
a(n) = (n-1)!*(n-1) + 2*a(n-1) with a(1) = 1.
a(n) = Sum_{k=1..n} k*A140709(n,k).
(1 + x + 2*x^2 + 4*x^3 + 8*x^4 + ...)*(1 + 2*x + 6*x^2 + 24*x^3 + 120*x^4 + ...) = (1 + 3*x + 10*x^2 + 38*x^3 + 172*x^4 + ...) which is (Sum_{n>=0} A011782(n)*x^n) * (Sum_{n>=0} A000142(n+1)*x^n) = Sum_{n>=0} a(n+1)*x^n. - Gary W. Adamson, Feb 24 2012
a(n) = Sum_{j=0..n} (j+1)!*A011782(n-j) = (n+1)! + Sum_{j=0..n-1} 2^(n-k-1)*(j+1)!. - G. C. Greubel, May 03 2021
D-finite with recurrence a(n) +(-n-3)*a(n-1) +3*n*a(n-2) +2*(-n+2)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

cp's OEIS Frontend

A132371 a(n) = n! - Sum_{j=1..n-1} j!.

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