A134432 Sum of entries in all the arrangements of the set {1,2,...,n} (to n=0 there corresponds the empty set).

0, 1, 9, 66, 490, 3915, 34251, 328804, 3452436, 39456405, 488273005, 6510306726, 93097386174, 1421850988831, 23105078568495, 398118276872520, 7251440043035176, 139227648826275369, 2810658160680434001, 59519819873232720010, 1319356007189991960210

Offset: 0

Author

Emeric Deutsch, Nov 16 2007

Keywords

• nonn

Comments

Appears to be the binomial transform of A001286 (filled with the appropriate two leading zeros), shifted one index left. - R. J. Mathar, Apr 04 2012

Examples

			a(2)=9 because the arrangements of {1,2} are (empty), 1, 2, 12 and 21.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..447

Crossrefs

Cf. A000522, A001286, A134431, A271703, A271705, A326659.

Programs

Magma

[Binomial(n+1,2)*(&+[Factorial(j)*Binomial(n-1, j-1): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jan 09 2022

Maple

Q[0]:=1: for n to 17 do Q[n]:=sort(simplify(Q[n-1]+t^n*x*(diff(x*Q[n-1], x))), t) end do: for n from 0 to 17 do P[n]:=sort(subs(x=1,Q[n])) end do: seq(subs(t =1,diff(P[n],t)),n=0..17);
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, [t!, 0],
      b(n-1, t)+(p-> p+[0, n*p[1]])(b(n-1, t+1)))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 19 2020

Mathematica

(* First program *)
b[n_, s_, t_]:= b[n, s, t] = If[n==0, t! x^s, b[n-1, s, t] + b[n-1, s+n, t+1]];
T[n_]:= T[n]= Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]] @ b[n, 0, 0];
a[n_] := Sum[k T[n][[k+1]], {k, 0, n(n+1)/2}];
a /@ Range[0, 20] (* Jean-François Alcover, Feb 19 2020, after Alois P. Heinz *)
(* Second program *)
a[n_]:= ((n+1)/2)*Sum[j*j!*Binomial[n,j], {j,0,n}];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jan 09 2022 *)

Sage

[((n+1)/2)*sum( j*factorial(j)*binomial(n, j) for j in (0..n) ) for n in (0..30)] # G. C. Greubel, Jan 09 2022

Formula

a(n) = Sum_{k=0..n*(n+1)/2} k*A134431(n,k).

a(n) = (d/dt)P[n](t) evaluated at t=1; here P[n](t)=Q[n](t,1) where the polynomials Q[n](t,x) are defined by Q[0]=1 and Q[n]=Q[n-1] + xt^n (d/dx)xQ[n-1]. (Q[n](t,x) is the bivariate generating polynomial of the arrangements of {1,2,...,n}, where t (x) marks the sum (number) of the entries; for example, Q[2](t,x) = 1 + tx + t^2*x + 2t^3*x^2, corresponding to: empty, 1, 2, 12 and 21, respectively.)

E.g.f.: exp(x)*x*(2 + x - x^2) / (2*(1 - x)^3). - Ilya Gutkovskiy, Jun 02 2020

From G. C. Greubel, Jan 09 2022: (Start)

a(n) = A271705(n+1, 2).

a(n) = ((n+1)/2) * Sum_{j=0..n} j * j! * binomial(n, j).

a(n) = (1/n!)*binomial(n+1, 2) * Sum_{j=0..n} (j!)^2 * A271703(n, j). (End)

D-finite with recurrence (-n+1)*a(n) +(n+1)^2*a(n-1) -n*(n+1)*a(n-2)=0. - R. J. Mathar, Jul 26 2022

Extensions

More terms from Alois P. Heinz, Dec 22 2017