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

0,3

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

Links

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

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

Example

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

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A055148 A014830 A048439 * A098107 A226201 A091647

Adjacent sequences: A134429 A134430 A134431 * A134433 A134434 A134435

Keyword

nonn

Author

Emeric Deutsch, Nov 16 2007

Extensions

More terms from Alois P. Heinz, Dec 22 2017

Status

approved