A134432 - OEIS

%I #30 Sep 08 2022 08:45:32

%S 0,1,9,66,490,3915,34251,328804,3452436,39456405,488273005,6510306726,

%T 93097386174,1421850988831,23105078568495,398118276872520,

%U 7251440043035176,139227648826275369,2810658160680434001,59519819873232720010,1319356007189991960210

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

%C 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

%H Alois P. Heinz, <a href="/A134432/b134432.txt">Table of n, a(n) for n = 0..447</a>

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

%F 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.)

%F E.g.f.: exp(x)*x*(2 + x - x^2) / (2*(1 - x)^3). - _Ilya Gutkovskiy_, Jun 02 2020

%F From _G. C. Greubel_, Jan 09 2022: (Start)

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

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

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

%F 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

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

%p 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);

%p # second Maple program:

%p b:= proc(n, t) option remember; `if`(n=0, [t!, 0],

%p       b(n-1, t)+(p-> p+[0, n*p[1]])(b(n-1, t+1)))

%p     end:

%p a:= n-> b(n, 0)[2]:

%p seq(a(n), n=0..23);  # _Alois P. Heinz_, Feb 19 2020

%t (* First program *)

%t 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 T[n_]:= T[n]= Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]] @ b[n, 0, 0];

%t a[n_] := Sum[k T[n][[k+1]], {k, 0, n(n+1)/2}];

%t a /@ Range[0, 20] (* _Jean-François Alcover_, Feb 19 2020, after _Alois P. Heinz_ *)

%t (* Second program *)

%t a[n_]:= ((n+1)/2)*Sum[j*j!*Binomial[n,j], {j,0,n}];

%t Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Jan 09 2022 *)

%o (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

%o (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

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

%K nonn

%O 0,3

%A _Emeric Deutsch_, Nov 16 2007

%E More terms from _Alois P. Heinz_, Dec 22 2017