A203152
(n-1)-st elementary symmetric function of {1, 2, 2, 3, 3, 4, 4, 5, 5, ..., floor(1+n/2)}.
Original entry on oeis.org
1, 3, 8, 28, 96, 420, 1824, 9696, 51360, 322560, 2021760, 14670720, 106323840, 875992320, 7211151360, 66526064640, 613365903360, 6265340928000, 63970228224000, 716840699904000, 8030097782784000, 97954524315648000
Offset: 1
Let esf abbreviate "elementary symmetric function". Then
0th esf of {1}: 1;
1st esf of {1,2}: 1+2 = 3;
2nd esf of {1,2,2} is 1*2 + 1*2 + 2*2 = 8.
-
SymmPolyn := proc(L::list,n::integer)
local c,a,sel;
a :=0 ;
sel := combinat[choose](nops(L),n) ;
for c in sel do
a := a+mul(L[e],e=c) ;
end do:
a;
end proc:
A203152 := proc(n)
local k ;
L := [seq(floor(1+k/2),k=1..n)] ;
SymmPolyn(L,n-1) ;
end proc: # R. J. Mathar, Sep 23 2016
-
f[k_] := Floor[(k + 2)/2]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}] (* A203152 *)
A203154
(n-1)-st elementary symmetric function of {2, 3, 3, 4, 4, 5, 5,...,Floor[(n+4)/2]}.
Original entry on oeis.org
1, 5, 21, 102, 480, 2688, 14880, 96480, 622080, 4613760, 34110720, 285586560, 2386298880, 22289541120, 207921530880, 2145056256000, 22108972032000, 249782787072000, 2820035699712000, 34637103857664000, 425205351825408000
Offset: 1
Let esf abbreviate "elementary symmetric function". Then
0th esf of {2}: 1
1st esf of {2,3}: 2+3=5;
2nd esf of {2,3,3} is 2*3+2*3+3*3=21.
-
SymmPolyn := proc(L::list,n::integer)
local c,a,sel;
a :=0 ;
sel := combinat[choose](nops(L),n) ;
for c in sel do
a := a+mul(L[e],e=c) ;
end do:
a;
end proc:
A203154 := proc(n)
[seq(floor((k+4)/2),k=1..n)] ;
SymmPolyn(%,n-1);
end proc: # R. J. Mathar, Sep 23 2016
# second Maple program:
f:= proc(n) local L,x;
if n::odd then L:= `*`(x+2,seq((x+i)$2, i=3..2+n/2))
else L:= `*`(seq((x+i)*(x+i+1),i=2..1+n/2))
fi;
coeff(L,x,1);
end proc:
map(f, [$1..50]); # Robert Israel, Nov 27 2017
-
f[k_] := Floor[(k + 4)/2]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}] (* A203154 *)
A203155
(n-1)-st elementary symmetric function of {3, 3, 4, 4, 5, 5,..., Floor[(n+5)/2]}.
Original entry on oeis.org
1, 6, 33, 168, 984, 5640, 37440, 246240, 1853280, 13880160, 117391680, 989936640, 9315855360, 87500528640, 907925760000, 9408462336000, 106785133056000, 1210848984576000, 14928525545472000, 183922359312384000, 2448351304261632000
Offset: 1
Let esf abbreviate "elementary symmetric function". Then
0th esf of {3}: 1
1st esf of {3,3}: 3+3=6
2nd esf of {3,3,4} is 3*3+3*4+3*4=33
-
f[k_] := Floor[(k + 5)/2]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}] (* A203155 *)
Showing 1-3 of 3 results.