A021424 -id:A021424 - cp's OEIS frontend

A016209 Expansion of 1/((1-x)(1-3x)(1-5x)).

1, 9, 58, 330, 1771, 9219, 47188, 239220, 1205941, 6059229, 30384718, 152189310, 761743711, 3811110039, 19062724648, 95335146600, 476740303081, 2383895225649, 11920057258978, 59602029687090

Offset: 0

N. J. A. Sloane

For a combinatorial interpretation following from a(n) = A039755(n+2,2) = h^{(3)}A039755.%20-%20_Wolfdieter%20Lang">n, the complete homogeneous symmetric function of degree n in the symbols {1, 3, 5} see A039755. - _Wolfdieter Lang, May 26 2017

			a(2) = h^{(3)}_2 = 1^2 + 3^2 + 5^2 + 1^1*(3^1 + 5^1) + 3^1*5^1 = 58. - _Wolfdieter Lang_, May 26 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

Cf. A016218, A016208, A000392, A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A016256, A039755, A021424.

[(5^(n+2)-2*3^(n+2)+1)/8: n in [0..20]]; // Vincenzo Librandi, Sep 17 2011

A016209 := proc(n) (5^(n+2)-2*3^(n+2)+1)/8; end proc: # R. J. Mathar, Mar 22 2011

Join[{a=1,b=9},Table[c=8*b-15*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2011 *)
CoefficientList[Series[1/((1-x)(1-3x)(1-5x)),{x,0,30}],x] (* or *) LinearRecurrence[ {9,-23,15},{1,9,58},30] (* Harvey P. Dale, Feb 20 2020 *)

PARI
```
a(n)=if(n<0,0,n+=2; (5^n-2*3^n+1)/8)
```

a(n) = A039755(n+2, 2).
a(n) = (5^(n+2) - 2*3^(n+2)+1)/8 = a(n-1) + A005059(n+1) = 8*a(n-1) - 15*a(n-2) + 1 = (A003463(n+2) - A003462(n+2))/2. - Henry Bottomley, Jun 06 2000
G.f.: 1/((1-x)(1-3*x)(1-5*x)). See the name.
E.g.f.: (25*exp(5*x) - 18*exp(3*x) + exp(x))/8, from the e.g.f. of the third column (k=2) of A039755. - Wolfdieter Lang, May 26 2017

A286719 Column k=4 of triangle A039755, Sheffer(exp(x), (exp(2*x) - 1)/2).

Original entry on oeis.org

1, 25, 395, 5075, 58086, 618870, 6289690, 61885450, 595122671, 5629238615, 52605474285, 487197745125, 4481780785756, 41018845739260, 373968405050180, 3399402534376100, 30830907772159341, 279134548584080805, 2523817507756513375, 22795663165336810375, 205730405672107235426, 1855561201430080303250, 16727971116048518559870, 150747219419372400319950, 1358093516662781192486011

Offset: 0

Wolfdieter Lang, May 26 2017

For a combinatorial interpretation following from the g.f. and the a(n) = h^{(5)}_n formula below see A039755.

			a(2) =  h^{(5)}_2 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 + 1^1*(3^1 + 5^1 + 7^1 + 9^1) + 3^1*(5^1 + 7^1 + 9^1) + 5^1*(7^1 + 9^1) + 7^1*9^1 = 165 + 230 = 395. The multichoose(5, 2) = binomial(6, 2) = 15 polytopes are five squares and ten rectangles of total area 165 and 230, respectively.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (25,-230,950,-1689,945).

Cf. A003462 (k=1), A016209 (k=2), A021424 (k=3), A039755.

Vec(1 / ((1 - x)*(1 - 3*x)*(1 - 5*x)*(1 - 7*x)*(1 - 9*x)) + O(x^40)) \\ Colin Barker, Dec 23 2017

a(n) =  A039755(n+4,4),  n >= 0.
G.f.: 1/Product_{j=0..4}(1 - (1+2*j)*x).
E.g.f.: (d^4/dx^4)(exp(x)*((exp(2*x)-1)/2)^4/4!) = (2187/128)*exp(9*x) - (2401/96)*exp(7*x) + (625/64)*exp(5*x) - (27/32)*exp(3*x) + (1/384)*exp(x).
a(n) = h^{(5)}_n, the complete homogeneous symmetric function of degree n of the five symbols 1, 3, 5, 7, 9.
From Colin Barker, Dec 23 2017: (Start)
G.f.: 1 / ((1 - x)*(1 - 3*x)*(1 - 5*x)*(1 - 7*x)*(1 - 9*x)).
a(n) = (1 - 4*3^(4+n) + 6*5^(4+n) - 4*7^(4+n) + 9^(4+n)) / 384.
a(n) = 25*a(n-1) - 230*a(n-2) + 950*a(n-3) - 1689*a(n-4) + 945*a(n-5) for n>4.
(End)

cp's OEIS Frontend

A016209 Expansion of 1/((1-x)(1-3x)(1-5x)).

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