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

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

0,2

Comments

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

Links

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).

Formula

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)

Example

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.

Prog

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

Crossrefs

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

Sequence in context: A306322 A344733 A028130 * A174515 A028116 A028075

Adjacent sequences: A286716 A286717 A286718 * A286720 A286721 A286722

Keyword

nonn,easy

Author

Wolfdieter Lang, May 26 2017

Status

approved