A060190 A column and diagonal of A060187 (k=4).

1, 76, 1682, 23548, 259723, 2485288, 21707972, 178300904, 1403080725, 10708911188, 79944249686, 587172549764, 4261002128223, 30644790782352, 218917362275080, 1556000598766224, 11017646288488233, 77790282457881756

Offset: 4

Author

N. J. A. Sloane, Mar 20 2001

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 4..1000
• P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 19 (1921), 305-340; Coll. Papers II, pp. 267-302.
• Index entries for linear recurrences with constant coefficients, signature (30,-385,2776,-12418,35908,-67818,82552,-62109,26190,-4725).

Crossrefs

Cf. A060187.

Programs

Magma

[(&+[(-1)^k*Binomial(n,k)*(7-2*k)^(n-1): k in [0..3]]): n in [4..40]]; // G. C. Greubel, Aug 01 2024

Maple

r := proc(n, k) option remember;
if n = 0 then if k = 0 then 1 else 0 fi else
(2*(n-k)+1)*r(n-1, k-1) + (2*k+1)*r(n-1, k) fi end:
A060189 := n -> r(n-1, 3): seq(A060189(n), n = 4..21); # Peter Luschny, May 06 2013

Mathematica

r[n_, k_] := r[n, k] = If[n == 0, If[k == 0, 1, 0], (2*(n-k)+1)*r[n-1, k-1] + (2*k+1)*r[n-1, k]]; A060189[n_] := r[n-1, 3]; Table[A060189[n], {n, 4, 21}] (* Jean-François Alcover, Dec 03 2013, translated from Peter Luschny's program *)
A060190[n_]:= Sum[(-1)^k*Binomial[n,k]*(7-2*k)^(n-1), {k,0,3}];
Table[A060190[n], {n,4,40}] (* G. C. Greubel, Aug 01 2024 *)

SageMath

[sum((-1)^k*binomial(n,k)*(7-2*k)^(n-1) for k in range(4)) for n in range(4,40)] # G. C. Greubel, Aug 01 2024

Formula

From Wolfdieter Lang, Apr 17 2017: (Start)

a(n) = A060187(n, 4), n >= 4, and 0 for n < 4,

a(n) = A060187(n, n-3), n >= 4, and 0 for n < 4.

O.g.f.: x^4*(1 + 46*x - 213*x^2 - 428*x^3 + 2295*x^4 - 1794*x^5 - 675*x^6) / Product_{j=0..3} (1 - (1+2*j)*x)^(4-j).

E.g.f.: (exp(7*x) - 7*x*exp(5*x) + (21*x^2/2)*exp(3*x) - (7*x^3/3!)*exp(x) - 1)/7. (End)

a(n) = Sum_{k=0..3} (-1)^k*binomial(n,k)*(7-2*k)^(n-1). - G. C. Greubel, Aug 01 2024

Extensions

More terms from Vladeta Jovovic, Mar 20 2001