OFFSET
1,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (210, -17985, 836310, -23627805, 429628026, -5189886205, 42366601950, -235447933875, 889918833750, -2267731621875, 3835990781250, -4208760859375, 2865761718750, -1098193359375, 180878906250).
FORMULA
a(n) = 70^n - (4*n + 1)*35^n + binomial(4*n+1, 2)*15^n - binomial(4*n+1, 3)*5^n + binomial(4*n+1, 4). - Andrew Howroyd, May 07 2020
From G. C. Greubel, Sep 09 2022: (Start)
a(n) = Sum_{j=0..4} (-1)^j*binomial(4*n+1, j)*binomial(8-j, 4)^n.
G.f.: x^2*(1 +13630*x +2073340*x^2 -60833350*x^3 -1182529995*x^4 +34295189100*x^5 -173276304000*x^6 -651083647500*x^7 +5378182646875*x^8 -9980105906250*x^9 -2825648437500*x^10 +19397519531250*x^11 +3165380859375*x^12)/( Product_{j=0..4} (1 - binomial(j+4,4)*x)^(5-j) ).
E.g.f.: exp(70*x) -(1+140*x)*exp(35*x) +150*x*(1+12*x)*exp(15*x) -(50/3)*x*(3 +48*x +80*x^2)*exp(5*x) +(1/3)*x*(15 +174*x +176*x^2 +32*x^3)*exp(x). (End)
MATHEMATICA
Table[Sum[(-1)^j*Binomial[4*n+1, j]*Binomial[8-j, 4]^n, {j, 0, 4}], {n, 30}] (* G. C. Greubel, Sep 09 2022 *)
PROG
(PARI) a(n) = {70^n - (4*n + 1)*35^n + binomial(4*n+1, 2)*15^n - binomial(4*n+1, 3)*5^n + binomial(4*n+1, 4)} \\ Andrew Howroyd, May 07 2020
(Magma) [(&+[(-1)^j*Binomial(4*n+1, j)*Binomial(8-j, 4)^n: j in [0..4]]): n in [1..30]]; // G. C. Greubel, Sep 09 2022
(SageMath)
def A151642(n): return sum((-1)^j*binomial(4*n+1, j)*binomial(8-j, 4)^n for j in (0..4))
[A151642(n) for n in (1..30)] # G. C. Greubel, Sep 09 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, May 29 2009
EXTENSIONS
Terms a(8) and beyond from Andrew Howroyd, May 07 2020
STATUS
approved