A025992 Expansion of 1/((1-2*x)*(1-5*x)*(1-7*x)*(1-8*x)).

1, 22, 313, 3666, 38493, 377286, 3529681, 31947322, 282198565, 2447183310, 20920905369, 176852694018, 1481626607917, 12322682753494, 101879323774177, 838170485025354, 6867569457133749, 56077266261254238

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Bruno Berselli, May 09 2013: (Start)

a(n) - 2*a(n-1), for n>0, gives A019928 (after 1);

a(n) - 5*a(n-1), for n>0, gives A016311 (after 1);

a(n) - 7*a(n-1), for n>0, gives A016297 (after 1);

a(n) - 8*a(n-1), for n>0, gives A016296 (after 1);

a(n) - 7*a(n-1) + 10*a(n-2), for n>1, gives A016177 (after 15);

a(n) - 9*a(n-1) + 14*a(n-2), for n>1, gives A016162 (after 13);

a(n) - 10*a(n-1) + 16*a(n-2), for n>1, gives A016161 (after 12);

a(n) - 12*a(n-1) + 35*a(n-2), for n>1, gives A016131 (after 10);

a(n) - 13*a(n-1) + 40*a(n-2), for n>1, gives A016130 (after 9);

a(n) - 15*a(n-1) + 56*a(n-2), for n>1, gives A016127 (after 7);

a(n) - 20*a(n-1) +131*a(n-2) -280*a(n-3), for n>2, gives A000079 (after 4);

a(n) - 17*a(n-1) +86*a(n-2) -112*a(n-3), for n>2, gives A000351 (after 25);

a(n) - 15*a(n-1) +66*a(n-2) -80*a(n-3), for n>2, gives A000420 (after 49);

a(n) - 14*a(n-1) +59*a(n-2) -70*a(n-3), for n>2, gives A001018 (after 64),

and naturally: a(n) - 22*a(n-1) + 171*a(n-2) - 542*a(n-3) + 560*a(n-4), for n>3, gives 0 (see Harvey P. Dale in Formula lines). (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (22,-171,542,-560).

Crossrefs

Cf. A000079, A000351, A000420, A001018, A016127, A016130, A016131, A016161, A016162, A016177, A016296, A016297, A016311, A019928.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!(1/((1-2*x)*(1-5*x)*(1-7*x)*(1-8*x)))); // Bruno Berselli, May 09 2013

Mathematica

CoefficientList[Series[1/((1-2x)(1-5x)(1-7x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[ {22,-171,542,-560},{1,22,313,3666},30] (* Harvey P. Dale, Jan 29 2013 *)

PARI

a(n) = n+=3; (5*8^n-9*7^n+5*5^n-2^n)/90 \\ Charles R Greathouse IV, Oct 03 2016

Python

def A025992(n): return (5*pow(8,n+3)-9*pow(7,n+3)+pow(5,n+4)-pow(2,n+3))//90
print([A025992(n) for n in range(41)]) # G. C. Greubel, Dec 27 2024

Formula

a(0)=1, a(1)=22, a(2)=313, a(3)=3666, a(n) = 22*a(n-1) - 171*a(n-2) + 542*a(n-3) - 560*a(n-4). - Harvey P. Dale, Jan 29 2013

a(n) = (5*8^(n+3) - 9*7^(n+3) + 5^(n+4) - 2^(n+3))/90. - Yahia Kahloune, May 07 2013

E.g.f.: (1/90)*(-8*exp(2*x) + 625*exp(5*x) - 3087*exp(7*x) + 2560*exp(8*x)). - G. C. Greubel, Dec 27 2024