A113630 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8.

1, 45, 4097, 83653, 757305, 4272461, 17736745, 59409477, 169826513, 429794605, 987654321, 2098573445, 4178995657, 7879732173, 14181546905, 24517448581, 40926266145, 66242446637, 104327377633, 160347899205, 241108033241

Offset: 0

Author

Jonathan Vos Post, Jan 14 2006

Keywords

• easy
• nonn

Comments

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7*x^6 + 8*x^7 + 9*x^8 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 = (x^10 - 1)/(x-1).

Examples

			a(3) = 1 + 2*3 + 3*3^2 + 4*3^3 + 5*3^4 + 6*3^5 + 7*3^6 + 8*3^7 + 9*3^8 = 83653 is prime.
a(5) = 1 + 2*5 + 3*5^2 + 4*5^3 + 5*5^4 + 6*5^5 + 7*5^6 + 8*5^7 + 9*5^8 = 4272461 is prime.
a(8) = 1 + 2*8 + 3*8^2 + 4*8^3 + 5*8^4 + 6*8^5 + 7*8^6 + 8*8^7 + 9*8^8 = 169826513 is prime.
a(23) = 1 + 2*23 + 3*23^2 + 4*23^3 + 5*23^4 + 6*23^5 + 7*23^6 + 8*23^7 + 9*23^8 = 733113789893 is prime.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A000012, A005408, A056109, A056578, A056579.

Cf. A249951 (primes), A068475.

Programs

Haskell

a113630 n = sum $ zipWith (*) [1..9] $ iterate (* n) 1
-- Reinhard Zumkeller, Nov 22 2014

Magma

[1+2*n+3*n^2+4*n^3+5*n^4+6*n^5+7*n^6+8*n^7+9*n^8: n in [0..20]]; // Vincenzo Librandi, Nov 09 2014

Mathematica

CoefficientList[Series[(5 x^8 + 1548 x^7 + 31360 x^6 + 129620 x^5 + 148266 x^4 + 48316 x^3 + 3728 x^2 + 36 x + 1) / (1 - x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 09 2014 *)
With[{c=Total[Table[k n^(k-1),{k,9}]]},Table[c,{n,0,30}]] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{1,45,4097,83653,757305,4272461,17736745,59409477,169826513},30] (* Harvey P. Dale, Jul 18 2017 *)

PARI

vector(100,n,1 + 2*(n-1)+ 3*(n-1)^2 + 4*(n-1)^3 + 5*(n-1)^4 + 6*(n-1)^5 + 7*(n-1)^6 + 8*(n-1)^7 + 9*(n-1)^8) \\ Derek Orr, Nov 09 2014

Python

A113630_list, m = [1], [362880, -1229760, 1607760, -1011480, 309816, -40752, 1584, -4, 1]
for _ in range(10**3):
    for i in range(8):
        m[i+1]+= m[i]
    A113630_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

Formula

a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8.

G.f.: -(5*x^8 +1548*x^7 +31360*x^6 +129620*x^5 +148266*x^4 +48316*x^3 +3728*x^2 +36*x +1) / (x -1)^9. - Colin Barker, May 08 2013