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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.

%I A113630 #24 Feb 02 2025 13:46:38
%S A113630 1,45,4097,83653,757305,4272461,17736745,59409477,169826513,429794605,
%T A113630 987654321,2098573445,4178995657,7879732173,14181546905,24517448581,
%U A113630 40926266145,66242446637,104327377633,160347899205,241108033241
%N 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.
%C A113630 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).
%H A113630 Chai Wah Wu, <a href="/A113630/b113630.txt">Table of n, a(n) for n = 0..1000</a>
%H A113630 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A113630 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.
%F A113630 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
%e A113630 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.
%e A113630 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.
%e A113630 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.
%e A113630 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.
%t A113630 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 *)
%t A113630 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 *)
%o A113630 (Python)
%o A113630 A113630_list, m = [1], [362880, -1229760, 1607760, -1011480, 309816, -40752, 1584, -4, 1]
%o A113630 for _ in range(10**3):
%o A113630     for i in range(8):
%o A113630         m[i+1]+= m[i]
%o A113630     A113630_list.append(m[-1]) # _Chai Wah Wu_, Nov 09 2014
%o A113630 (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
%o A113630 (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
%o A113630 (Haskell)
%o A113630 a113630 n = sum $ zipWith (*) [1..9] $ iterate (* n) 1
%o A113630 -- _Reinhard Zumkeller_, Nov 22 2014
%Y A113630 Cf. A000012, A005408, A056109, A056578, A056579.
%Y A113630 Cf. A249951 (primes), A068475.
%K A113630 easy,nonn
%O A113630 0,2
%A A113630 _Jonathan Vos Post_, Jan 14 2006