A007487 Sum of 9th powers. (Formerly M5460)

0, 1, 513, 20196, 282340, 2235465, 12313161, 52666768, 186884496, 574304985, 1574304985, 3932252676, 9092033028, 19696532401, 40357579185, 78800938560, 147520415296, 266108291793, 464467582161, 787155279940, 1299155279940, 2093435326521, 3300704544313

Offset

0,3

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 815.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

B. Berselli, A description of the recursive method in Formula lines (second formula): website Matem@ticamente (in Italian).

Eric Weisstein's World of Mathematics, Power Sum.

Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Formula

a(n) = n^2*(n+1)^2*(n^2+n-1)*(2*n^4+4*n^3-n^2-3*n+3)/20 (see MathWorld, Power Sum, formula 39). a(n) = n*A000542(n) - Sum_{i=0..n-1} A000542(i). - Bruno Berselli, Apr 26 2010

G.f.: x*(1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(1-x)^11. a(n) = a(-n-1). - Bruno Berselli, Aug 23 2011

a(n) = -Sum_{j=1..9} j*s(n+1,n+1-j)*S(n+9-j,n), where s(n,k) and S(n,k) are the Stirling numbers of the first kind and the second kind, respectively. - Mircea Merca, Jan 25 2014

a(n) = (16/5)*A000217(n)^5 - 4*A000217(n)^4 + (12/5)*A000217(n)^3 - (3/5)*A000217(n)^2. - Michael Raney, Mar 14 2016

a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n > 10. - Wesley Ivan Hurt, Dec 21 2016

Maple

[seq(add(i^9, i=1..n), n=0..40)];
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^9 od: seq(a[n], n=0..22); # Zerinvary Lajos, Feb 22 2008

Mathematica

lst={}; s=0; Do[s=s+n^9; AppendTo[lst, s], {n, 10^2}]; lst..or..Table[Sum[k^9, {k, 1, n}], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Range[0, 30]^9] (* Harvey P. Dale, Oct 09 2016 *)

Prog

(Magma) [&+[n^9: n in [0..m]]: m in [0..22]]; // Bruno Berselli, Aug 23 2011
(Python)
A007487_list, m = [0], [362880, -1451520, 2328480, -1905120, 834120, -186480, 18150, -510, 1, 0, 0]
for _ in range(10**2):
....for i in range(10):
........m[i+1]+= m[i]
....A007487_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014
(PARI) a(n)=n^2*(n+1)^2*(n^2+n-1)*(2*n^4+4*n^3-n^2-3*n+3)/20 \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Row 9 of array A103438.

Cf. A000217, A000542.

Sequence in context: A013957 A294304 A036087 * A023878 A301553 A297494

Adjacent sequences: A007484 A007485 A007486 * A007488 A007489 A007490

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved