A056468 -id:A056468 - cp's OEIS frontend

A001014 Sixth powers: a(n) = n^6.

0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, 85766121, 113379904, 148035889, 191102976, 244140625, 308915776, 387420489, 481890304

Offset: 0

N. J. A. Sloane

Numbers both square and cubic. - Patrick De Geest
Totally multiplicative sequence with a(p) = p^6 for prime p. - Jaroslav Krizek, Nov 01 2009
Numbers n for which the order of the torsion subgroup of the elliptic curve y^2 = x^3 + n is t = 6, cf. Gebel link. - Artur Jasinski, Jun 30 2010
Note that Sum_{n>=1} 1/a(n) = Pi^6 / 945. - Mohammad K. Azarian, Nov 01 2011
The binomial transform yields A056468. The inverse binomial transform yields the (finite) 0, 1, 62, 540, ..., 720, the 6th row in A019538 and A131689. - R. J. Mathar, Jan 16 2013
For n > 0, a(n) is the largest number k such that k + n^3 divides k^2 + n^3. - Derek Orr, Oct 01 2014

			The 6th powers of the first few integers are: 0^6 = 0 = a(0), 1^6 = 1 = a(1), 2^6 = 64 = a(2), 3^6 = 9^3 = 729 = a(3), 4^6 = 2^12 = 4096 = a(4), 5^6 = 25^3 = 15625 = a(5), etc.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity, eq. (6.37).
Granino A. Korn and Theresa M.Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), p. 982.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..500
Henry Bottomley, Illustration of initial terms
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
Richard J. Mathar, Construction of Bhaskara pairs, arXiv:1703.01677 [math.NT], 2017.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Subsequence of A201217.
Cf. A000540 (partial sums), A022522 (first differences), A008292.
Intersection of A000290 (squares) and A000578 (cubes).
Cf. A002604 (n^6+1), A123866 (n^6-1), A013664 (zeta(6)), A275703 (eta(6)).
Cf. A003358 - A003368 (sums of 2, ..., 12 positive sixth powers).

a001014 n = a001014_list !! n
a001014_list = map (^ 6) [0..] -- Reinhard Zumkeller, Dec 04 2011

Table[n^6, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)

A001014(n):=n^6$
makelist(A001014(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

A001014(n)=n^6 \\ Charles R Greathouse IV, Sep 24 2015

A001014 = lambda n: n**6 # M. F. Hasler, Jul 03 2025

a(n) = A123866(n) + 1 = A002604(n) - 1.
G.f.: -x*(1+x)*(x^4+56*x^3+246*x^2+56*x+1) / (x-1)^7. - Simon Plouffe in his 1992 dissertation
Multiplicative with a(p^e) = p^(6e). - David W. Wilson, Aug 01 2001
E.g.f.: (x + 31x^2 + 90x^3 + 65x^4 + 15x^5 + x^6)*exp(x).  Generally, the e.g.f. for n^m is Sum_{k=1..m} A008277(m,k)*x^k*exp(x). - Geoffrey Critzer, Aug 25 2013
From Ant King, Sep 23 2013: (Start)
Signature {7, -21, 35, -35, 21, -7, 1}.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + 720. (End)
a(n) == 1 (mod 7) if gcd(n, 7) = 1, otherwise a(n) == 0 (mod 7). See A109720. - Jake Lawrence, May 28 2016
From Ilya Gutkovskiy, Jul 06 2016: (Start)
Dirichlet g.f.: zeta(s-6).
Sum_{n>=1} 1/a(n) = Pi^6/945 = A013664. (End)
a(n) = Sum_{k=1..6} Eulerian(6, k)*binomial(n+6-k, 6), with Eulerian(6, k) = A008292(6, k) (the numbers are 1, 57, 302, 302, 57, 1) for n >= 0. Worpitzki's identity for powers of 6. See. e.g., Graham et al., eq. (6, 37) (using A173018, the row reversed version of A123125). - Wolfdieter Lang, Jul 17 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 31*zeta(6)/32 = 31*Pi^6/30240 (A275703). - Amiram Eldar, Oct 08 2020
From Amiram Eldar, Jan 20 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = (cosh(Pi)-cos(sqrt(3)*Pi))*sinh(Pi)/(2*Pi^3).
Product_{n>=2} (1 - 1/a(n)) = cosh(sqrt(3)*Pi/2)^2/(6*Pi^2). (End)

Comments from 2010 - 2011 edited by M. F. Hasler, Jul 05 2024

A084641 Binomial transform of n^7.

Original entry on oeis.org

0, 1, 130, 2574, 25904, 183200, 1040112, 5076400, 22171648, 88915968, 333209600, 1181548544, 4001402880, 13033885696, 41061830656, 125666611200, 374947708928, 1093874155520, 3128047828992, 8785866391552, 24280799641600, 66124498599936, 177683966197760

Offset: 0

Paul Barry, Jun 08 2003

The binomial transforms of n, n^2, n^3, n^4, n^5, n^6 are A001787, A001788, A058645, A058649, A059338, A056468 respectively.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Cf. A001015, A001787, A001788, A058645, A058649, A059338, A056468.

[n^2*(n^5+21*n^4+105*n^3+35*n^2-210*n+112)*2^(n-7): n in [0..40]]; // G. C. Greubel, Mar 20 2023

LinearRecurrence[{16,-112,448,-1120,1792,-1792,1024,-256}, {0,1,130, 2574,25904,183200,1040112,5076400}, 41] (* Amiram Eldar, Nov 26 2021 *)

[n^2*(n^5+21*n^4+105*n^3+35*n^2-210*n+112)*2^(n-7) for n in range(41)] # G. C. Greubel, Mar 20 2023

a(n) = n^2*(n^5 + 21*n^4 + 105*n^3 + 35*n^2 - 210*n + 112)*2^(n-7).
a(n) = Sum_{k=0..n} C(n, k)*k^7.
G.f.: x*(1+114*x+606*x^2-1168*x^3-96*x^4+816*x^5-272*x^6)/(1-2*x)^8. - Colin Barker, Sep 20 2012

A349706 Square array T(n,k) = Sum_{j=0..k} binomial(k,j) * j^n for n and k >= 0, read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 1, 4, 8, 0, 1, 6, 12, 16, 0, 1, 10, 24, 32, 32, 0, 1, 18, 54, 80, 80, 64, 0, 1, 34, 132, 224, 240, 192, 128, 0, 1, 66, 342, 680, 800, 672, 448, 256, 0, 1, 130, 924, 2192, 2880, 2592, 1792, 1024, 512, 0, 1, 258, 2574, 7400, 11000, 10752, 7840, 4608, 2304, 1024

Offset: 0

Michel Marcus, Nov 26 2021

			Square array begins:
  1 2  4   8   16    32
  0 1  4  12   32    80
  0 1  6  24   80   240
  0 1 10  54  224   800
  0 1 18 132  680  2880
  0 1 34 342 2192 11000

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Renate Golombek, Aufgabe 1088, El. Math., 49 (1994), 126-127.
Simsek Yilmaz, New families of special numbers for computing negative order Euler numbers and related numbers and polynomials, Applicable Analysis and Discrete Mathematics 2018 Volume 12, Issue 1, Pages: 1-35. See B(n,k).

Rows n=0..7 give A000079, A001787, A001788, A058645, A058649, A059338, A056468, A084641.
Main diagonal gives A072034.
Cf. A209849.

T[n_, k_] := Sum[Binomial[k, j] * If[j == n == 0, 1, j^n], {j, 0, k}]; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 26 2021 *)

T(n,k) = sum(j=0, k, binomial(k,j)*j^n);

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A001014 Sixth powers: a(n) = n^6.

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A084641 Binomial transform of n^7.

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A349706 Square array T(n,k) = Sum_{j=0..k} binomial(k,j) * j^n for n and k >= 0, read by ascending antidiagonals.

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