login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A303675 Triangle read by rows: coefficients in the sum of odd powers as expressed by Faulhaber's theorem, T(n, k) for n >= 1, 1 <= k <= n. 2
1, 6, 1, 120, 30, 1, 5040, 1680, 126, 1, 362880, 151200, 17640, 510, 1, 39916800, 19958400, 3160080, 168960, 2046, 1, 6227020800, 3632428800, 726485760, 57657600, 1561560, 8190, 1, 1307674368000, 871782912000, 210680870400, 22313491200, 988107120, 14217840, 32766, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
T(n,k) are the coefficients in an identity due to Faulhaber: Sum_{j=0..n} j^(2*m-1) = Sum_{k=1..m} T(m,k) binomial(n+k, 2*k). See the Knuth reference, page 10.
More explicitly, Faulhaber's theorem asserts that, given integers n >= 0, m >= 1 and odd, Sum_{k=1..n} k^m = Sum_{k=1..(m+1)/2} C(n+k,n-k)*[(1/k)*Sum_{j=0..k-1} (-1)^j*C(2*k,j)*(k-j)^(m+1)]. The coefficients T(m, k) are indicated by square brackets. Sums similar to this inner part are A304330, A304334, A304336; however, these triangles are (0,0)-based and lead to equivalent but slightly more systematic representations. - Peter Luschny, May 12 2018
REFERENCES
John H. Conway and Richard Guy, The Book of Numbers, Springer (1996), p. 107.
LINKS
Donald E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:9207222 [math.CA], 1992.
FORMULA
T(n, k) = (2*(n-k)+1)!*A008957(n, k), n >= 1, 1 <= k <= n.
T(n, k) = (1/m)*Sum_{j=0..m} (-1)^j*binomial(2*m,j)*(m-j)^(2*n) where m = n-k+1. - Peter Luschny, May 09 2018
EXAMPLE
The triangle begins (see the Knuth reference p. 10):
1;
6, 1;
120, 30, 1;
5040, 1680, 126, 1;
362880, 151200, 17640, 510, 1;
39916800, 19958400, 3160080, 168960, 2046, 1;
6227020800, 3632428800, 726485760, 57657600, 1561560, 8190, 1;
.
Let S(n, m) = Sum_{j=1..n} j^m. Faulhaber's formula gives for m = 7 (m odd!):
F(n, 7) = 5040*C(n+4, 8) + 1680*C(n+3, 6) + 126*C(n+2, 4) + C(n+1, 2).
Faulhaber's theorem asserts that for all n >= 1 S(n, 7) = F(n, 7).
If n = 43 the common value is 1600620805036.
MAPLE
T := proc(n, k) local m; m := n-k;
2*(2*m+1)!*add((-1)^(j+m)*(j+1)^(2*n)/((j+m+2)!*(m-j)!), j=0..m) end:
seq(seq(T(n, k), k=1..n), n=1..8); # Peter Luschny, May 09 2018
MATHEMATICA
(* After Peter Luschny's above formula. *)
T[n_, k_] := (1/(n-k+1))*Sum[(-1)^j*Binomial[2*(n-k+1), j]*((n-k+1) - j)^(2*n), {j, 0, n-k+1}]; Column[Table[T[n, k], {n, 1, 10}, {k, 1, n}], Center]
PROG
(Sage)
def A303675(n, k): return factorial(2*(n-k)+1)*A008957(n, k)
for n in (1..7): print([A303675(n, k) for k in (1..n)]) # Peter Luschny, May 10 2018
CROSSREFS
First column is a bisection of A000142, second column is a bisection of A001720.
Row sums give A100868.
Sequence in context: A365908 A331557 A352058 * A266302 A352012 A183284
KEYWORD
nonn,tabl
AUTHOR
Kolosov Petro, May 08 2018
EXTENSIONS
New name by Peter Luschny, May 10 2018
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 15 10:24 EDT 2024. Contains 371681 sequences. (Running on oeis4.)