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.

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

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

Table of n, a(n) for n=1..36.

Donald E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:9207222 [math.CA], 1992.

Petro Kolosov, Polynomial identities involving central factorial numbers, GitHub, 2024. See pp. 3, 6.

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.

Cf. A008955, A008957, A036969 and A304330, A304334, A304336.

Sequence in context: A365908 A331557 A352058 * A266302 A352012 A183284

Adjacent sequences: A303672 A303673 A303674 * A303676 A303677 A303678

Keyword

nonn,tabl

Author

Kolosov Petro, May 08 2018

Extensions

New name by Peter Luschny, May 10 2018

Status

approved