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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.

%I A303675 #91 Dec 28 2024 11:11:24
%S A303675 1,6,1,120,30,1,5040,1680,126,1,362880,151200,17640,510,1,39916800,
%T A303675 19958400,3160080,168960,2046,1,6227020800,3632428800,726485760,
%U A303675 57657600,1561560,8190,1,1307674368000,871782912000,210680870400,22313491200,988107120,14217840,32766,1
%N 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.
%C A303675 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.
%C A303675 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
%D A303675 John H. Conway and Richard Guy, The Book of Numbers, Springer (1996), p. 107.
%H A303675 Donald E. Knuth, <a href="https://arxiv.org/abs/math/9207222">Johann Faulhaber and Sums of Powers</a>, arXiv:9207222 [math.CA], 1992.
%H A303675 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/PolynomialIdentitiesInvolvingCentralFactorialNumbers.pdf">Polynomial identities involving central factorial numbers</a>, GitHub, 2024. See pp. 3, 6.
%F A303675 T(n, k) = (2*(n-k)+1)!*A008957(n, k), n >= 1, 1 <= k <= n.
%F A303675 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
%e A303675 The triangle begins (see the Knuth reference p. 10):
%e A303675          1;
%e A303675          6,          1;
%e A303675        120,         30,         1;
%e A303675       5040,       1680,       126,        1;
%e A303675     362880,     151200,     17640,      510,       1;
%e A303675   39916800,   19958400,   3160080,   168960,    2046,    1;
%e A303675 6227020800, 3632428800, 726485760, 57657600, 1561560, 8190, 1;
%e A303675 .
%e A303675 Let S(n, m) = Sum_{j=1..n} j^m. Faulhaber's formula gives for m = 7 (m odd!):
%e A303675 F(n, 7) = 5040*C(n+4, 8) + 1680*C(n+3, 6) + 126*C(n+2, 4) + C(n+1, 2).
%e A303675 Faulhaber's theorem asserts that for all n >= 1 S(n, 7) = F(n, 7).
%e A303675 If n = 43 the common value is 1600620805036.
%p A303675 T := proc(n,k) local m; m := n-k;
%p A303675 2*(2*m+1)!*add((-1)^(j+m)*(j+1)^(2*n)/((j+m+2)!*(m-j)!), j=0..m) end:
%p A303675 seq(seq(T(n, k), k=1..n), n=1..8); # _Peter Luschny_, May 09 2018
%t A303675 (* After _Peter Luschny_'s above formula. *)
%t A303675 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]
%o A303675 (Sage)
%o A303675 def A303675(n, k): return factorial(2*(n-k)+1)*A008957(n, k)
%o A303675 for n in (1..7): print([A303675(n, k) for k in (1..n)]) # _Peter Luschny_, May 10 2018
%Y A303675 First column is a bisection of A000142, second column is a bisection of A001720.
%Y A303675 Row sums give A100868.
%Y A303675 Cf. A008955, A008957, A036969 and A304330, A304334, A304336.
%K A303675 nonn,tabl
%O A303675 1,2
%A A303675 _Kolosov Petro_, May 08 2018
%E A303675 New name by _Peter Luschny_, May 10 2018