1,5

Rosinger connects this sequence to Weisstein's Faulhaber's Formula page. Rosinger also discusses, without reference to OEIS, (1.1) A000217 Triangular numbers: a(n) = C(n+1,2) = n*(n+1)/2 = 0+1+2+...+n; (1.2) A000330 Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n*(n+1)*(2n+1)/6; (1.4) A033312 n! - 1 [with different offset and the formula 1*1! + 2*2! + 3*3! + ...]; (1.4) A007489 Sum_{k=1..n} k!. - Jonathan Vos Post, Feb 22 2007

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 106, 1996.

R. Mestrovic, A congruence modulo n^3 involving two consecutive sums of powers and its applications, arXiv:1211.4570 [math.NT], 2012. - From N. J. A. Sloane, Jan 03 2013

R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

Elemer E. Rosinger, Synthesizing Sums, arXiv:math/0702605 [math.GM], 2007.

Eric Weisstein's World of Mathematics, Power Sum

Eric Weisstein's World of Mathematics, Faulhaber's Formula.

T(n, k) = T(n-1, k-1) * (n-1) * A064538(n-1) / (k*A064538(n-2)) for k>1; T(n, 1) = A064538(n-1) - Sum_{k=2..n} T(n, k) for n>1; T(1, 1)=1.

Triangle T(n, k) begins:

n\k 1 2 3 4 5 6 7 8 9 10 ...

1: 1

2: 1 1

3: 1 3 2

4: 0 1 2 1

5: -1 0 10 15 6

6: 0 -1 0 5 6 2

7: 1 0 -7 0 21 21 6

8: 0 2 0 -7 0 14 12 3

9: -3 0 20 0 -42 0 60 45 10

10: 0 -3 0 10 0 -14 0 15 10 2

... Reformatted. - Wolfdieter Lang, Feb 02 2015

For example row n=7: partial sums of 6th powers (A000540)

1^6+2^6+...+m^6 = (m-7*m^3+21*m^5+21*m^6+6*m^7)/42.

T := proc(n, k) option remember; local A, B;

A := proc(n) option remember; denom((bernoulli(n+1, x)-bernoulli(n+1))/(n+1)) end:

B := proc(n) option remember; add(T(n, j), j=2..n) end;

if k>1 then T(n-1, k-1)*(n-1)*A(n-1)/(k*A(n-2)) elif n>1 then A(n-1)-B(n) else 1 fi end: seq(print(seq(T(n, k), k=1..n)), n=1..10); # Peter Luschny, Feb 02 2015

# Alternative:

A079618row := proc(n) bernoulli(n, x); (subs(x=x+1, %)-subs(x=1, %))/n;

seq(coeff(numer(%), x, k), k=1..n) end:

seq(A079618row(n), n=1..13); # Peter Luschny, Jul 14 2020

T[n_, k_] := T[n, k] = Module[{A, B}, A[m_] := A[m] = Denominator[ Together[ (BernoulliB[m+1, x] - BernoulliB[m+1])/(m+1)]]; B[m_] := B[m] = Sum[T[m, j], {j, 2, m}]; Which[k>1, T[n-1, k-1]*(n-1)*A[n-1]/(k*A[n-2]), n>1, A[n-1] - B[n], True, 1]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Sep 04 2015, after Peter Luschny *)

(PARI) row(p) = {v = vector(p+1, k, (-1)^(k==p)*binomial(p+1, k)*bernfrac(p+1-k))/(p+1); lcmd = lcm(vector(#v, k, denominator(v[k]))); v*lcmd; }

tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Feb 16 2016

Cf. A064538, A000217, A000330, A007489, A033312.

Sequence in context: A292795 A295028 A294201 * A151844 A286223 A341163

Adjacent sequences: A079615 A079616 A079617 * A079619 A079620 A079621

sign,tabl

Henry Bottomley, Jan 29 2003

Edited. Offset corrected from 0 to 1. Typo in formula corrected. - Wolfdieter Lang, Feb 02 2015

approved