A079618 Triangle of coefficients in polynomials for partial sums of powers, scaled to produce integers: Sum_{i=1..m} i^(n-1) = Sum_{k=1..n} T(n,k)*m^k/A064538(n-1).

1, 1, 1, 1, 3, 2, 0, 1, 2, 1, -1, 0, 10, 15, 6, 0, -1, 0, 5, 6, 2, 1, 0, -7, 0, 21, 21, 6, 0, 2, 0, -7, 0, 14, 12, 3, -3, 0, 20, 0, -42, 0, 60, 45, 10, 0, -3, 0, 10, 0, -14, 0, 15, 10, 2, 5, 0, -33, 0, 66, 0, -66, 0, 55, 33, 6, 0, 10, 0, -33, 0, 44, 0, -33, 0, 22, 12, 2, -691, 0, 4550, 0, -9009, 0, 8580, 0, -5005, 0, 2730, 1365, 210

Offset: 1

Author

Henry Bottomley, Jan 29 2003

Keywords

• sign
• tabl
• changed

Comments

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

Examples

			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.
		

References

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

Links

• 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.

Crossrefs

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

Programs

Maple

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

Mathematica

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

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.

Extensions

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