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A079618
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Triangle of coefficients in polynomials for partial sums of powers, scaled to produce integers: sum_i{1<=i<=m}i^(n-1) = sum_k{1<=k<=n}T(n,k)*m^k/A064538(n-1).
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1
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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
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OFFSET
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1,5
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COMMENTS
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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 of k!, k=1..n. - Jonathan Vos Post, Feb 22 2007
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REFERENCES
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Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 106, 1996.
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LINKS
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Eric Weisstein's World of Mathematics, Power Sum
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FORMULA
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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<=k<=n} T(n, k) for n>1; T(1, 1)=1.
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EXAMPLE
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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
For example row n=7: partial sums of 6th powers (A000540)
1^6+2^6+...+m^6 = (m-7m^3+21m^5+21m^6+6m^7)/42.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Edited. Offset corrected from 0 to 1. Typo in formula corrected. - Wolfdieter Lang, Feb 02 2015
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STATUS
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approved
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