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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).

%I A079618 #45 Aug 22 2025 17:19:07
%S A079618 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,
%T A079618 -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,
%U A079618 -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
%N 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).
%C A079618 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
%D A079618 Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 106, 1996.
%H A079618 R. Mestrovic, <a href="http://arxiv.org/abs/1211.4570">A congruence modulo n^3 involving two consecutive sums of powers and its applications</a>, arXiv:1211.4570 [math.NT], 2012. - From _N. J. A. Sloane_, Jan 03 2013
%H A079618 R. Mestrovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mestrovic/mes4.html">On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers</a>, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
%H A079618 Elemer E. Rosinger, <a href="http://arXiv.org/abs/math.GM/0702605">Synthesizing Sums</a>, arXiv:math/0702605 [math.GM], 2007.
%H A079618 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerSum.html">Power Sum</a>
%H A079618 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FaulhabersFormula.html">Faulhaber's Formula.</a>
%F A079618 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.
%e A079618 Triangle T(n, k) begins:
%e A079618 n\k 1   2   3   4   5    6   7   8   9 10 ...
%e A079618 1:  1
%e A079618 2:  1   1
%e A079618 3:  1   3   2
%e A079618 4:  0   1   2   1
%e A079618 5: -1   0  10  15   6
%e A079618 6:  0  -1   0   5   6    2
%e A079618 7:  1   0  -7   0  21   21   6
%e A079618 8:  0   2   0  -7   0   14  12   3
%e A079618 9: -3   0  20   0 -42    0  60  45  10
%e A079618 10: 0  -3   0  10   0  -14   0  15  10  2
%e A079618 ... Reformatted. - _Wolfdieter Lang_, Feb 02 2015
%e A079618 For example row n=7: partial sums of 6th powers (A000540)
%e A079618   1^6+2^6+...+m^6 = (m-7*m^3+21*m^5+21*m^6+6*m^7)/42.
%p A079618 T := proc(n, k) option remember; local A, B;
%p A079618 A := proc(n) option remember; denom((bernoulli(n+1,x)-bernoulli(n+1))/(n+1)) end:
%p A079618 B := proc(n) option remember; add(T(n,j),j=2..n) end;
%p A079618 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
%p A079618 # Alternative:
%p A079618 A079618row := proc(n) bernoulli(n,x); (subs(x=x+1,%)-subs(x=1,%))/n;
%p A079618 seq(coeff(numer(%),x,k), k=1..n) end:
%p A079618 seq(A079618row(n), n=1..13); # _Peter Luschny_, Jul 14 2020
%t A079618 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_ *)
%o A079618 (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;}
%o A079618 tabl(nn) = for (n=0, nn, print(row(n))); \\ _Michel Marcus_, Feb 16 2016
%Y A079618 Cf. A064538, A000217, A000330, A007489, A033312.
%K A079618 sign,tabl
%O A079618 1,5
%A A079618 _Henry Bottomley_, Jan 29 2003
%E A079618 Edited. Offset corrected from 0 to 1. Typo in formula corrected. - _Wolfdieter Lang_, Feb 02 2015