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Search: id:A079618
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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
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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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 (jvospost3(AT)gmail.com), 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
Elemer E. Rosinger, Synthesizing Sums, 21 Feb 2007.
Eric Weisstein's World of Mathematics, Faulhaber's Formula.
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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); T(1, 1)=1.
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EXAMPLE
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Rows start: 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; etc. For example, 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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CROSSREFS
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Cf. A000217, A000330, A007489, A033312.
Sequence in context: A131290 A116604 A138741 this_sequence A117406 A151844 A008783
Adjacent sequences: A079615 A079616 A079617 this_sequence A079619 A079620 A079621
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KEYWORD
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sign,tabl
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), Jan 29 2003
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