A002681 Numerators of coefficients for repeated integration. (Formerly M5136 N2227)

1, -1, 1, -23, 263, -133787, 157009, -16215071, 2689453969, -26893118531, 5600751928169, -3340626516019229, 885646796787371, -859202038021848149, 2766671664340938282413, -319473088311274492668499, 436677987276721765221113, -191960665849028069896950959123

Offset

0,4

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..17.

H. E. Salzer, Coefficients for repeated integration with central differences, Journal of Mathematics and Physics, 28 (1949), 54-61.

Formula

a(n) is the numerator of ((n+1)/2)M(n) + (2n+2)M(n+1), where M(n) = (2/(2n+1)!)*Integral_{t=0..1} (t*Product_{k=1..n} (t^2 - k^2)). - Emeric Deutsch, Jan 25 2005

Maple

M:=n->(2/(2*n+1)!)*int(t*product(t^2-k^2, k=1..n), t=0..1): A:=n->((n+1)/2)*M(n)+(2*n+2)*M(n+1): seq(numer(A(n)), n=0..18); # Emeric Deutsch, Jan 25 2005

Mathematica

M[n_] := (2/(2n+1)!) Integrate[t Product[t^2-k^2, {k, 1, n}], {t, 0, 1}];
A[n_] := ((n+1)/2) M[n] + (2n+2) M[n+1];
Table[Numerator[A[n]], {n, 0, 18}] (* Jean-François Alcover, Oct 04 2021, after Maple code *)

Crossrefs

Cf. A002195, A002196, A002682.

Sequence in context: A042018 A125411 A140620 * A240876 A142220 A257930

Adjacent sequences: A002678 A002679 A002680 * A002682 A002683 A002684

Keyword

sign,frac

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Jan 25 2005

Status

approved