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Search: a002551 -id:a002551
Displaying 11-13 of 13 results found.
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A002550
Denominators of coefficients of log(1+x)/sqrt(1+x).
(Formerly M5139 N2229)
+10
2
1, 1, 24, 12, 640, 1920, 107520, 1792, 2064384, 10321920, 43253760, 64880640, 5398069248, 4198498304, 503819796480, 62977474560, 16610786017280, 4271344975872, 649244436332544, 41618233098240, 27967452642017280
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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
Vincenzo Librandi, Table of n, a(n) for n = 1..200
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables). [Annotated scanned copy]
MATHEMATICA
Denominator[CoefficientList[Series[Log[1 + x]/Sqrt[1 + x]/x, {x, 0, 40}], x]] (* Vincenzo Librandi, Mar 25 2014 *)
CROSSREFS
Cf. A002549.
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Benoit Cloitre, Mar 29 2002
STATUS
approved
A002554
Numerators of coefficients for numerical differentiation.
(Formerly M4034 N1676)
+10
2
1, -5, 259, -3229, 117469, -7156487, 2430898831, -60997921, 141433003757, -25587296781661, 51270597630767, -6791120985104747, 3400039831130408821, -15317460638921852507, 25789165074168004597399, -1550286106708510672406629, 24823277118070193095631689
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
Ruperto Corso, Table of n, a(n) for n = 1..387
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 23.
FORMULA
a(n) is the numerator of (-1)^(n-1)*Cn-1{1^2..(2n-1)^2}/((2n)!*2^(2n-3)), where Cn{1^2..(2n+1)^2} equals 1 when n=0, otherwise it is the sum of the products of all possible combinations, of size n, of the numbers (2k+1)^2 with k=0,1,...,n. - Ruperto Corso, Dec 15 2011
a(n) = numerator(A001824(n-1)*(-1)^(n-1)/(2^(2*n-3)*(2*n)!)). - Sean A. Irvine, Mar 29 2014
MAPLE
with(combinat):
a:=n->add(mul(k, k=j), j=choose([seq((2*i-1)^2, i=1..n)], n-1))*(-1)^(n-1)/(2^(2*n-3)*(2*n)!):
seq(numer(a(n)), n=1..20); # Ruperto Corso, Dec 15 2011
CROSSREFS
Cf. A001824, A002555.
KEYWORD
sign,frac
AUTHOR
N. J. A. Sloane
EXTENSIONS
Corrected and extended by Ruperto Corso, Dec 15 2011
STATUS
approved
A002553
Coefficients for numerical differentiation.
(Formerly M5166 N2243)
+10
0
1, 24, 640, 7168, 294912, 2883584, 54525952, 167772160, 36507222016, 326417514496, 5772436045824, 50577534877696, 1759218604441600, 15199648742375424, 261208778387488768, 2233785415175766016, 101457092405402533888
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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..16.
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 23, (see denominators of numbers named M(1,2k+1)).
FORMULA
a(n) = denom(A001818(n)*(-1)^(n-1)/(2^(2*n)*(2*n+1)!)). - Sean A. Irvine, Mar 29 2014
a(n) is the denominator of(-1)^(n-1)*Cn-1{1^2..(2n-1)^2}/((2n+1)!*2^(2n)), where Cn{1^2..(2n+1)^2} is equal to 1 when n=0, otherwise, it is the sum of the products of all possible combinations, of size n, of the numbers (2k+1)^2 with k=0,1,..,n. - Sean A. Irvine, after Ruperto Corso, Mar 29 2014
MAPLE
with(combinat): a:=n->add(mul(k, k=j), j=choose([seq((2*i-1)^2, i=1..n)], n))*(-1)^(n-1)/(2^(2*n)*(2*n+1)!):seq(a(n), n=0..20); # Sean A. Irvine, after Ruperto Corso
CROSSREFS
Cf. A001818, A002555.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Sean A. Irvine, Mar 29 2014
STATUS
approved
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