keyword:sign - cp's OEIS frontend

A049347 Period 3: repeat [1, -1, 0].

1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0

Offset: 0

Wolfdieter Lang

G.f. 1/cyclotomic(3, x) (the third cyclotomic polynomial).
Self-convolution yields (-1)^n*A099254(n). - R. J. Mathar, Apr 06 2008
Hankel transform of A099324. - Paul Barry, Aug 10 2009
A057083(n) = p(-1)  where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0..n. - Michael Somos, Apr 29 2012
a(n) appears, together with b(n) = A099837(n+3) in the formula 2*exp(2*Pi*n*I/3) = b(n) + a(n)*sqrt(3)*I, n >= 0, with I = sqrt(-1). See A164116 for the case N=5. - Wolfdieter Lang, Feb 27 2014
The binomial transform is 1, 0, -1, -1, 0, 1, 1, 0, -1, -1.. (see A010891). The inverse binom. transform is 1, -2, 3, -3, 0, 9, -27, 54, -81.. (see A057682). - R. J. Mathar, Feb 25 2023

			G.f. = 1 - x + x^3 - x^4 + x^6 - x^7 + x^9 - x^10 + x^12 - x^13 + x^15 + ...

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 175.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012.
Elena Barcucci, Antonio Bernini, Stefano Bilotta and Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (-1,-1).
Index entries for sequences related to Chebyshev polynomials.
Index to sequences related to inverse of cyclotomic polynomials

Cf. A001006, A010892, A057078, A057083, A102283, A106510, A130716, A143818, A194770.

Alternating row sums of A049310 (Chebyshev-S). [Wolfdieter Lang, Nov 04 2011]

&cat[[1,-1,0]: n in [0..90]]; // Vincenzo Librandi, Apr 03 2014

A049347 := proc(n)
    op(modp(n,3)+1,[1,-1,0]) ;
end proc:
seq(A049347(n),n=0..100) ; # R. J. Mathar, Aug 06 2016

Flatten[Table[{1, -1, 0}, {27}]] (* Alonso del Arte, Feb 07 2013 *)
CoefficientList[Series[1/Cyclotomic[3, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)
LinearRecurrence[{-1, -1}, {1, -1}, 90] (* Ray Chandler, Sep 15 2015 *)
Table[DirichletCharacter[3, 2, n + 1], {n, 0, 29}] (* Steven Foster Clark, May 29 2019 *)
Table[Mod[n + 2, 3] - 1, {n, 0, 20}] (* Michael Somos, Sep 24 2019 *)
Table[ChebyshevU[n, -1/2], {n, 0, 20}] (* Eric W. Weisstein, Jan 09 2024 *)
ChebyshevU[Range[0, 20], -1/2] (* Eric W. Weisstein, Jan 09 2024 *)

A049347(n) := block(
        [1,-1,0][1+mod(n,3)]
)$ /* R. J. Mathar, Mar 19 2012 */

{a(n) = n++; kronecker( -3, n)} /* Michael Somos, Oct 03 2006 */

{a(n) = [1, -1, 0][n%3 + 1]} /* Michael Somos, Oct 15 2008 */

a(n)=(n+2)%3-1 /* Jaume Oliver Lafont, Mar 24 2009 */

def A049347():
    x, y = 1, -1
    while True:
        yield x
        x, y = y, -x - y
a = A049347(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013

G.f.: 1/(1+x+x^2).
a(n) = +1 if n mod 3 = 0, a(n) = -1 if n mod 3 = 1, else 0.
a(n) = S(n, -1) = U(n, -1/2) (Chebyshev's U(n, x) polynomials.)
a(n) = 2*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/3. - Paul Barry, Mar 15 2004
a(n) = Sum_{k >= 0} (-1)^(n-k)*C(n-k, k).
Given g.f. A(x), then B(x) = x * A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v + 2*u*v. - Michael Somos, Oct 03 2006
Euler transform of length 3 sequence [-1, 0, 1]. - Michael Somos, Oct 03 2006
a(n) = b(n+1) where b(n) is multiplicative with b(3^e) = 0^e, b(p^e) = 1 if p == 1 (mod 3), b(p^e) = (-1)^e if p == 2 (mod 3). - Michael Somos, Oct 03 2006
From Michael Somos, Oct 03 2006: (Start)
G.f.: (1 - x) /(1 - x^3).
a(n) = -a(1-n) = -a(n-1) - a(n-2) = a(n-3). (End)
From Michael Somos, Apr 29 2012: (Start)
G.f.: 1 / (1 + x / ( 1 - x / (1 + x))).
a(n) = (-1)^n * A010892(n).
a(n) * n! = A194770(n+1).
Revert transform of A001006. Convolution inverse of A130716. MOBIUS transform of A002324. EULER transform is A111317. BIN1 transform of itself. STIRLING transform is A143818(n+2). (End)
a(-n) = A057078(n). a(n) = A106510(n+1) unless n=0. - Michael Somos, Oct 15 2008
G.f. A(x) = 1/(1+x+x^2) = Q(0); Q(k) =  1- x/(1 - x^2/(x^2 - 1 + x/(x - 1 + x^2/(x^2 - 1/Q(k+1))))); (continued fraction 3 kind, 5-step ). - Sergei N. Gladkovskii, Jun 19 2012
a(n) = -1 + floor(67/333*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013
a(n) = -1 + floor(19/26*3^(n+1)) mod 3. - Hieronymus Fischer, Jan 03 2013
a(n) = ceiling((n-1)/3) - ceiling(n/3) + floor(n/3) - floor((n-1)/3). - Wesley Ivan Hurt, Dec 06 2013
a(n) = n + 1 - 3*floor((n+2)/3). - Mircea Merca, Feb 04 2014
a(n) = A102283(n+1) for all n in Z. - Michael Somos, Sep 24 2019
E.g.f.: exp(-x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Oct 26 2022

Edited by Charles R Greathouse IV, Mar 23 2010

A000587 Rao Uppuluri-Carpenter numbers (or complementary Bell numbers): e.g.f. = exp(1 - exp(x)).

Original entry on oeis.org

1, -1, 0, 1, 1, -2, -9, -9, 50, 267, 413, -2180, -17731, -50533, 110176, 1966797, 9938669, 8638718, -278475061, -2540956509, -9816860358, 27172288399, 725503033401, 5592543175252, 15823587507881, -168392610536153, -2848115497132448, -20819319685262839

Offset: 0

N. J. A. Sloane

Alternating row sums of Stirling2 triangle A048993.
Related to the matrix-exponential of the Pascal-matrix, see A000110 and A011971. - Gottfried Helms, Apr 08 2007
Closely linked to A000110 and especially the contribution there of Jonathan R. Love (japanada11(AT)yahoo.ca), Feb 22 2007, by offering what is a complementary finding.
Number of set partitions of 1..n with an even number of parts, minus the number of such partitions with an odd number of parts. - Franklin T. Adams-Watters, May 04 2010
After -2, the smallest prime is a(36) = -1454252568471818731501051, no others through a(100).  What is the first prime >0 in the sequence? - Jonathan Vos Post, Feb 02 2011
a(723) ~ 1.9*10^1265 is almost certainly prime. - D. S. McNeil, Feb 02 2011
Stirling transform of a(n) = [1, -1, 0, 1, 1, ...] is A033999(n) = [1, -1, 1, -1, 1, ...]. - Michael Somos, Mar 28 2012
Negated coefficients in the asymptotic expansion: A005165(n)/n! ~ 1 - 1/n + 1/n^2 + 0/n^3 - 1/n^4 - 1/n^5 + 2/n^6 + 9/n^7 + 9/n^8 - 50/n^9 - 267/n^10 - 413/n^11 + O(1/n^12), starting from the O(1/n) term. - Vladimir Reshetnikov, Nov 09 2016
Named after Venkata Ramamohana Rao Uppuluri and John A. Carpenter of the Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee. They are called "Rényi numbers" by Fekete (1999), after the Hungarian mathematician Alfréd Rényi (1921-1970). - Amiram Eldar, Mar 11 2022

			G.f. = 1 - x + x^3 + x^4 - 2*x^5 - 9*x^6 - 9*x^7 + 50*x^8 + 267*x^9 + 413*x^10 - ...

N. A. Kolokolnikova, Relations between sums of certain special numbers (Russian), in Asymptotic and enumeration problems of combinatorial analysis, pp. 117-124, Krasnojarsk. Gos. Univ., Krasnoyarsk, 1976.
Alfréd Rényi, Új modszerek es eredmenyek a kombinatorikus analfzisben. I. MTA III Oszt. Ivozl., Vol. 16 (1966), pp. 7-105.
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).
M. V. Subbarao and A. Verma, Some remarks on a product expansion. An unexplored partition function, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (Gainesville, FL, 1999), pp. 267-283, Kluwer, Dordrecht, 2001.

Alois P. Heinz, Table of n, a(n) for n = 0..595 (first 101 terms from T. D. Noe)
M. Aguiar and A. Lauve, The characteristic polynomial of the Adams operators on graded connected Hopf algebras, 2014. See Example 31. - _N. J. A. Sloane_, May 24 2014
W. Asakly, A. Blecher, C. Brennan, A. Knopfmacher, T. Mansour, and S. Wagner, Set partition asymptotics and a conjecture of Gould and Quaintance, Journal of Mathematical Analysis and Applications, Volume 416, Issue 2 (15 August 2014), Pages 672-682.
Tewodros Amdeberhan, Valerio de Angelis and Victor H. Moll, Complementary Bell numbers: arithmetical properties and Wilf's conjecture.
S. Barbero, U. Cerruti, and N. Murru, A Generalization of the Binomial Interpolated Operator and its Action on Linear Recurrent Sequences, J. Int. Seq., Vol. 13 (2010), Article 10.9.7.
R. E. Beard, On the Coefficients in the Expansion of e^(e^t) and e^(-e^t), J. Institute of Actuaries, Vol. 76 (1950), pp. 152-163. [Annotated scanned copy]
Pascal Caron, Jean-Gabriel Luque, Ludovic Mignot, and Bruno Patrou, State complexity of catenation combined with a boolean operation: a unified approach, arXiv:1505.03474 [cs.FL], 2015.
Valerio De Angelis and Dominic Marcello, Wilf's Conjecture, The American Mathematical Monthly, Vol. 123, No. 6 (2016), pp. 557-573.
S. de Wannemacker, T. Laffey and R. Osburn, On a conjecture of Wilf, arXiv:math/0608085 [math.NT], 2006-2007.
Branko Dragovich, On Summation of p-Adic Series, arXiv:1702.02569 [math.NT], 2017.
Branko Dragovich, Andrei Yu. Khrennikov, and Natasa Z. Misic, Summation of p-Adic Functional Series in Integer Points, arXiv:1508.05079, 2015
B. Dragovich and N. Z. Misic, p-Adic invariant summation of some p-adic functional series, P-Adic Numbers, Ultrametric Analysis, and Applications, Volume 6, Issue 4 (October 2014), pp. 275-283.
Antal E. Fekete, Apropos Bell and Stirling Numbers, Crux Mathematicorum, Vol. 25, No. 5 (1999), pp. 274-281.
B. Harris and L. Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Ill. J. Math., Vol. 12 (1968), pp. 264-277.
M. Klazar, Counting even and odd partitions, Amer. Math. Monthly, Vol. 110, No. 6 (2003), pp. 527-532.
M. Klazar, Bell numbers, their relatives and algebraic differential equations, J. Combin. Theory, A 102 (2003), 63-87.
A. Knopfmacher and M. E. Mays, Graph compositions I: Basic enumerations, Integers, Vol. 1 (2001), Article A4. (See the first two columns of the table on p. 9.)
Vaclav Kotesovec, Plot of |a(n)/n!|^(1/n) / |exp(1/W(-n))/W(-n)| for n = 1..40000, where W is the LambertW function.
Peter J. Larcombe, Jack Sutton, and James Stanton, A note on the constant 1/e, Palest. J. Math. (2023) Vol. 12, No. 2, 609-619. See p. 617.
J. W. Layman and C. L. Prather, Generalized Bell numbers and zeros of successive derivatives of an entire function, Journal of Mathematical Analysis and Applications, Volume 96, Issue 1 (15 October 1983), Pages 42-51.
Toufik Mansour and Mark Shattuck, Counting subword patterns in permutations arising as flattened partitions of sets, Appl. Anal. Disc. Math. (2022), OnLine-First (00):9-9.
T. Mansour, M. Shattuck and D. G. L. Wang, Recurrence relations for patterns of type (2, 1) in flattened permutations, arXiv preprint arXiv:1306.3355 [math.CO], 2013.
S. Ramanujan, Notebook entry.
V. R. Rao Uppuluri and J. A. Carpenter, Numbers generated by the function exp(1-e^x), Fib. Quart., Vol. 7, No. 4 (1969), pp. 437-448.
Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, Combinatorial algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
Frank Ruskey, Jennifer Woodcock and Yuji Yamauchi, Counting and computing the Rand and block distances of pairs of set partitions, Journal of Discrete Algorithms, Volume 16 (October 2012), Pages 236-248. [_N. J. A. Sloane_, Oct 03 2012]
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq., Vol. 14 (2011), Article 11.9.7.
D. Subedi, Complementary Bell Numbers and p-adic Series, J. Int. Seq., Vol. 17 (2014), Article 14.3.1.
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Complementary Bell Number.
D. Wuilquin, Letters to N. J. A. Sloane, August 1984.
Yifan Yang, On a multiplicative partition function, Electron. J. Combin., Vol. 8, No. 1 (2001), Research Paper 19.

Cf. A000110, A011971 (base triangle PE), A078937 (PE^2).

Cf. A007318, A153229, A213170.

a000587 n = a000587_list !! n
a000587_list = 1 : f a007318_tabl [1] where
   f (bs:bss) xs = y : f bss (y : xs) where y = - sum (zipWith (*) xs bs)
-- Reinhard Zumkeller, Mar 04 2014

b:= proc(n, t) option remember; `if`(n=0, 1-2*t,
      add(b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 28 2016

Table[ -1 * Sum[ (-1)^( k + 1) StirlingS2[ n, k ], {k, 0, n} ], {n, 0, 40} ]
With[{nn=30},CoefficientList[Series[Exp[1-Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 04 2011 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ 1 - Exp[x]], {x, 0, n}]]; (* Michael Somos, May 27 2014 *)
a[ n_] := If[ n < 0, 0, With[{m = n + 1}, SeriesCoefficient[ Series[ Nest[ x Factor[ 1 - # /. x -> x / (1 - x)] &, 0, m], {x, 0, m}], {x, 0, m}]]]; (* Michael Somos, May 27 2014 *)
Table[BellB[n, -1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)
b[1] = 1; k = 1; Flatten[{1, Table[Do[j = k; k -= b[m]; b[m] = j;, {m, 1, n-1}]; b[n] = k; k*(-1)^n, {n, 1, 40}]}] (* Vaclav Kotesovec, Sep 09 2019 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( 1 - exp( x + x * O(x^n))), n))}; /* Michael Somos, Mar 14 2011 */

{a(n) = local(A); if( n<0, 0, n++; A = O(x); for( k=1, n, A = x - x * subst(A, x, x / (1 - x))); polcoeff( A, n))}; /* Michael Somos, Mar 14 2011 */

Vec(serlaplace(exp(1 - exp(x+O(x^99))))) /* Joerg Arndt, Apr 01 2011 */

a(n)=round(exp(1)*suminf(k=0,(-1)^k*k^n/k!))
vector(20,n,a(n-1)) \\ Derek Orr, Sep 19 2014 -- a direct approach

x='x+O('x^66); Vec(serlaplace(exp(1 - exp(x)))) \\ Michel Marcus, Sep 19 2014

# The objective of this implementation is efficiency.
# n -> [a(0), a(1), ..., a(n)] for n > 0.
def A000587_list(n):
    A = [0 for i in range(n)]
    A[n-1] = 1
    R = [1]
    for j in range(0, n):
        A[n-1-j] = -A[n-1]
        for k in range(n-j, n):
            A[k] += A[k-1]
        R.append(A[n-1])
    return R
# Peter Luschny, Apr 18 2011

# Python 3.2 or higher required
from itertools import accumulate
A000587, blist, b = [1,-1], [1], -1
for _ in range(30):
    blist = list(accumulate([b]+blist))
    b = -blist[-1]
    A000587.append(b) # Chai Wah Wu, Sep 19 2014

expnums(26, -1) # Zerinvary Lajos, May 15 2009

a(n) = e*Sum_{k>=0} (-1)^k*k^n/k!. - Benoit Cloitre, Jan 28 2003
E.g.f.: exp(1 - e^x).
a(n) = Sum_{k=0..n} (-1)^k S2(n, k), where S2(i, j) are the Stirling numbers of second kind A008277.
G.f.: (x/(1-x))*A(x/(1-x)) = 1 - A(x); the binomial transform equals the negative of the sequence shifted one place left. - Paul D. Hanna, Dec 08 2003
With different signs: g.f.: Sum_{k>=0} x^k/Product_{L=1..k} (1 + L*x).
Recurrence: a(n) = -Sum_{i=0..n-1} a(i)*C(n-1, i). - Ralf Stephan, Feb 24 2005
Let P be the lower-triangular Pascal-matrix, PE = exp(P-I) a matrix-exponential in exact integer arithmetic (or PE = lim exp(P)/exp(1) as limit of the exponential); then a(n) = PE^-1 [n,1]. - Gottfried Helms, Apr 08 2007
Take the series 0^n/0! - 1^n/1! + 2^n/2! - 3^n/3! + 4^n/4! + ... If n=0 then the result will be 1/e, where e = 2.718281828... If n=1, the result will be -1/e. If n=2, the result will be 0 (i.e., 0/e). As we continue for higher natural number values of n sequence for the Roa Uppuluri-Carpenter numbers is generated in the numerator, i.e., 1/e, -1/e, 0/e, 1/e, 1/e, -2/e, -9/e, -9/e, 50/e, 267/e, ... . - Peter Collins (pcolins(AT)eircom.net), Jun 04 2007
The sequence (-1)^n*a(n), with general term Sum_{k=0..n} (-1)^(n-k)*S2(n, k), has e.g.f. exp(1-exp(-x)). It also has Hankel transform (-1)^C(n+1,2)*A000178(n) and binomial transform A109747. - Paul Barry, Mar 31 2008
G.f.: 1 / (1 + x / (1 - x / (1 + x / (1 - 2*x / (1 + x / (1 - 3*x / (1 + x / ...))))))). - Michael Somos, May 12 2012
From Sergei N. Gladkovskii, Sep 28 2012 to Feb 07 2014: (Start)
Continued fractions:
G.f.: -1/U(0) where U(k) = x*k - 1 - x + x^2*(k+1)/U(k+1).
G.f.: 1/(U(0)+x) where U(k) = 1 + x - x*(k+1)/(1 + x/U(k+1)).
G.f.: 1+x/G(0) where G(k) = x*k - 1 + x^2*(k+1)/G(k+1).
G.f.: (1 - G(0))/(x+1) where G(k) = 1 - 1/(1-k*x)/(1-x/(x+1/G(k+1) )).
G.f.: 1 + x/(G(0)-x) where G(k) = x*k + 2*x - 1 - x*(x*k+x-1)/G(k+1).
G.f.: G(0)/(1+x), where G(k) = 1-x^2*(k+1)/(x^2*(k+1)+(x*k-1-x)*(x*k-1)/G(k+1)).
(End)
a(n) = B_n(-1), where B_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Oct 20 2015
From Mélika Tebni, May 20 2022: (Start)
a(n) = Sum_{k=0..n} (-1)^k*Bell(k)*A129062(n, k).
a(n) = Sum_{k=0..n} (-1)^k*k!*A130191(n, k). (End)

A316524 Signed sum over the prime indices of n.

Original entry on oeis.org

0, 1, 2, 0, 3, -1, 4, 1, 0, -2, 5, 2, 6, -3, -1, 0, 7, 1, 8, 3, -2, -4, 9, -1, 0, -5, 2, 4, 10, 2, 11, 1, -3, -6, -1, 0, 12, -7, -4, -2, 13, 3, 14, 5, 3, -8, 15, 2, 0, 1, -5, 6, 16, -1, -2, -3, -6, -9, 17, -1, 18, -10, 4, 0, -3, 4, 19, 7, -7, 2, 20, 1, 21, -11, 2, 8, -1, 5, 22, 3, 0, -12, 23, -2, -4, -13, -8, -4, 24

Offset: 1

Gus Wiseman, Jul 05 2018

sign

If n = prime(x_1) * prime(x_2) * prime(x_3) * ... * prime(x_k) then a(n) = x_1 - x_2 + x_3 - ... + (-1)^(k-1) x_k, where the x_i are weakly increasing positive integers.

The value of a(n) depends only on the squarefree part of n, A007913(n). - Antti Karttunen, May 06 2022

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000607, A007913, A071321, A100118, A175352, A316313, A316523.
Cf. A027746, A112798, A119899 (positions of negative terms).
Cf. A344616 (absolute values), A344617 (signs).

Table[Sum[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]][[k]]*(-1)^(k-1),{k,PrimeOmega[n]}],{n,100}]

a(n) = {my(f = factor(n), vp = []); for (k=1, #f~, for( j=1, f[k,2], vp = concat (vp, primepi(f[k,1])));); sum(k=1, #vp, vp[k]*(-1)^(k+1));} \\ Michel Marcus, Jul 06 2018

from sympy import factorint, primepi
def A316524(n):
    fs = [primepi(p) for p in factorint(n,multiple=True)]
    return sum(fs[::2])-sum(fs[1::2]) # Chai Wah Wu, Aug 23 2021

a(n) = A344616(n) * A344617(n) = a(A007913(n)). - Antti Karttunen, May 06 2022

More terms from Antti Karttunen, May 06 2022

A164555 Numerators of the "original" Bernoulli numbers; also the numerators of the Bernoulli polynomials at x=1.

Original entry on oeis.org

1, 1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051

Offset: 0

Paul Curtz, Aug 15 2009

Apart from a sign flip in a(1), the same as A027641.
a(n) is also the numerator of the n-th term of the binomial transform of the sequence of Bernoulli numbers, i.e., of the sequence of fractions A027641(n)/A027642(n).
a(n)/A027642(n) with e.g.f. x/(1-exp(-x)) is the a-sequence for the Sheffer matrix A094645, see the W. Lang link under A006232 for Sheffer a- and z-sequences. - Wolfdieter Lang, Jun 20 2011
a(n)/A027642(n) also give the row sums of the rational triangle of the coefficients of the Bernoulli polynomials A053382/A053383 (falling powers) or A196838/A196839 (rising powers). - Wolfdieter Lang, Oct 25 2011
Given M = the beheaded Pascal's triangle, A074909; with B_n as a vector V, with numerators shown: (1, 1, 1, ...). Then M*V = [1, 2, 3, 4, 5, ...]. If the sign in a(1) is negative in V, then M*V = [1, 0, 0, 0, ...]. - Gary W. Adamson, Mar 09 2012
One might interpret the term "'original' Bernoulli numbers" as numbers given by the e.g.f. x/(1-exp(-x)). - Peter Luschny, Jun 17 2012
Let B(n) = a(n)/A027642(n) then B(n) = Integral_{x=0..1} F_n(x) where F_n(x) are the signed Fubini polynomials F_n(x) = Sum_{k=0..n} (-1)^n*Stirling2(n,k)*k!*(-x)^k (see illustration). - Peter Luschny, Jan 09 2017

			From _Peter Luschny_, Aug 13 2017: (Start)
Integral_{x=0..1} 1 = 1,
Integral_{x=0..1} x = 1/2,
Integral_{x=0..1} 2*x^2 - x = 1/6,
Integral_{x=0..1} 6*x^3 - 6*x^2 + x = 0,
Integral_{x=0..1} 24*x^4 - 36*x^3 + 14*x^2 - x = -1/30,
Integral_{x=0..1} 120*x^5 - 240*x^4 + 150*x^3 - 30*x^2 + x = 0,
...
Integral_{x=0..1} Sum_{k=0..n} (-1)^n*Stirling2(n,k)*k!*(-x)^k = Bernoulli(n). (End)

Jacob Bernoulli, Ars Conjectandi, Basel: Thurneysen Brothers, 1713. See page 97.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 106-108.

Seiichi Manyama, Table of n, a(n) for n = 0..629
Peter Luschny, Illustration of the first terms.
Peter Luschny, The Bernoulli Manifesto, 2013.
Tom Rike, Sums of powers and Bernoulli numbers.

Cf. A027641, A027642, A006232, A053382, A053383, A074909, A094645, A196838, A196839.

Cf. A242179, A106831, A000142.

a164555 n = a164555_list !! n
a164555_list = 1 : map (numerator . sum) (zipWith (zipWith (%))
   (zipWith (map . (*)) (tail a000142_list) a242179_tabf) a106831_tabf)
-- Reinhard Zumkeller, Jul 04 2014

A164555 := proc(n) if n <= 2 then 1; else numer(bernoulli(n)) ; fi; end: # R. J. Mathar, Aug 26 2009
seq(numer(n!*coeff(series(t/(1-exp(-t)),t,n+2),t,n)),n=0..40); # Peter Luschny, Jun 17 2012

CoefficientList[ Series[ x/(1 - Exp[-x]), {x, 0, 40}], x]*Range[0, 40]! // Numerator (* Jean-François Alcover, Mar 04 2013 *)
Table[Numerator[BernoulliB[n,1]], {n, 0, 40}] (* Vaclav Kotesovec, Jan 03 2021 *)

a = lambda n: bernoulli_polynomial(1,n).numerator()
[a(n) for n in (0..40)] # Peter Luschny, Jan 09 2017

a(n) = numerator(B(n)) with B(n) = Sum_{k=0..n} (-1)^(n-k) * C(n+1, k+1) * S(n+k, k) / C(n+k, k) and S the Stirling set numbers. - Peter Luschny, Jun 25 2016
a(n) = numerator(n*EulerPolynomial(n-1, 1)/(2*(2^n-1))) for n>=1. - Peter Luschny, Sep 01 2017
From Artur Jasinski, Jan 01 2021: (Start)
a(n) = numerator(-2*cos(Pi*n/2)*Gamma(n+1)*zeta(n)/(2*Pi)^n) for n != 1.
a(n) = numerator(-n*zeta(1-n)) for n >= 1. In the case n = 0 take the limit. (End)

Edited and extended by R. J. Mathar, Sep 03 2009

Name extended by Peter Luschny, Jan 09 2017

A033880 Abundance of n, or (sum of divisors of n) - 2n.

Original entry on oeis.org

-1, -1, -2, -1, -4, 0, -6, -1, -5, -2, -10, 4, -12, -4, -6, -1, -16, 3, -18, 2, -10, -8, -22, 12, -19, -10, -14, 0, -28, 12, -30, -1, -18, -14, -22, 19, -36, -16, -22, 10, -40, 12, -42, -4, -12, -20, -46, 28, -41, -7, -30, -6, -52, 12, -38, 8, -34, -26, -58, 48, -60, -28, -22

Offset: 1

N. J. A. Sloane

For no known n is a(n) = 1. If there is such an n it must be greater than 10^35 and have seven or more distinct prime factors (Hagis and Cohen 1982). - Jonathan Vos Post, May 01 2011
a(n) = -1 iff n is a power of 2. a(n) = 1 - n iff n is prime. - Omar E. Pol, Jan 30 2014 [If a(n) = -1 then n is called a least deficient number or an almost perfect number. All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number that is not a power of 2. See A000079. - Jianing Song, Oct 13 2019]
According to Deléglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480 (cf. A302991). Since the perfect numbers having density 0, the deficient numbers have density 0.7520 < 1 - A(2) < 0.7526 (cf. A318172). - Daniel Forgues, Oct 10 2015
2-abundance of n, a special case of the k-abundance of n, defined as (sum of divisors of n) - k*n, k >= 1. - Daniel Forgues, Oct 24 2015
Not to be confused with the abundancy of n, defined as (sum of divisors of n) / n. (Cf. A017665 / A017666.) - Daniel Forgues, Oct 25 2015

			For n = 10 the divisors of 10 are 1, 2, 5, 10. The sum of proper divisors of 10 minus 10 is 1 + 2 + 5 - 10 = -2, so the abundance of 10 is a(10) = -2. - _Omar E. Pol_, Dec 27 2013

Richard K. Guy, "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." Section B2 in Unsolved Problems in Number Theory, 2nd ed., New York: Springer-Verlag, pp. 45-53, 1994.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

J. G. Wurtzel, Table of n, a(n) for n = 1..10000 [This replaces an earlier b-file computed by T. D. Noe]
Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.
Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), 137-143.
Peter Hagis Jr. and Graeme L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
Eric Weisstein's World of Mathematics, Abundance.
Eric Weisstein's World of Mathematics, Abundancy.
Eric Weisstein's World of Mathematics, Quasiperfect Number.

Equals -A033879.
Cf. A000203, A001065. A005843.
Lists of positions where certain values occur: A005100 (a(n) < 0), A000396 (a(n) = 0) and A005101 (a(n) > 0), A023197 (a(n) >= n), A028982 (a(n) odd).
Cf. A072691, A302991, A318172.

[SumOfDivisors(n)-2*n: n in [1..100]]; // Vincenzo Librandi, Oct 11 2015

Maple
```
with(numtheory); n->sigma(n) - 2*n;
```

Array[Total[Divisors[#]]-2#&,70] (* Harvey P. Dale, Sep 16 2011 *)
Table[DivisorSigma[1, n] - 2*n, {n, 1, 70}] (* Amiram Eldar, Jun 09 2022 *)

a(n)=sigma(n)-2*n \\ Charles R Greathouse IV, Nov 20 2012

from sympy import divisor_sigma
def A033880(n): return divisor_sigma(n)-(n<<1) # Chai Wah Wu, Apr 12 2024

[sigma(n, 1)-2*n for n in range(1, 64)] # Stefano Spezia, Jul 18 2025

a(n) = A000203(n) - A005843(n). - Omar E. Pol, Dec 14 2008
a(n) = A001065(n) - n. - Omar E. Pol, Dec 27 2013
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/12 - 1 = -0.1775329665... . - Amiram Eldar, Apr 06 2024

Definition corrected Jul 04 2005

A036263 Second differences of primes.

Original entry on oeis.org

1, 0, 2, -2, 2, -2, 2, 2, -4, 4, -2, -2, 2, 2, 0, -4, 4, -2, -2, 4, -2, 2, 2, -4, -2, 2, -2, 2, 10, -10, 2, -4, 8, -8, 4, 0, -2, 2, 0, -4, 8, -8, 2, -2, 10, 0, -8, -2, 2, 2, -4, 8, -4, 0, 0, -4, 4, -2, -2, 8, 4, -10, -2, 2, 10, -8, 4, -8, 2, 2, 2, -2, 0, -2, 2, 2, -4, 4, 2, -8, 8, -8, 4, -2, 2, 2, -4, -2, 2, 8, -4

Offset: 1

N. J. A. Sloane

			a(3) = 5 + 11 - 2*7 = 16 - 14 = 2.

Edward Bernstein, Table of n, a(n) for n = 1..100000 (Terms 1 to 10000 from T. D. Noe)
R. G. Batchko, A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes, arXiv preprint arXiv:1405.2900 [math.GM], 2014. See Table 2.

Cf. A001223, A036262, A051635, A006562, A051634, A147812, A147813, A182394, A079054, A064113 (places of zeros).

For records see A293154, A293155.

a036263 n = a036263_list !! (n-1)
a036263_list = zipWith (-) (tail a001223_list) a001223_list
-- Reinhard Zumkeller, Oct 29 2011

A036263:=n->ithprime(n) + ithprime(n+2) - 2*ithprime(n+1); seq(A036263(n), n=1..100); # Wesley Ivan Hurt, Apr 01 2014

Table[Prime[n - 1] + Prime[n + 1] - 2*Prime[n], {n, 2, 105}]
Differences[Prime[Range[100]], 2] (* Harvey P. Dale, Oct 14 2012 *)

for(n=2,100,print1(prime(n+2)-2*prime(n+1)+prime(n)","))

from sympy import prime
def A036263(n): return prime(n)-(prime(n+1)<<1)+prime(n+2) # Chai Wah Wu, Sep 28 2024

a(A064113(n)) = 0. - Reinhard Zumkeller, Jan 20 2012
a(n) = prime(n) + prime(n+2) - 2*prime(n+1). - Thomas Ordowski, Jul 21 2012
Conjecture: |a(1)| + |a(2)| + ... + |a(n)| ~ prime(n). - Thomas Ordowski, Jul 21 2012
a(n) = A001223(n+1) - A001223(n). - R. J. Mathar, Sep 19 2013
Sum_{i = 1..n - 1} a(i) = A046933(n), n >= 1. - Daniel Forgues, Apr 15 2014
Sum_{i = 2..n - 1} a(i) = prime(n + 1) - prime(n) - 2; Sum_{i = 2..n - 1} a(i) = 0 whenever prime(n) is a lesser of twin primes. - Hamdi Murat Yildirim, Jun 24 2014

A056594 Period 4: repeat [1,0,-1,0]; expansion of 1/(1 + x^2).

Original entry on oeis.org

1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0

Offset: 0

Wolfdieter Lang, Aug 04 2000

G.f. is inverse of cyclotomic(4,x). Unsigned: A000035(n+1).
Real part of i^n and imaginary part of i^(n+1), i=sqrt(-1). - Reinhard Zumkeller, Jul 22 2007
The BINOMIAL transform generates A009116(n); the inverse BINOMIAL transform generates (-1)^n*A009116(n). - R. J. Mathar, Apr 07 2008
a(n-1), n >= 1, is the nontrivial Dirichlet character modulo 4, called Chi_2(4;n) (the trivial one is Chi_1(4;n) given by periodic(1,0) = A000035(n)). See the Apostol reference, p. 139, the k = 4, phi(k) = 2 table. - Wolfdieter Lang, Jun 21 2011
a(n-1), n >= 1, is the character of the Dirichlet beta function. - Daniel Forgues, Sep 15 2012
a(n-1), n >= 1, is also the (strongly) multiplicative function h(n) of Theorem 5.12, p. 150, of the Niven-Zuckerman reference. See the formula section. This function h(n) can be employed to count the integer solutions to n = x^2 + y^2. See A002654 for a comment with the formula. - Wolfdieter Lang, Apr 19 2013
This sequence is duplicated in A101455 but with offset 1. - Gary Detlefs, Oct 04 2013
For n >= 2 this gives the determinant of the bipartite graph with 2*n nodes and the adjacency matrix A(n) with elements A(n;1,2) = 1 = A(n;n,n-1), and for 1 < i < n  A(n;i,i+1) = 1 = A(n;i,i-1), otherwise 0. - Wolfdieter Lang, Jun 25 2023

			With a(n-1) = h(n) of Niven-Zuckerman: a(62) = h(63) = h(3^2*7^1) = (-1)^(2*1)*(-1)^(1*3) = -1 = h(3)^2*h(7) = a(2)^2*a(6) = (-1)^2*(-1) = -1. - _Wolfdieter Lang_, Apr 19 2013

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.
I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley (1980), p. 150.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:1 at page 300.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry and Nikolaos Pantelidis, On pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays, J Algebr Comb 54, 399-423 (2021).
PeakMath, L-functions and the Langlands program (RH Saga S1E2), YouTube video (2024).
Eric Weisstein's World of Mathematics, Kronecker Symbol.
Wikipedia, Dirichlet beta function.
Wikipedia, Kronecker Symbol.
Index entries for linear recurrences with constant coefficients, signature (0,-1).
Index entries for sequences related to Chebyshev polynomials.
Index to sequences related to inverse of cyclotomic polynomials.

Cf. A049310, A074661, A131852, A002654, A146559 (binomial transform).

&cat[ [1, 0, -1, 0]: n in [0..23] ]; // Bruno Berselli, Feb 08 2011

A056594 := n->(1-irem(n,2))*(-1)^iquo(n,2); # Peter Luschny, Jul 27 2011

CoefficientList[Series[1/(1 + x^2), {x, 0, 50}], x]
a[n_]:= KroneckerSymbol[-4,n+1];Table[a[n],{n,0,93}] (* Thanks to Jean-François Alcover. - Wolfdieter Lang, May 31 2013 *)
CoefficientList[Series[1/Cyclotomic[4, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)

A056594(n) := block(
        [1,0,-1,0][1+mod(n,4)]
)$ /* R. J. Mathar, Mar 19 2012 */

PARI
```
{a(n) = real( I^n )}
```
PARI
```
{a(n) = kronecker(-4, n+1) }
```

def A056594(n): return (1,0,-1,0)[n&3] # Chai Wah Wu, Sep 23 2023

G.f.: 1/(1+x^2).
E.g.f.: cos(x).
a(n) = (1/2)*((-i)^n + i^n), where i = sqrt(-1). - Mitch Harris, Apr 19 2005
a(n) = (1/2)*((-1)^(n+floor(n/2)) + (-1)^floor(n/2)).
Recurrence: a(n)=a(n-4), a(0)=1, a(1)=0, a(2)=-1, a(3)=0.
a(n) = T(n, 0) = A053120(n, 0); T(n, x) Chebyshev polynomials of the first kind. - Wolfdieter Lang, Aug 21 2009
a(n) = S(n, 0) = A049310(n, 0); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind.
Sum_{k>=0} a(k)/(k+1) = Pi/4. - Jaume Oliver Lafont, Mar 30 2010
a(n) = Sum_{k=0..n} A101950(n,k)*(-1)^k. - Philippe Deléham, Feb 10 2012
a(n) = (1/2)*(1 + (-1)^n)*(-1)^(n/2). - Bruno Berselli, Mar 13 2012
a(0) = 1, a(n-1) = 0 if n is even, a(n-1) = Product_{j=1..m} (-1)^(e_j*(p_j-1)/2) if the odd n-1 = p_1^(e_1)*p_2^(e_2)*...*p_m^(e_m) with distinct odd primes p_j, j=1..m. See the function h(n) of Theorem 5.12 of the Niven-Zuckerman reference. - Wolfdieter Lang, Apr 19 2013
a(n) = (-4/(n+1)), n >= 0, where (k/n) is the Kronecker symbol. See the Eric Weisstein and Wikipedia links. Thanks to Wesley Ivan Hurt. - Wolfdieter Lang, May 31 2013
a(n) = R(n,0)/2 with the row polynomials R of A127672. This follows from the product of the zeros of R, and the formula Product_{k=0..n-1} 2*cos((2*k+1)*Pi/(2*n)) = (1 + (-1)^n)*(-1)^(n/2), n >= 1 (see the Gradstein and Ryshik reference, p. 63, 1.396 4., with x = sqrt(-1)). - Wolfdieter Lang, Oct 21 2013
a(n) = Sum_{k=0..n} i^(k*(k+1)), where i=sqrt(-1). - Bruno Berselli, Mar 11 2015
Dirichlet g.f. of a(n) shifted right: L(chi_2(4),s) = beta(s) = (1-2^(-s))*(d.g.f. of A034947), see comments by Lang and Forgues. - Ralf Stephan, Mar 27 2015
a(n) = cos(n*Pi/2). - Ridouane Oudra, Sep 29 2024

A006352 Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).

Original entry on oeis.org

1, -24, -72, -96, -168, -144, -288, -192, -360, -312, -432, -288, -672, -336, -576, -576, -744, -432, -936, -480, -1008, -768, -864, -576, -1440, -744, -1008, -960, -1344, -720, -1728, -768, -1512, -1152, -1296, -1152, -2184, -912, -1440, -1344, -2160, -1008, -2304, -1056, -2016, -1872, -1728

Offset: 0

N. J. A. Sloane

Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

The series Q(q), R(q) are modular forms, but P(q) is not. - Michael Somos, May 18 2017

			G.f. = 1 - 24*x - 72*x^2 - 96*x^3 - 168*x^4 - 144*x^5 - 288*x^6 + ...

R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see pp. 111 and 113.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 19, Eq. (17).

T. D. Noe, Table of n, a(n) for n = 0..1000
F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201-210.
J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms.
Heng Huat Chan, Shaun Cooper, and Pee Choon Toh, Ramanujan's Eisenstein series and powers of Dedekind's eta-function, Journal of the London Mathematical Society 75.1 (2007): 225-242. See P(q).
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998.
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.
V. P. Varin, Special solutions to Chazy equation, Comput. Math. Math. Phys. 57, No. 2, 211-235 (2017), eq (75).
Eric Weisstein's World of Mathematics, Eisenstein Series.
Index entries for sequences related to Eisenstein series.

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).

Cf. A000594 (Delta), A076835, A145155 (Delta').

E := proc(k) local n,t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n,n=1..60); series(t1,q,60); end; E(2);

a[n_] := -24*DivisorSigma[1, n]; a[0] = 1; Table[a[n], {n, 0, 46}] (* Jean-François Alcover, Dec 12 2012 *)
a[ n_] := If[ n < 1, Boole[n == 0], -24 DivisorSigma[ 1, n]]; (* Michael Somos, Apr 08 2015 *)

{a(n) = if( n<1, n==0, -24 * sigma(n))}; /* Michael Somos, Apr 09 2003 */

from sympy import divisor_sigma
def a(n): return 1 if n == 0 else -24 * divisor_sigma(n)
[a(n) for n in range(51)]  # Indranil Ghosh, Jul 15 2017

a(n) = -24*sigma(n) = -24*A000203(n), for n>0.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 + 4*u2^2 + 9*u3^2 + 36*u6^2 - 8*u1*u2 + 6*u1*u3 + 24*u2*u6 - 72*u3*u6. - Michael Somos, May 29 2005
G.f.: 1 - 24*sum(k>=1, k*x^k/(1 - x^k)).
G.f.: 1 + 24 *x*deriv(eta(x))/eta(x) where eta(x) = prod(n>=1, 1-x^n); (cf. A000203). - Joerg Arndt, Sep 28 2012
G.f.: 1 - 24*x/(1-x) + 48*x^2/(Q(0) -  2*x^2 + 2*x), where Q(k)= (2*x^(k+2) - x - 1)*k - 1 - 2*x + 3*x^(k+2) - x*(k+1)*(k+3)*(1-x^(k+2))^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 16 2013
G.f.: q*Delta'/Delta where Delta is the generating function of Ramanujan's tau function (A000594). - Seiichi Manyama, Jul 15 2017

A006232 Numerators of Cauchy numbers of first type.

Original entry on oeis.org

1, 1, -1, 1, -19, 9, -863, 1375, -33953, 57281, -3250433, 1891755, -13695779093, 24466579093, -132282840127, 240208245823, -111956703448001, 4573423873125, -30342376302478019, 56310194579604163

Offset: 0

N. J. A. Sloane

The corresponding denominators are given in A006233.
-a(n+1), n>=0, also numerators from e.g.f. 1/x-1/log(1+x), with denominators A075178(n). |a(n+1)|, n>=0, numerators from e.g.f. 1/x+1/log(1-x) with denominators A075178(n). For formula of unsigned a(n) see A075178.
The signed rationals a(n)/A006233(n) provide the a-sequence for the Stirling2 Sheffer matrix A048993. See the W. Lang link concerning Sheffer a- and z-sequences.
Cauchy numbers of the first type are also called Bernoulli numbers of the second kind.
Named after the French mathematician, engineer and physicist Augustin-Louis Cauchy (1789-1857). - Amiram Eldar, Jun 17 2021

			1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90, ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
Harold Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, Cambridge, 1946, p. 259.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
Arnold Adelberg, 2-adic congruences of Norland numbers and of Bernoulli numbers of the second kind, J. Number Theory, Vol. 73, No. 1 (1998), pp. 47-58.
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products, (1980), page 2 (formula 0.131).
L. B. W. Jolley, Summation of Series, Dover, (1961) (formula 70).
Wolfdieter Lang, Sheffer a- and z-sequences.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Hong-Mei Liu, Shu-Hua Qi and Shu-Yan Ding, Some Recurrence Relations for Cauchy Numbers of the First Kind, JIS, Vol. 13 (2010), Article 10.3.8.
Rui-Li Liu and Feng-Zhen Zhao, Log-concavity of two sequences related to Cauchy numbers of two kinds, Online Journal of Analytic Combinatorics, Issue 14 (2019), #09.
Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math., Vol. 306, No. 16 (2006), pp. 1906-1920.
Eric Weisstein's World of Mathematics, Bernoulli Numbers of the Second Kind.
Ming Wu and Hao Pan, Sums of products of Bernoulli numbers of the second kind, Fib. Quart., Vol. 45, No. 2 (2007), pp. 146-150.
Feng-Zhen Zhao, Sums of products of Cauchy numbers, Discrete Math., Vol. 309, No. 12 (2009), pp. 3830-3842.

Cf. A006233 (denominators), A002206, A002207, A002208, A002209, A002657, A002790.

[Numerator((&+[StirlingFirst(n,k)/(k+1): k in [0..n]])): n in [0..20]]; // G. C. Greubel, Nov 13 2018

seq(numer(add(stirling1(n,k)/(k+1),k=0..n)),n=0..20); # Peter Luschny, Apr 28 2009

a[n_] := Numerator[ Sum[ StirlingS1[n, k]/(k + 1), {k, 0, n}]]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 03 2011, after Maple *)
a[n_] := Numerator[ Integrate[ Gamma[x+1]/Gamma[x-n+1], {x, 0, 1}]]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jul 29 2013 *)
a[ n_] := If[ n < 0, 0, (-1)^n Numerator @ Integrate[ Pochhammer[ -x, n], {x, 0, 1}]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, Numerator [ n! SeriesCoefficient[ x / Log[ 1 + x], {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *)
Join[{1}, Array[Numerator[(1/#) Integrate[Product[(x - k), {k, 0, # - 1}], {x, 0, 1}]] &, 25]] (* Michael De Vlieger, Nov 13 2018 *)

for(n=0,20, print1(numerator( sum(k=0,n, stirling(n, k, 1)/(k+1)) ), ", ")) \\ G. C. Greubel, Nov 13 2018

# Results are abs values
from fractions import gcd
aa,n,sden = [0,1],1,1
while n < 20:
    j,snom,sden,a = 1,0,(n+1)*sden,0
    while j < len(aa):
        snom,j = snom+aa[j]*(sden//(j+1)),j+1
    nom,den = snom,sden
    print(n,nom//gcd(nom,den))
    aa,j = aa+[-aa[j-1]],j-1
    while j > 0:
        aa[j],j = n*aa[j]-aa[j-1],j-1
    n = n+1 # A.H.M. Smeets, Nov 14 2018

from fractions import Fraction
from sympy.functions.combinatorial.numbers import stirling
def A006232(n): return sum(Fraction(stirling(n,k,kind=1,signed=True),k+1) for k in range(n+1)).numerator # Chai Wah Wu, Jul 09 2023

def A006232_list(len):
    f, R, C = 1, [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n, 0, -1):
            C[k] = -C[k-1] * k / (k + 1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append((C[0]*f).numerator())
        f *= n
    return R
print(A006232_list(20)) # Peter Luschny, Feb 19 2016

Numerator of integral of x(x-1)...(x-n+1) from 0 to 1.
E.g.f.: x/log(1+x). (Note: the numerator of the coefficient of x^n/n! is a(n) - Michael Somos, Jul 12 2014)
Numerator of Sum_{k=0..n} A048994(n,k)/(k+1). - Peter Luschny, Apr 28 2009
Sum_{k=1..n} 1/k = C + log(n) + 1/(2n) + Sum_{k=2..inf} |a(n)|/A075178(n-1) * 1/(n*(n+1)*...*(n+k-1)) (section 0.131 in Gradshteyn and Ryzhik tables). - Ralf Stephan, Jul 12 2014
a(n) = numerator(f(n) * n!), where f(0) = 1, f(n) = Sum_{k=0..n-1} (-1)^(n-k+1) * f(k) / (n-k+1). - Daniel Suteu, Feb 23 2018
Sum_{k = 1..n} (1/k) = A001620 + log(n) + 1/(2n) - Sum_{k >= 2} abs((a(k)/A006233(k)/k/(Product_{j = 0..k-1} (n-j)))), (see I. S. Gradsteyn, I. M. Ryzhik). - A.H.M. Smeets, Nov 14 2018

A001057 Canonical enumeration of integers: interleaved positive and negative integers with zero prepended.

Original entry on oeis.org

0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 9, -9, 10, -10, 11, -11, 12, -12, 13, -13, 14, -14, 15, -15, 16, -16, 17, -17, 18, -18, 19, -19, 20, -20, 21, -21, 22, -22, 23, -23, 24, -24, 25, -25, 26, -26, 27, -27, 28, -28, 29, -29, 30, -30, 31, -31

Offset: 0

N. J. A. Sloane

Go forwards and backwards with increasing step sizes. - Daniele Parisse and Franco Virga, Jun 06 2005
The partial sums of the divergent series 1 - 2 + 3 - 4 + ... give this sequence. Euler summed it to 1/4 which was one of the first examples of summing divergent series. - Michael Somos, May 22 2007
From Peter Luschny, Jul 12 2009: (Start)
The general formula for alternating sums of powers is in terms of the Swiss-Knife polynomials P(n,x) A153641 2^(-n-1)(P(n,1)-(-1)^k P(n,2k+1)). Thus
a(k) = 2^(-2)(P(1,1)-(-1)^k P(1,2k+1)). (End)
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=4, a(n-3)=(-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 26 2010
Cantor ordering of the integers producing a 1-1 and onto correspondence between the natural numbers and the integers showing that the set Z of integers has the same cardinality as the set N of natural numbers. The cardinal of N is the first transfinite cardinal aleph_null (or aleph_naught), which is the cardinality of a given infinite set if and only if it is countably infinite (denumerable), i.e., it can be put in 1-1 and onto correspondence (with a proper Cantor ordering) with the natural numbers. - Daniel Forgues, Jan 23 2010
a(n) is the determinant of the (n+2) X (n+2) (0,1)-Toeplitz matrix M satisfying: M(i,j)=0 iff i=j or i=j-1. The matrix M arises in the variation of ménage problem where not a round table, but one side of a rectangular table is  considered (see comments of Vladimir Shevelev in A000271). Namely M(i,j) defines the class of permutations p of 1,2,...,n+2 such that p(i)<>i and p(i)<>i+1 for i=1,2,...,n+1, and p(n+2)<>n+2. And a(n) is also the difference between the number of even and odd such permutations. - Dmitry Efimov, Mar 02 2017

			G.f. = x - x^2 + 2*x^3 - 2*x^4 + 3*x^5 - 3*x^6 + 4*x^7 - 4*x^8 + 5*x^9 - 5*x^10 + ...

T. D. Noe, Table of n, a(n) for n = 0..1000
D. Efimov, Determinants of generalized binary band matrices, arXiv:1702.05655 [math.RA], 2017.
G. Myerson and A. J. van der Poorten, Some problems concerning recurrence sequences, Amer. Math. Monthly 102 (1995), no. 8, 698-705.
Omar E. Pol, Illustration of initial terms of A001057, A005132, A000217
Wikipedia, 1 - 2 + 3 - 4 + ...
Index entries for linear recurrences with constant coefficients, signature (-1,1,1).

Cf. A008619, A004526, A166711, A166871, A130472 (negation), A142150 (partial sums), A010551 (partial products for n > 0).

Alternating row sums of A104578 are a(n+1), for n >= 0.

a001057 n = (n' + m) * (-1) ^ (1 - m) where (n',m) = divMod n 2
a001057_list = 0 : concatMap (\x -> [x,-x]) [1..]
-- Reinhard Zumkeller, Apr 02 2012

a := n -> (1-(-1)^n*(2*n+1))/4; # Peter Luschny, Jul 12 2009

Join[{0},Riffle[Range[35],-Range[35]]] (* Harvey P. Dale, Sep 21 2011 *)
a[ n_] := -(-1)^n Ceiling[n/2]; (* Michael Somos, Jun 05 2013 *)
LinearRecurrence[{-1, 1, 1}, {0, 1, -1}, 63] (* Jean-François Alcover, Jan 07 2019 *)

{a(n) = if( n%2, n\2 + 1, -n/2)}; /* Michael Somos, Jul 20 1999 */

def a(n): return n//2 + 1 if n%2 else -n//2
print([a(n) for n in range(63)]) # Michael S. Branicky, Jul 14 2022

Euler transform of [-1, 2] is sequence a(n+1). - Michael Somos, Jun 11 2003
G.f.: x / ((1 + x) * (1 - x^2)). - Michael Somos, Jul 20 1999
E.g.f.: (exp(x) - (1 - 2*x) * exp(-x)) / 4. - Michael Somos, Jun 11 2003
a(n) = 1 - 2*a(n-1) -a(n-2); a(2*n) = -n, a(2*n+1) = n+1. - Michael Somos, Jul 20 1999
|a(n+1)| = A008619(n). |a(n-1)| = A004526(n). - Michael Somos, Jul 20 1999
a(n) = -a(n-1) + a(n-2) + a(n-3). a(n) = (-1)^(n+1) * floor((n+1) / 2). - Michael Somos, Jun 11 2003
a(1) = 1, a(n) = a(n-1)+n or a(n-1)-n whichever is closer to 0 on the number line. Or abs(a(n)) = min{abs(a(n-1)+n), abs(a(n-1)-n)}. - Amarnath Murthy, Jul 01 2003
a(n) = Sum_{k=0..n} k*(-1)^(k+1). - Paul Barry, Aug 20 2003
a(n) = (1-(2n+1)*(-1)^n)/4. - Paul Barry, Feb 02 2004
a(0) = 0; a(n) = (-1)^(n-1) * (n-|a(n-1)|) for n >= 1. - Rick L. Shepherd, Jul 14 2004
a(n) = a(n-1)-n*(-1)^n, a(0)=0; or a(n) = -a(n-1)+(1-(-1)^n)/2, a(0)=0. - Daniele Parisse and Franco Virga, Jun 06 2005
a(n) = ceiling(n/2) * (-1)^(n+1), n >= 0. - Franklin T. Adams-Watters, Nov 25 2011 (corrected by Daniel Forgues, Jul 21 2012)
a(n) = a(-1-n) for all n in Z. - Michael Somos, Jun 05 2013
Sum_{n>=1} 1/a(n) = 0. - Jaume Oliver Lafont, Jul 14 2017

Thanks to Michael Somos for helpful comments.

Name edited by Franklin T. Adams-Watters, Jan 30 2012

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