A000748 -id:A000748 - cp's OEIS frontend

A007653 Coefficients of L-series for elliptic curve "37a1": y^2 + y = x^3 - x.

1, -2, -3, 2, -2, 6, -1, 0, 6, 4, -5, -6, -2, 2, 6, -4, 0, -12, 0, -4, 3, 10, 2, 0, -1, 4, -9, -2, 6, -12, -4, 8, 15, 0, 2, 12, -1, 0, 6, 0, -9, -6, 2, -10, -12, -4, -9, 12, -6, 2, 0, -4, 1, 18, 10, 0, 0, -12, 8, 12, -8, 8, -6, -8, 4, -30, 8, 0, -6, -4, 9, 0, -1, 2, 3, 0, 5, -12, 4, 8, 9, 18, -15, 6, 0, -4, -18, 0, 4, 24, 2, 4, 12, 18, 0

Offset: 1

N. J. A. Sloane

G.f. is Fourier series of a weight 2 level 37 modular cusp form.

			G.f. = q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + 6*q^9 + 4*q^10 - 5*q^11 + ...

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

Robin Visser, Table of n, a(n) for n = 1..10000
N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 57.
LMFDB, Elliptic Curve 37a1.
Don Zagier, Modular points, modular curves, modular surfaces and modular forms, Arbeitstagung Bonn 1984: Proceedings of the meeting held by the Max-Planck-Institut für Mathematik, Bonn June 15-22, 1984. Springer Berlin Heidelberg, 1985. See Eq. (8).

Cf. A000748, A045866, A045867.

A := Basis( CuspForms( Gamma0(37), 2), 72); A[1] - 2*A[2]; /* Michael Somos, Jan 02 2017 */

{a(n) = if( n<1, 0, ellak( ellinit([ 0, 0, -1, -1, 0]), n))}; /* Michael Somos, Mar 04 2011 */

{a(n) = if( n<1, 0, qfrep([ 2, 1, 0, 1; 1, 8, 1, -3; 0, 1, 10, 2; 1, -3, 2, 12 ], n, 1)[n] - qfrep([ 4, 1, 2, 1; 1, 4, 1, 0; 2, 1, 6, -2; 1, 0, -2, 20 ], n, 1)[n])}; /* Michael Somos, Apr 02 2006 */

def a(n):
    return EllipticCurve("37a1").an(n)  # Robin Visser, Aug 02 2023

a(3^n) = A000748(n).
a(n) = (A045866(n) - A045867(n)) / 2.
a(n) is multiplicative with a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) where a(p) = p+1 - number of solutions of y^2 + y = x^3 - x modulo p including the point at infinity. - Michael Somos, Mar 03 2011
G.f. is a period 1 Fourier series which satisfies f(-1 / (37 t)) = -37 (t/i)^2 f(t) where q = exp(2 Pi i t).

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 22 2000

A132677 Period 3: repeat [1, 2, -3].

Original entry on oeis.org

1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3

Offset: 0

Paul Curtz, Nov 15 2007

a(n) is proportional to its 6n-th differences.

Nonsimple continued fraction expansion of 1+sqrt(2/5) = 1.63245553... (see A010494). - R. J. Mathar, Mar 08 2012

Index entries for linear recurrences with constant coefficients, signature (-1,-1).

Cf. A000748, A010494, A049347, A101544.

&cat [[1, 2, -3]^^30]; // Wesley Ivan Hurt, Jul 01 2016

seq(op([1, 2, -3]), n=0..50); # Wesley Ivan Hurt, Jul 01 2016
A132677 := proc(n)
    op(1+modp(n,3),[1,2,-3]) ;
end proc: # R. J. Mathar, Mar 14 2025

PadRight[{}, 100, {1, 2, -3}] (* Wesley Ivan Hurt, Jul 01 2016 *)

a(n)=[1,2,-3][1+n%3] \\ Jaume Oliver Lafont, Mar 24 2009

G.f.: (1+3*x)/(1+x+x^2). - Jaume Oliver Lafont, Mar 24 2009
a(n) = cos(2*Pi*n/3) + 5*sin(2*Pi*n/3)/sqrt(3). - R. J. Mathar, Oct 08 2011
a(n) + a(n-1) + a(n-2) = 0 for n > 1, a(n) = a(n-3) for n > 2. - Wesley Ivan Hurt, Jul 01 2016

A094715 a(n) = Sum_{2i+3j=n, 0<=i<=n, 0<=j<=n} n!/( (2i)!(3*j)! ).

Original entry on oeis.org

1, 0, 1, 1, 1, 10, 2, 35, 29, 85, 211, 220, 926, 1001, 3095, 5461, 9829, 25126, 37130, 97223, 164921, 349525, 728575, 1309528, 2973350, 5326685, 11450531, 22369621, 43942081, 91869970, 174174002, 365088395, 708653429, 1431655765, 2884834891

Offset: 0

Benoit Cloitre, May 23 2004

nonn

Average of binomial and inverse binomial transform of {1, 0, 0, 1, 0, 0, 1, ...}. - Paul Barry, Jan 04 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000748, A010892, A024493, A094717.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-2*x^2-x^3+x^4-x^5)/((1-2*x)*(1-x+x^2)*(1+3*x+3*x^2)) )); // G. C. Greubel, Feb 13 2023

A094715_list := proc(n) local i; (exp(z)+2*exp(-z/2)*cos(z*sqrt(3/4)))*cosh(z)/3;  series(%,z,n+2): seq(i!*coeff(%,z,i),i=0..n) end: A094715_list(34); # Peter Luschny, Jul 10 2012

Table[(1/6)*(Boole[n==0] +2^n +2*ChebyshevU[n,1/2] -ChebyshevU[n-1, 1/2] +2*3^(n/2)*ChebyshevU[n, -Sqrt[3]/2] +3^((n+1)/2)*ChebyshevU[n- 1, -Sqrt[3]/2]), {n,0,50}] (* G. C. Greubel, Feb 13 2023 *)

a(n)=sum(i=0,n,sum(j=0,n,if(n-2*i-3*j,0,n!/(2*i)!/(3*j)!)))

def A094715_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x^2-x^3+x^4-x^5)/((1-2*x)*(1-x+x^2)*(1+3*x+3*x^2)) ).list()
A094715_list(50) # G. C. Greubel, Feb 13 2023

Limit_{n --> oo} a(n)/2^n = 1/6.
G.f.: (1-2*x^2-x^3+x^4-x^5)/((1-2*x)*(1-x+x^2)*(1+3*x+3*x^2)). - Vladeta Jovovic, May 23 2004
a(n) = (1/3)*Sum_{k=0..floor(n/2)} C(n, 2*k)*(2*cos(2*Pi*(n-2*k)/3) + 1). - Paul Barry, Jan 04 2005 [corrected by Jason Yuen, Aug 28 2024]
E.g.f.: (exp(z) + 2*exp(-z/2)*cos(z*sqrt(3/4)))*cosh(z)/3. - Peter Luschny, Jul 10 2012
a(n) = (1/6)*([n=0] + 2^n + 2*A010892(n) - A010892(n-1) + 2*A000748(n) + 3*A000748(n-1)). - G. C. Greubel, Feb 13 2023

A168615 Inverse binomial transform of A169609, or of A144437 preceded by 1.

Original entry on oeis.org

1, 2, -2, 0, 6, -18, 36, -54, 54, 0, -162, 486, -972, 1458, -1458, 0, 4374, -13122, 26244, -39366, 39366, 0, -118098, 354294, -708588, 1062882, -1062882, 0, 3188646, -9565938, 19131876, -28697814, 28697814, 0, -86093442, 258280326, -516560652

Offset: 0

Paul Curtz, Dec 01 2009

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-3).

Cf. A169609, A144437, A000748, A057083, A057682.

[ n le 2 select n else n eq 3 select -2 else -3*Self(n-1)-3*Self(n-2): n in [1..37] ]; // Klaus Brockhaus, Dec 03 2009

Join[{1,2,-2}, LinearRecurrence[{-3, -3}, {0, 6}, 25]] (* G. C. Greubel, Jul 27 2016 *)
LinearRecurrence[{-3,-3},{1,2,-2},40] (* Harvey P. Dale, Jul 21 2024 *)

a(n) = -3*a(n-1) - 3*a(n-2) for n > 2; a(0) = 1, a(1) = 2, a(2) = -2.
a(n) = 2*A123877(n-1), n>0.
G.f.: 1+2*x*(1+2*x)/(1+3*x+3*x^2).
a(6*m + 3) = 0, m>=0. - G. C. Greubel, Jul 27 2016

Edited and extended by Klaus Brockhaus, Dec 03 2009

A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, -1, 3, -2, -1, 1, 0, -2, 5, -3, -1, 1, 1, -2, -2, 7, -4, -1, 1, -1, 5, -7, -1, 9, -5, -1, 1, 0, -3, 12, -15, 1, 11, -6, -1, 1, 1, -3, -3, 21, -26, 4, 13, -7, -1, 1, -1, 7, -15, 3, 31, -40, 8, 15, -8, -1, 1, 0, -4, 22, -42

Offset: 0

Philippe Deléham, Nov 12 2011

			Triangle begins :
1
-1, 1
0, -1, 1
1, -1, -1, 1
-1, 3, -2, -1, 1
0, -2, 5, -3, -1, 1
1, -2, -2, 7, -4, -1, 1
-1, 5, -7, -1, 9, -5, -1, 1

Cf. A026729, A063967, A129267, A176971 (diagonals sums).

T(n,k)=T(n-1,k-1)+T(n-2,k-1)-T(n-1,k)-T(n-2,k), T(0,0)=1.
G.f.: 1/(1-(y-1)*x-(y-1)*x^2).
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000748(n), A108520(n), A049347(n), A000007(n), A000045(n+1), A002605(n+1), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for x = -2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.

A141532 Inverse binomial transform of A141425.

Original entry on oeis.org

1, 1, 1, -2, 4, -8, 7, 22, -125, 376, -878, 1756, -3143, 5188, -8189, 13102, -22928, 45856, -101549, 232618, -524285, 1137148, -2362874, 4725748, -9185771, 17574376, -33554429, 64717378, -127043276, 254086552, -515347553, 1052218462, -2147483645

Offset: 0

Paul Curtz, Aug 12 2008

sign

This is the inverse binomial transform of A141425 if interpreted with offset 0.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-15,-20,-15,-6).

Cf. A000748, A049347.

I:=[1,1,-2,4,-8]; [1] cat [n le 5 select I[n] else -6*Self(n-1) -15*Self(n-2) -20*Self(n-3) -15*Self(n-4) -6*Self(n-5): n in [1..40]]; // G. C. Greubel, Mar 30 2021

LinearRecurrence[{-6,-15,-20,-15,-6}, {1,1,1,-2,4,-8}, 40] (* G. C. Greubel, Mar 30 2021 *)

def A141532_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1 +7*x +22*x^2 +39*x^3 +42*x^4 +27*x^5)/((1+x+x^2)*(1+3*x+3*x^2)*(1+2*x)) ).list()
A141532_list(40) # G. C. Greubel, Mar 30 2021

G.f.: (1 +7*x +22*x^2 +39*x^3 +42*x^4 +27*x^5)/((1+x+x^2)*(1+3*x+3*x^2)*(1+2*x)). - R. J. Mathar, Nov 11 2008
From G. C. Greubel, Mar 30 2021: (Start)
a(n) = (9/2)*[n=0] + (-2)^(n-1) - (3/2)*( ChebyshevU(n, -1/2) + 2*ChebyshevU(n-1, -1/2) + 3^((n-1)/2)*(sqrt(3)*ChebyshevU(n, -sqrt(3)/2) + 2*ChebyshevU(n-1, -sqrt(3)/2)) ).
a(n) = (9/2)*[n=0] + (-2)^(n-1) - (3/2)*(A049347(n) + 2*A049347(n-1) + A000748(n) + 2*A000748(n-1) ). (End)

Extended by R. J. Mathar, Nov 11 2008

A368149 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 3, 10, 10, 4, 5, 20, 31, 20, 5, 8, 40, 78, 76, 35, 6, 13, 76, 184, 232, 161, 56, 7, 21, 142, 406, 636, 582, 308, 84, 8, 34, 260, 861, 1604, 1831, 1296, 546, 120, 9, 55, 470, 1766, 3820, 5215, 4630, 2640, 912, 165, 10, 89, 840, 3533, 8696

Offset: 1

Clark Kimberling, Dec 25 2023

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
   1
   1    2
   2    4    3
   3   10   10    4
   5   20   31   20    5
   8   40   78   76   35    6
  13   76  184  232  161   56   7
  21  142  406  636  582  308  84  8
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 10*x^2 + 4*x^3, so (T(4,k)) = (3,10,10,4), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Cf. A000045 (column 1); A000027 (p(n,n-1)); A000244 (row sums), (p(n,1)); A033999 (alternating row sums), (p(n,-1)); A116415 (p(n,2)), A000748, (p(n,-2)); A152268, (p(n,3)); A190969, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150.

p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 - x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A379825 a(n) = [x^n] x/(12x^2 - 6x + 1).

Original entry on oeis.org

0, 1, 6, 24, 72, 144, 0, -1728, -10368, -41472, -124416, -248832, 0, 2985984, 17915904, 71663616, 214990848, 429981696, 0, -5159780352, -30958682112, -123834728448, -371504185344, -743008370688, 0, 8916100448256, 53496602689536, 213986410758144, 641959232274432

Offset: 0

Peter Luschny, Jan 04 2025

Index entries for linear recurrences with constant coefficients, signature (6,-12).

Cf. A057083, A000748, A190965.

w := sqrt(-3): a := n -> (w/6)*((3 - w)^n - (3 + w)^n):
seq(simplify(a(n)), n = 0..28);
# Alternative:
a := proc(n) option remember; if n < 2 then n else 6*(a(n - 1) - 2*a(n - 2)) fi end:
seq(a(n), n = 0..28);

LinearRecurrence[{6,-12},{0,1},29] (* James C. McMahon, Jan 05 2025 *)

a(n) = n! * [x^n] exp(3*x)*sin(sqrt(3)*x)/sqrt(3).
a(n) = (w/6)*((3 - w)^n - (3 + w)^n) where w = sqrt(-3).
a(n) = 6*a(n - 1) - 12*a(n - 2) for n >= 2.
a(n) = 2^n*3^((n - 1)/2)*sin((Pi*n)/6).
a(n) = 2^(n-1)*A057083(n-1) = 2^(n-1)*3^((n-1)/2)*ChebyshevU(n-1, sqrt(3)/2) for n >= 1.

cp's OEIS Frontend

A007653 Coefficients of L-series for elliptic curve "37a1": y^2 + y = x^3 - x.

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A132677 Period 3: repeat [1, 2, -3].

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A094715 a(n) = Sum_{2*i+3*j=n, 0<=i<=n, 0<=j<=n} n!/( (2*i)!*(3*j)! ).

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A168615 Inverse binomial transform of A169609, or of A144437 preceded by 1.

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A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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Examples

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A141532 Inverse binomial transform of A141425.

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A368149 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.

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A094715 a(n) = Sum_{2i+3j=n, 0<=i<=n, 0<=j<=n} n!/( (2i)!(3*j)! ).

A368149 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.

A379825 a(n) = [x^n] x/(12x^2 - 6x + 1).