A062157 -id:A062157 - cp's OEIS frontend

A291954 Triangle read by rows: T(n,k) = T(n-k,k-1) - T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

1, 0, 1, 0, -1, 0, 1, 1, 0, -1, -1, 0, 1, 0, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, -1, 0, 0, 1, 0, -1, 0, -1, 0, 2, 1, 0, 1, 1, -1, -1, 0, -1, -1, 1, 0, 0, 1, 0, -2, -1, 0, -1, 0, 2, 1, 0, 1, 1, -2, 0, 1, 0, -1, -1, 2, 1, -1, 0, 1, 0, -2, -1, 0, 0, -1, 0, 3, 1, -1, 0, 1

Offset: 0

Seiichi Manyama, Sep 06 2017

			First few rows are:
  1;
  0,  1;
  0, -1;
  0,  1,  1;
  0, -1, -1;
  0,  1,  0;
  0, -1,  0,  1;
  0,  1,  1, -1;
  0, -1, -1,  0;
  0,  1,  0, -1;
  0, -1,  0,  2, 1.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A003406.
Columns 0-1 give A000007, A062157.
Cf. A008289, A291895, A291929.

G.f. of column k: x^(k*(k+1)/2) / Product_{j=1..k} (1+x^j).

A063221 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 57 ).

Original entry on oeis.org

3, 10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98, 106, 110, 118, 122, 130, 134, 142, 146, 154, 158, 166, 170, 178, 182, 190, 194, 202, 206, 214, 218, 226, 230, 238, 242, 250, 254, 262, 266, 274, 278, 286, 290, 298

Offset: 1

N. J. A. Sloane, Jul 10 2001

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))
William A. Stein, The modular forms database
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A007310.

Essentially the same as A091999.

Seems to be 6n - 3 + A062157(n-1). - Ralf Stephan, Feb 16 2004
G.f.: 3x-2*x^2*(-5-2*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Jul 15 2015
a(n) = 2*A007310(n) for n>1. - Philippe Deléham, Nov 23 2016

A117997 Sum_{d|n} a(d) = n for n = 3^m (m >= 0) and for other n the sum is zero; i.e., the Möbius transform of [1, 0, 3, 0, 0, 0, 0, 0, 9, 0,...].

Original entry on oeis.org

1, -1, 2, 0, -1, -2, -1, 0, 6, 1, -1, 0, -1, 1, -2, 0, -1, -6, -1, 0, -2, 1, -1, 0, 0, 1, 18, 0, -1, 2, -1, 0, -2, 1, 1, 0, -1, 1, -2, 0, -1, 2, -1, 0, -6, 1, -1, 0, 0, 0, -2, 0, -1, -18, 1, 0, -2, 1, -1, 0, -1, 1, -6, 0, 1, 2, -1, 0, -2, -1, -1, 0, -1, 1, 0, 0, 1, 2, -1, 0, 54, 1, -1, 0, 1, 1, -2, 0, -1, 6, 1, 0, -2, 1, 1, 0, -1, 0, -6, 0, -1, 2, -1, 0, 2

Offset: 1

Paul D. Hanna, Apr 08 2006

sign

From Petros Hadjicostas, Jul 26 2020: (Start)
For p prime >= 2, Petrogradsky (2003) defined the multiplicative functions 1_p and mu_p in the following way:
1_p(n) = 1 when gcd(n,p) = 1 and 1_p(n) = 1 - p when gcd(n,p) = p;
mu_p(n) = mu(n) when gcd(n,p) = 1 and mu_p(n) = mu(m)*(p^s - p^(s-1)) when n = m*p^s with gcd(m,p) = 1 and s >= 1.
We have 1_2(n) = A062157(n), 1_3(n) = A061347(n), A067856(n) = mu_2(n), and a(n) = mu_3(n) for n >= 1.
Some of the results by other contributors here and in A067856 can be generalized:
(i) Rogel's (1897) formula for A067856 becomes Sum_{d | n} 1_p(d) * mu_p(n/d) = 0 for n > 1. Thus, 1_p is the Dirichlet inverse of mu_p.
(ii) R. J. Mathar's Dirichlet g.f. for mu_p becomes 1/(zeta(s) * (1 - p^(1-s))). The Dirichlet g.f. for 1_p is zeta(s) * (1 - p^(1-s)).
(iii) Benoit Cloitre's formula becomes 1 = Sum_{k=1..n} mu_p(k)*g_p(n/k), where g_p(x) = floor(x) - p*floor(x/p) = floor(x) mod p.
(iv) Paul D. Hanna's formula becomes Sum_{n >= 1} (mu_p(n)/n)*log((1 - x^(n*p))/(1 - x^n)) = x.
(v) The definition in the name of the sequence a(n) generalizes to Sum_{d | n} mu_p(d) = n, if n = p^s for s >= 0, and = 0, otherwise. Thus, mu_p(n) = Sum_{p^k | n, k >= 0} mu(n/p^k)*p^k. That is, (mu_p(n): n >= 1) is the Möbius transform of the sequence (b_p(n): n >= 1), where b_p(n) = p^k, if n = p^k for k >= 0, and b_p(n) = 0, otherwise.
(vi) We have the Lambert series Sum_{n >= 1} mu_p(n)*x^n/(1 - x^n) = Sum_{k >= 0} p^k*x^(p^k) = x + p*x^p + p^2*x^(p^2) + ..., which generalizes one of the formulas by Peter Bala in A067856.
(vii) By differentiating both sides of (iv) w.r.t. x and multiplying both sides by x, we get Sum_{n >= 1} mu_p(n)*(x^n + 2*x^(2*n) + ... + (p-1)*x^(n*(p-1)))/(1 + x^n + x^(2*n) + ... + x^(n*(p-1))) = x, which generalizes another one of Peter Bala's formulas in A067856. It can be thought as a "generalized Lambert series".
(viii) Dividing both sides of (vi) by x and integrating w.r.t. x from 0 to y, we get -Sum_{n >= 1} (mu_p(n)/n)*log(1 - y^n) = Sum_{k >= 0} y^(p^k) = y + y^p + y^(p^2) + y^(p^3) + ...
(ix) Obviously, f(n) = Sum_{d | n} 1_p(n/d)*g(d) if and only if g(n) = Sum_{d | n} mu_p(n/d)*f(d). (End)

Antti Karttunen, Table of n, a(n) for n = 1..59049
Peter Bala, A signed Dirichlet product of arithmetical functions.
V. M. Petrogradsky, Witt's formula for restricted Lie algebras, Advances in Applied Mathematics, 30 (2003), 219-227.
Eric Weisstein's World of Mathematics, Möbius Transform.

Cf. A061347, A062157, A067856.

{a(n)=if(n==1,1,-n*polcoeff(x+sum(k=1,n-1,a(k)/k*subst(log(1+x+x^2+x*O(x^n)),x,x^k+x*O(x^n))),n))}

A117997(n) = sumdiv(n,d,moebius(n/d)*if((3^valuation(d,3))==d,d,0)); \\ Antti Karttunen, Jan 15 2025

G.f.: x = Sum_{n >= 1} (a(n)/n)*log(1 + x^n + x^(2*n)).
1 = Sum_{k=1..n} a(k)*g(n/k), where g(x) = floor(x) - 3*floor(x/3). [Benoit Cloitre, Nov 11 2010]
From Petros Hadjicostas, Jul 26 2020: (Start)
a(n) = Sum_{3^k | n, k >= 0} mu(n/3^k)*3^k.
Dirichlet g.f.: 1/(zeta(s)*(1 - 3^(1-s))).
The sequence is the Dirichlet inverse of A061347.
Sum_{n >= 1} a(n)*x^n/(1 - x^n) = x + 3*x^3 + 9*x^9 + 27*x^27 + 81*x^81 + ...
Sum_{n >= 1} a(n)*(x^n + 2*x^(2*n))/(1 + x^n + x^(2*n)) = x.
-Sum_{n >= 1} (a(n)/n)*log(1 - x^n) = x + x^3 + x^9 + x^27 + x^81 + ... (End)

Offset changed to 1 by Petros Hadjicostas, Jul 26 2020

A118209 Expansion of Sum_{k>=1} lambda(k) * k * x^k/(1 + x^k) where lambda(k) is the Liouville function, A008836.

Original entry on oeis.org

1, -3, -2, 5, -4, 6, -6, -11, 7, 12, -10, -10, -12, 18, 8, 21, -16, -21, -18, -20, 12, 30, -22, 22, 21, 36, -20, -30, -28, -24, -30, -43, 20, 48, 24, 35, -36, 54, 24, 44, -40, -36, -42, -50, -28, 66, -46, -42, 43, -63, 32, -60, -52, 60, 40, 66, 36, 84, -58, 40, -60, 90, -42, 85, 48, -60, -66, -80, 44, -72, -70, -77, -72, 108, -42

Offset: 1

Stuart Clary, Apr 15 2006

Related to the logarithmic derivative of A118207(x) and A118208(x).

Related to a signed variant of A022998 via Mobius inversion. - R. J. Mathar, Jul 03 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008836, A022998, A028260, A061019, A061020, A062157, A117212, A118207, A118208.

nmax = 80; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; Drop[ CoefficientList[ Series[ Sum[ lambda[k] k x^k/(1 + x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ]
f[p_, e_] := (p*(-p)^e+1)/(p+1); f[2, e_] := ((-1)^e*2^(e+2) - 1)/3; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Aug 12 2023 *)

a(n) = sumdiv(n, d, (-1)^(n/d - 1)*(-1)^vecsum(factor(d)[,2])*d) \\ Michel Marcus, Dec 10 2016

a(n) = Sum_{d|n} (-1)^(n/d - 1)*lambda(d)*d, Dirichlet convolution of A061019 and A062157.
G.f.: A(x) is x times the logarithmic derivative of A118207(x).
G.f.: A(x) = A061020(x) - 2 A061020(x^2).
Dirichlet g.f.: zeta(s)*zeta(2s-2)*(1-2^(1-s))/zeta(s-1). - R. J. Mathar, Jul 03 2011
a(n) > 0 for n in A028260. - Michel Marcus, Dec 10 2016
Multiplicative with a(2^e) = ((-1)^e*2^(e+2) - 1)/3, and a(p^e) = (p*(-p)^e+1)/(p+1) for an odd prime p. - Amiram Eldar, Aug 12 2023

A274073 a(n) = 6^n-(-1)^n.

Original entry on oeis.org

0, 7, 35, 217, 1295, 7777, 46655, 279937, 1679615, 10077697, 60466175, 362797057, 2176782335, 13060694017, 78364164095, 470184984577, 2821109907455, 16926659444737, 101559956668415, 609359740010497, 3656158440062975, 21936950640377857, 131621703842267135

Offset: 0

Colin Barker, Jun 09 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,6).

Cf. A015540.

Sequences of the type k^n-(-1)^n: A062157 (k=0), A010673 (k=1), A062510 (k=2), A105723 (k=3), A247281 (k=4), A274072 (k=5), this sequence (k=6).

concat(0, Vec(7*x/((1+x)*(1-6*x)) + O(x^30)))

O.g.f.: 7*x/((1+x)*(1-6*x)).
E.g.f.: exp(6*x) - exp(-x).
a(n) = 5*a(n-1) + 6*a(n-2) for n>1.
a(n) = 7*A015540(n).

A021499 Decimal expansion of 1/495.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0

Offset: 0

N. J. A. Sloane

0, then repeat 0, 2. [Arkadiusz Wesolowski, Jul 22 2011]

Index entries for linear recurrences with constant coefficients, signature (0,1).

A010673 shifted right.

Join[{0,0},RealDigits[1/495,10,120][[1]]] (* or *) PadRight[{0},120,{2,0}] (* Harvey P. Dale, Dec 13 2020 *)

a(n) = 1 + (-1)^n - 2*0^n = (-1 - A033999(n))*A062157(n). [Arkadiusz Wesolowski, Jul 22 2011]

G.f.: 2*x^2/(1-x^2). - Bruno Berselli, Jul 22 2011

A274072 a(n) = 5^n-(-1)^n.

Original entry on oeis.org

0, 6, 24, 126, 624, 3126, 15624, 78126, 390624, 1953126, 9765624, 48828126, 244140624, 1220703126, 6103515624, 30517578126, 152587890624, 762939453126, 3814697265624, 19073486328126, 95367431640624, 476837158203126, 2384185791015624, 11920928955078126

Offset: 0

Colin Barker, Jun 09 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,5).

Cf. A015531.

Sequences of the type k^n-(-1)^n: A062157 (k=0), A010673 (k=1), A062510 (k=2), A105723 (k=3), A247281 (k=4), this sequence (k=5), A274073 (k=6).

LinearRecurrence[{4, 5}, {0, 6}, 30] (* Paolo Xausa, Oct 21 2024 *)

concat(0, Vec(6*x/((1+x)*(1-5*x)) + O(x^30)))

O.g.f.: 6*x/((1+x)*(1-5*x)).
E.g.f.: exp(5*x) - exp(-x).
a(n) = 4*a(n-1) + 5*a(n-2) for n>1.
a(n) = 6*A015531(n).

cp's OEIS Frontend

A291954 Triangle read by rows: T(n,k) = T(n-k,k-1) - T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A063221 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 57 ).

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A117997 Sum_{d|n} a(d) = n for n = 3^m (m >= 0) and for other n the sum is zero; i.e., the Möbius transform of [1, 0, 3, 0, 0, 0, 0, 0, 9, 0,...].

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A118209 Expansion of Sum_{k>=1} lambda(k) * k * x^k/(1 + x^k) where lambda(k) is the Liouville function, A008836.

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A274073 a(n) = 6^n-(-1)^n.

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A021499 Decimal expansion of 1/495.

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A274072 a(n) = 5^n-(-1)^n.

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