A006502 -id:A006502 - cp's OEIS frontend

A130654 Exponent m such that 2^m = A092505(n) = A002430(n) / A046990(n).

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

Offset: 1

Alexander Adamchuk, Jun 20 2007, Jun 23 2007

nonn

Conjecture: A092505(n) is always a power of 2. a(n) = Log[ 2, A092505(n) ]. Note that a(n) = 0 iff n is a power of 2; or A002430(2^n) = A046990(2^n) and A092505(2^n) = 1. It appears that a(2k+1) = 1 for k>0. Note that least index k such that a(k) = n is {1, 3, 14, 60, ...} which apparently coincides with A006502(n) = {1, 3, 14, 60, 279, 1251, ...} Related to Fibonacci numbers (see Carlitz reference).
Least index k such that a(k) = n is listed in A131262(n) = {1, 3, 14, 60, 248, ...}. Conjecture: A131262(n) = Sigma(2^n)*EulerPhi(2^n) = 2^(2n) - Floor(2^n/2) = A062354(2^n). If this conjecture is true then a(1008) = 5 and a(n)<5 for all n<1008.
Positions of records indeed continue as 1, 3, 14, 60, 248, 1008, 4064, 16320, ..., strongly suggesting union of {1} and A171499. - Antti Karttunen, Jan 13 2019

			A092505(n) begins {1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 1, ...}.
Thus a(1) = Log[2,1] = 0, a(2) = Log[2,1] = 0, a(3) = Log[2,2] = 1.

Antti Karttunen, Table of n, a(n) for n = 1..16385
L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, Fib. Quart., 16 (1978), 216-226.

Cf. A092505 = A002430(n) / A046990(n), n>0. Cf. A002430 = Numerators in Taylor series for tan(x). Cf. A046990 = Numerators of Taylor series for log(1/cos(x)). Cf. A006502 = Related to Fibonacci numbers.
Cf. A131262 = Least index k such that A130654(k) = n. Cf. A062354 = Sigma(n)*EulerPhi(n).
Cf. also A171499.

a=Series[ Tan[x], {x,0,256} ]; b=Series[ Log[ 1/Cos[x] ], {x,0,256}]; Table[ Log[ 2, Numerator[ SeriesCoefficient[ a, 2n-1 ] ] / Numerator[ SeriesCoefficient[ b, 2n ] ] ], {n,1,128} ]

a(n) = Log[ 2, A092505(n) ]. a(n) = Log[ 2, A002430(n) / A046990(n) ] = A007814(A092505(n)).

A259708 Triangle T(n,k) (0 <= k <= n) giving coefficients of certain polynomials related to Fibonacci numbers.

Original entry on oeis.org

1, 0, 1, 1, -1, 2, 0, 3, 0, 3, 1, 0, 14, 4, 5, 0, 8, 22, 60, 22, 8, 1, 6, 99, 244, 279, 78, 13, 0, 21, 240, 1251, 2016, 1251, 240, 21, 1, 25, 715, 5245, 14209, 14083, 5329, 679, 34, 0, 55, 1828, 21532, 88060, 139930, 88060, 21532, 1828, 55, 1, 78, 4817, 83060, 507398, 1218920, 1219382, 507068, 83225, 4762, 89

Offset: 0

N. J. A. Sloane, Jul 05 2015

The terms are the coefficients of the polynomials given by r_0(x) = 1; r_1(x) = x; r_(n+1) = (n+1)*x*r_n(x) + x*(1-x)*(r_n)'(x) + (1 - x)^2*r_(n-1)(x). [Carlitz, (1.6)]. Note: Carlitz wrongly states r_1(x) = 1. - Eric M. Schmidt, Jul 10 2015

			Triangle begins:
1,
0,1,
1,-1,2,
0,3,0,3,
1,0,14,4,5,
0,8,22,60,22,8,
1,6,99,244,279,78,13,
0,21,240,1251,2016,1251,240,21,
...

Eric M. Schmidt, Rows n = 0..50, flattened
L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, Fib. Quart., 16 (1978), 217. (Annotated scanned copy)
L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, Fib. Quart., 16 (1978), 216-226.

Diagonals include A000045, A259709, A006502.

Cf. A000142 (row sums).

A259708  := proc(n,k)
    if k < 0 or k > n then
        0;
    elif k =0 and n =0 then
        1;
    else
        (n-k+1)*procname(n-1,k-1)+k*procname(n-1,k)+procname(n-2,k)-2*procname(n-2,k-1) + procname(n-2,k-2) ;
    end if ;
end proc: # R. J. Mathar, Jun 18 2019

T[n_, k_] := T[n, k] = If[k < 0 || k > n, 0, If[k == 0 && n == 0, 1, (n - k + 1) T[n - 1, k - 1] + k T[n - 1, k] + T[n - 2, k] - 2 T[n - 2, k - 1] + T[n - 2, k - 2]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 30 2020 *)

@CachedFunction
def T(n,k) :
    if n < 0 or k < 0 : return 0
    if n == 0 and k == 0 : return 1
    return (n-k+1)*T(n-1,k-1) + k*T(n-1,k) + T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2)
# Eric M. Schmidt, Jul 10 2015

T(0,0) = 1; T(n+1,k) = (n-k+2)*T(n,k-1) + k*T(n,k) + T(n-1,k) - 2*T(n-1,k-1) + T(n-1,k-2), where we put T(n,k) = 0 if n < 0 or k < 0. As special cases, T(n,n) = Fibonacci(n+1) and T(n,0) = 1 (n even) or 0 (n odd). - Rewritten by Eric M. Schmidt, Jul 10 2015

More terms from and name revised by Eric M. Schmidt, Jul 10 2015

A259709 a(n) = A259708(n,n-1).

Original entry on oeis.org

0, -1, 0, 4, 22, 78, 240, 679, 1828, 4762, 12132, 30432, 75504, 185875, 455040, 1109540, 2697698, 6545670, 15859128, 38384099, 92832716, 224399570, 542226888, 1309862496, 3163659744, 7640066759, 18448657344, 44545569508, 107553559150, 259675788510

Offset: 1

N. J. A. Sloane, Jul 05 2015

sign

Eric M. Schmidt, Table of n, a(n) for n = 1..300
L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, Fib. Quart., 16 (1978), 217. (Annotated scanned copy)
L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, Fib. Quart., 16 (1978), 216-226.

Cf. A259708.

Empirical: a(n) = 4*a(n-1) - 2*a(n-2) - 6*a(n-3) + 2*a(n-4) + 4*a(n-5) + a(n-6). - Eric M. Schmidt, Jul 10 2015

Empirical: G.f.: -x^2*(-1+2*x^2+4*x) / ( (x^2+2*x-1)*(x^2+x-1)^2 ). - R. J. Mathar, Jul 15 2015

More terms from Eric M. Schmidt, Jul 10 2015

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A130654 Exponent m such that 2^m = A092505(n) = A002430(n) / A046990(n).

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A259708 Triangle T(n,k) (0 <= k <= n) giving coefficients of certain polynomials related to Fibonacci numbers.

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A259709 a(n) = A259708(n,n-1).

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