A141024 A triangular sequence in which the Prime[n]^(2*n) is treated like a variable expansion: (1-Prime[n])^(2*n) with the base Prime[0] is defined as one (in the Goldbach tradition) to lower the coefficients: t(n,m)=(-1)^m*Prime[n]^(2*n - m)*Binomial[2*n, m].

1, 4, -4, 1, 81, -108, 54, -12, 1, 15625, -18750, 9375, -2500, 375, -30, 1, 5764801, -6588344, 3294172, -941192, 168070, -19208, 1372, -56, 1, 25937424601, -23579476910, 9646149645, -2338460520, 372027810, -40584852, 3074610, -159720, 5445, -110, 1, 23298085122481, -21505924728444

Offset

1,2

Comments

The row sums are nearly log -linear:

{1, 1, 16, 4096, 1679616, 10000000000, 8916100448256, 72057594037927936,

121439531096594251776, 1457498964228107529355264,

87732524600823436081182539776};

b = Table[Log[Sum[T[n, m], {m, 0, 2*n}]], {n, 0, 10}];

f[x_] = Fit[b, {1, x}, x]=-15.6357 + 6.93343 x.

If Prime[n+1] is used, the whole set of coefficients is shifted upward,

but is still near log-Linear.

These coefficients as polynomials are the result of treating a Fourier

expansion of frequency as a Zeta zero like function where the b[n]'s of the

Zeta zeros become phase like angular variables and drop out.

The power of two comes from the squaring necessary to isolate the primes as a variable. Basically the coefficient sequence is a quantum polynomial set

whose weights are the row sums.

The roots are 2*n multiplicity of primes: ( Mathematica fails here above 11

to give the Integer roots to the polynomials)

Table[Table[x /. NSolve[Sum[T[n, m]*x^m, {m, 0, 2*n}] == 0, x][[m + 1]], {m, 0, 2*n - 1}], {n, 0, 10}]

Links

Table of n, a(n) for n=1..38.

Formula

t(n,m)=(-1)^m*Prime[n]^(2*n - m)*Binomial[2*n, m].

Example

{1},
{4, -4, 1},
{81, -108, 54, -12, 1},
{15625, -18750, 9375, -2500, 375, -30, 1},
{5764801, -6588344, 3294172, -941192, 168070, -19208, 1372, -56,1}, {25937424601, -23579476910, 9646149645, -2338460520, 372027810, -40584852, 3074610, -159720, 5445, -110, 1},
{23298085122481, -21505924728444, 9098660462034, -2332989862060, 403786706895, -49696825464, 4459971516, -294064056, 14137695, -483340, 11154, -156, 1},
{168377826559400929, -138664092460683118, 53018623587908251, -12474970255978412, 2018009894349449, -237412928746994, 20948199595323, -1408282325736, 72485119707, -2842553714, 83604521, -1788332, 26299, -238,1},
{288441413567621167681, -242898032477996772784, 95880802293946094520,
-23549670738863953040, 4028233152700413020, -508829450867420592,
49097578592470408, -3691547262591760, 218578456337670, -10225892694160,
376743415048, -10815600432, 237184220, -3841040, 43320, -304, 1}, {3244150909895248285300369, -2538900712091933440669854,
938289393599192793291033, -217574352138943256415312, 35474079153088574415540,
-4318583549071652537544, 406823087956025239044, -30322217735852812848,
1812741277686852942, -87572042400331060, 3426732093925998, -108355165025328,
2748138243396, -55146586824, 856313460, -9928272, 80937, -414, 1},
{176994576151109753197786640401, -122065224931799829791576993380,
39986884029382702862757980590, -8273148419872283350915444260,
1212444164981282904875539245, -133786942342762251572473296,
11533357098513987204523560, -795403937828550841691280,
44570048240392935094770, -2049197620247951038840, 77728185595611935956,
-2436620238106957240, 63016040640697170, -1337210411473680, 23055351921960,
-318004854096, 3426776445, -27803460, 159790, -580, 1}

Mathematica

Clear[T, n, m] T[n_, m_] = If[n == 0, (-1)^m*Binomial[2*n, m], (-1)^m*Prime[n]^(2*n - m)*Binomial[2*n, m]]; a = Table[Table[T[n, m], {m, 0, 2*n}], {n, 0, 10}]; Flatten[a]

Crossrefs

Sequence in context: A067328 A111845 A120396 * A173210 A328922 A058888

Adjacent sequences: A141021 A141022 A141023 * A141025 A141026 A141027

Keyword

uned,tabf,sign

Author

Roger L. Bagula, Jul 29 2008

Status

approved