cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A353584 Denominators of coefficients c(n) in product expansion of 1 + tan x = Product_{k>=1} 1 + c(k)*x^k.

Original entry on oeis.org

1, 1, 3, 3, 15, 15, 315, 105, 567, 405, 155925, 467775, 1216215, 34749, 638512875, 638512875, 2170943775, 32564156625, 1856156927625, 3093594879375, 38979295480125, 5568470782875, 49308808782358125, 2900518163668125, 284473896821296875, 1232720219558953125
Offset: 1

Views

Author

M. F. Hasler, May 07 2022

Keywords

Comments

See the sequence of numerators, A353583, for references and more.

Examples

			1 + tan x = (1 + x)(1 + 1/3*x^3)(1 - 1/3*x^4/3)(1 + 7/15*x^5)(1 - 7/15*x^5)(...),
and this sequence lists the denominators of (1, 0, 1/3, -1/3, 7/15, -7/15, ...).

Crossrefs

Cf. A353583 (numerators)
Cf. A170918 / A170919 for a variant.

Programs

  • PARI
    t=1+tan(x+O(x)^29); vector(#t-1,n,c=polcoef(t,n);t/=1+c*x^n;denominator(c))

A353611 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + tan(x).

Original entry on oeis.org

1, 0, 2, -8, 56, -336, 3184, -27264, 309760, -3297280, 48104704, -624745472, 10591523840, -159594803200, 3133776259072, -56224864108544, 1249919350046720, -24600643845095424, 624022403933077504, -14094091678163140608, 381632216575339397120, -9516741266133420605440
Offset: 1

Views

Author

Ilya Gutkovskiy, May 07 2022

Keywords

Crossrefs

Cf. A000182, A006973, A009006, A137852, A353583, A353584, A353607, A353608, A353609, A353610.

Programs

  • Mathematica
    nn = 22; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Tan[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A353586 Numerators of coefficients c(n) in product expansion of (tan x)/x = Product_{k>=1} 1 + c(k)*x^(2k).

Original entry on oeis.org

1, 2, 1, 53, 91, 811, 73267, 35540711, 49830764, 34241488, 35249288479, 19259769465311, 732125336837021, 619038578481164306, 30015706187367326893, 16177789439326291541, 46789354983174555461, 498213391899375541476686, 248130101882943187003954597, 2572596069535443792125179632949
Offset: 1

Views

Author

M. F. Hasler, May 07 2022

Keywords

Comments

The coefficients of odd powers are zero since (tan x)/x is an even function.
See A353587 for the denominators, and A353583 (similar for 1 + tan x) for references and more.

Examples

			(tan x)/x = (1 + 1/3*x^2)(1 + 2/15*x^4)(1 + 1/105*x^6)(1 + 53/2835*x^8)...
and this sequence lists the numerators of (1/3, 2/15, 1/105, 53/2835, ...).

Crossrefs

Cf. A353587 (denominators); A353583 / A353584 (product expansion of 1 + tan x).
Cf. A170918 / A170919 for a variant.

Programs

  • PARI
    t=tan(x+O(x)^58)/x; vector(#t\2,n,c=polcoef(t,n*2);t/=1+c*x^(n*2);numerator(c))

A353587 Denominators of coefficients c(n) in product expansion of (tan x)/x = Product_{k>=1} 1 + c(k)*x^(2k).

Original entry on oeis.org

3, 15, 105, 2835, 66825, 3648645, 383107725, 97692469875, 1856156927625, 5568470782875, 9056719980433125, 33283445928091734375, 1298054391195577640625, 3952575621190533915703125, 367589532770719654160390625, 112527407991036628824609375, 3842566358093920359949921875
Offset: 1

Views

Author

M. F. Hasler, May 07 2022

Keywords

Comments

The coefficients of odd powers are zero since (tan x)/x is an even function.

Examples

			(tan x)/x = (1 + 1/3*x^2)(1 + 2/15*x^4)(1 + 1/105*x^6)(1 + 53/2835*x^8)...
and this sequence lists the denominators of (1/3, 2/15, 1/105, 53/2835, ...).

Crossrefs

Cf. A353586 (numerators); A353583 / A353584 (product expansion of 1 + tan x).
Cf. A170918 / A170919 for a variant.

Programs

  • PARI
    t=tan(x+O(x)^58)/x; vector(#t\2,n,c=polcoef(t,n*2);t/=1+c*x^(n*2);denominator(c))

A354065 Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + tan(x).

Original entry on oeis.org

1, -2, 2, -8, 56, -496, 3184, -22784, 273920, -4539136, 48104704, -506000384, 10591523840, -204528633856, 2888557717504, -53417657237504, 1249919350046720, -28453501844586496, 624022403933077504, -13729309300086800384, 372737701735949926400, -11010228423219933085696
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2022

Keywords

Crossrefs

Cf. A000182, A003707, A009006, A353583, A353584, A353611, A353911, A354055, A354056, A354063, A354064, A354066.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Tan[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

E.g.f.: Sum_{k>=1} mu(k) * log(1 + tan(x^k)) / k.

A170919 a(n) = denominator of the coefficient c(n) of x^n in (tan x)/Product_{k=1..n-1} 1 + c(k)*x^k, n = 1, 2, 3, ...

Original entry on oeis.org

1, 1, 3, 3, 5, 45, 105, 315, 2835, 14175, 5775, 467775, 6081075, 2837835, 212837625, 70945875, 3618239625, 97692469875, 206239658625, 9280784638125, 1031198293125, 142924083427125, 322279795963125, 101111706320625, 136968913284328125, 161872352063296875
Offset: 1

Views

Author

N. J. A. Sloane, Jan 30 2010

Keywords

Examples

			1, -1, 7/3, -14/3, 54/5, -1112/45, 6574/105, -48488/315, 1143731/2835, ...

Links

Crossrefs

Cf. A170918 (numerators), A170910-A170917.
Cf. A353583 / A353584 for power product expansion of 1 + tan x.
Cf. A353586 / A353587 for power product expansion of (tan x)/x.

Programs

  • Maple
    L := 28: g := NULL:
t := series(tan(x), x, L):
for n from 1 to L-2 do
   c := coeff(t, x, n);
   t := series(t/(1 + c*x^n), x, L);
   g := g, c;
od: map(denom, [g]); # Based on Maple in A170918. - Peter Luschny, Oct 05 2019

Extensions

Following a suggestion from Ilya Gutkovskiy, name corrected so that it matches the data by Peter Luschny, May 12 2022

A353911 Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + tan(x).

Original entry on oeis.org

1, -2, 2, -32, 56, -416, 3184, -85504, 309760, -4087552, 48104704, -546922496, 10591523840, -194387924992, 3133776259072, -129880886411264, 1249919350046720, -29073986250604544, 624022403933077504, -15137719350365519872, 381632216575339397120, -11149155036737662615552
Offset: 1

Views

Author

Ilya Gutkovskiy, May 10 2022

Keywords

Crossrefs

Cf. A000182, A009006, A353583, A353584, A353611, A353873, A353910, A353912.

Programs

  • Mathematica
    nn = 22; f[x_] := Product[1/(1 - a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Tan[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A354175 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + tan(x).

Original entry on oeis.org

1, 0, 2, -8, 56, -256, 3184, -36224, 273920, -2845696, 48104704, -676312064, 10591523840, -149454094336, 2888557717504, -72214957359104, 1249919350046720, -23620669488234496, 624022403933077504, -15637185047733469184, 372737701735949926400, -9655667879651150135296
Offset: 1

Views

Author

Ilya Gutkovskiy, May 18 2022

Keywords

Crossrefs

Cf. A000182, A009006, A067856, A353583, A353584, A353611, A353911, A354065, A354171, A354172, A354173, A354174, A354176.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = 2^(n + 1) (2^(n + 1) - 1) Abs[BernoulliB[n + 1]]/((n + 1) n!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 22}]

Formula

E.g.f.: Sum_{k>=1} A067856(k) * log(1 + tan(x^k)) / k.

A170918 a(n) = numerator of the coefficient c(n) of x^n in (tan x)/Product_{0 < k < n} 1 + c(k)*x^k, n = 1, 2, 3, ...

Original entry on oeis.org

1, -1, 7, -14, 54, -1112, 6574, -48488, 1143731, -14813072, 16252211, -3500388967, 125127865048, -158589803803, 33133618166566, -30512906279732, 4378989933312913, -330336346477870319, 1981395373839282068, -251479418962683770473, 79893293800974935213, -31493610597939643431532
Offset: 1

Views

Author

N. J. A. Sloane, Jan 30 2010

Keywords

Examples

			1, -1, 7/3, -14/3, 54/5, -1112/45, 6574/105, -48488/315, 1143731/2835, ...

Links

Crossrefs

Cf. A170919 (denominators), A170910-A170917.
Cf. A353583 / A353584 for power product expansion of 1 + tan x.
Cf. A353586 / A353587 for power product expansion of (tan x)/x.

Programs

  • Maple
    t1:=tan(x);
L:=100;
t0:=series(t1,x,L):
g:=[]; M:=40; t2:=t0:
for n from 1 to M do
t3:=coeff(t2,x,n); t2:=series(t2/(1+t3*x^n),x,L); g:=[op(g),t3];
od:
g;
g1:=map(numer,g);
g2:=map(denom,g);
  • PARI
    t=tan(x+O(x)^25); vector(#t,n,c=polcoef(t,n);t/=1+c*x^n;numerator(c)) \\ M. F. Hasler, May 07 2022

Extensions

Following a suggestion from Ilya Gutkovskiy, name corrected so that it matches the data by M. F. Hasler, May 07 2022

A170923 a(n) = denominator of the coefficient c(n) of x^n in sqrt(1+x)/Product_{k=1..n-1} 1 + c(k)*x^k, n = 1, 2, 3, ...

Original entry on oeis.org

2, 8, 8, 128, 32, 512, 128, 32768, 128, 32768, 2048, 2097152, 8192, 2097152, 32768, 2147483648, 131072, 16777216, 524288, 34359738368, 2097152, 8589934592, 8388608, 35184372088832, 524288, 549755813888, 33554432, 562949953421312, 536870912, 35184372088832
Offset: 1

Views

Author

N. J. A. Sloane, Jan 31 2010

Keywords

Links

Crossrefs

Cf. A170922 (numerators).
Cf. A353583 / A353584 for power product expansion of 1 + tan x.
Cf. A353586 / A353587 for power product expansion of (tan x)/x.

Programs

  • Maple
    L := 32: g := NULL:
t := series(sqrt(1+x), x, L):
for n from 1 to L-2 do
   c := coeff(t, x, n);
   t := series(t/(1 + c*x^n), x, L);
   g := g, c;
od: map(denom, [g]); # Peter Luschny, May 12 2022

Extensions

Following a suggestion from Ilya Gutkovskiy, name corrected so that it matches the data by Peter Luschny, May 12 2022
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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