A353587 -id:A353587 - cp's OEIS frontend

A353583 Numerators of coefficients c(n) in product expansion of 1 + tan x = Product_{k>=1} 1 + c(k)*x^k.

1, 0, 1, -1, 7, -7, 199, -71, 484, -368, 187909, -610103, 2068657, -63614, 1530164189, -1715846683, 7628902283, -125125345078, 9521826231889, -17921564328719, 291162274608871, -47147385565688, 552647133893696333, -36898601487519532, 4761064630028162378

Offset: 1

M. F. Hasler, May 07 2022

See A353584 for the denominators, and A353586 for the analog for (tan x)/x.

			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 numerators of (1, 0, 1/3, -1/3, 7/15, -7/15, ...).

Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
Giedrius Alkauskas, Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems, arXiv:1609.09842 [math.NT], 2016.
H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
Wolfdieter Lang, Recurrences for the general problem.

Cf. A353584 (denominators), A353586 / A353587 (similar for (tan x)/x).

Cf. A170918 / A170919 for a variant.

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

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

M. F. Hasler, May 07 2022

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.

			(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, ...).

Cf. A353587 (denominators); A353583 / A353584 (product expansion of 1 + tan x).

Cf. A170918 / A170919 for a variant.

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

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

N. J. A. Sloane, Jan 30 2010

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

Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
Giedrius Alkauskas, Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems, arXiv:1609.09842 [math.NT], 2016.
H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
Wolfdieter Lang, Recurrences for the general problem.

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.

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

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

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

N. J. A. Sloane, Jan 30 2010

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

Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
Giedrius Alkauskas, Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems, arXiv:1609.09842 [math.NT], 2016.
H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
Wolfdieter Lang, Recurrences for the general problem.

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.

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);

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

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

N. J. A. Sloane, Jan 31 2010

Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
Giedrius Alkauskas, Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems, arXiv:1609.09842 [math.NT], 2016.
H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
Wolfdieter Lang, Recurrences for the general problem.

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.

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

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

A170922 a(n) = numerator 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

1, -1, 1, -13, 3, -37, 9, -1861, 7, -1491, 93, -81001, 315, -69705, 1083, -63586357, 3855, -438821, 13797, -822684711, 49689, -186369117, 182361, -704368012465, 10485, -10165801275, 619549, -9738266477517, 9256395, -566066862375, 34636833, -140047960975823893

Offset: 1

N. J. A. Sloane, Jan 31 2010

			1/2, -1/8, 1/8, -13/128, 3/32, -37/512, 9/128, -1861/32768, ...

Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
Giedrius Alkauskas, Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems, arXiv:1609.09842 [math.NT], 2016.
H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
Wolfdieter Lang, Recurrences for the general problem.

Cf. A170923 (denominators).
Cf. A353583 / A353584 for power product expansion of 1 + tan x.
Cf. A353586 / A353587 for power product expansion of (tan x)/x.

L := 34: 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(numer, [g]); # Peter Luschny, May 12 2022

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

A170924 a(n) = numerator of the coefficient c(n) of x^n in (1/sqrt(1-x))/Product_{k=1..n-1} 1 + c(k)*x^k, n = 1, 2, 3, ...

Original entry on oeis.org

1, 3, 1, 27, 3, 39, 9, 2955, 7, 1737, 93, 88047, 315, 79779, 1083, 77010795, 3855, 488391, 13797, 905252529, 49689, 204066351, 182361, 756251509503, 10485, 10978530465, 619549, 10462007147787, 9256395, 603860858253, 34636833, 150202954242966315

Offset: 1

N. J. A. Sloane, Jan 31 2010

			1/2, 3/8, 1/8, 27/128, 3/32, 39/512, 9/128, 2955/32768, 7/128, ...

Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
Giedrius Alkauskas, Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems, arXiv:1609.09842 [math.NT], 2016.
H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
Wolfdieter Lang, Recurrences for the general problem.

Cf. A170923 (denominators).
Cf. A353583 / A353584 for power product expansion of 1 + tan x.
Cf. A353586 / A353587 for power product expansion of (tan x)/x.

L := 34: g := NULL:
t := series(1/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(numer, [g]); # Peter Luschny, May 12 2022

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

cp's OEIS Frontend

A353583 Numerators of coefficients c(n) in product expansion of 1 + tan x = Product_{k>=1} 1 + c(k)*x^k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

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, ...

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Extensions

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, ...

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Extensions

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, ...

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Extensions

A170922 a(n) = numerator 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, ...

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Extensions

A170924 a(n) = numerator of the coefficient c(n) of x^n in (1/sqrt(1-x))/Product_{k=1..n-1} 1 + c(k)*x^k, n = 1, 2, 3, ...

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Extensions