A353583
Numerators 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, 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
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.
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
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, ...).
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
(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, ...).
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
(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, ...).
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
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.
-
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
A170920
Write x*cot(x) = Product_{n>=1} (1 + g_n*x^(2*n)); a(n) = numerator(g_n).
Original entry on oeis.org
-1, -1, -1, -16, -91, -58844, -73267, -1196588, -49830764, -1715330699, -35249288479, -374085503198546, -732125336837021, -779432268293710651, -30015706187367326893, -183998031852529374082, -46789354983174555461, -115977125342651266593554, -248130101882943187003954597, -13171311382437535379302071714878
Offset: 1
-1/3, -1/45, -1/105, -16/4725, -91/66825, -58844/127702575, -73267/383107725, ...
-
t1:=x*cot(x);
L:=100;
t0:=series(t1, x, L);
g:=[];
M:=20; # number of terms to get
t2:=t0:
for n from 1 to M do
t3:=coeff(t2, x, 2*n); t2:=series(t2/(1+t3*x^(2*n)), x, L); g:=[op(g), t3];
od:
g;
g1:=map(numer, g);
g2:=map(denom, g);
Showing 1-6 of 6 results.
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