A004174 -id:A004174 - cp's OEIS frontend

A024235 Expansion of e.g.f. tan(x)*sin(x)/2 (even powers only).

0, 1, 2, 31, 692, 25261, 1351382, 99680491, 9695756072, 1202439837721, 185185594118762, 34674437196568951, 7757267081778543452, 2043536254646561946181, 626129820701814932734142, 220771946624511552276841411, 88759695789769644718332394832

Offset: 0

R. H. Hardin

From Peter Bala, Nov 10 2016: (Start)
This sequence gives the coefficients in an asymptotic expansion related to the constant Pi/8. Recall the Madhava-Gregory-Leibniz series Pi/4 = Sum_{k = 1..inf} (-1)^(k-1)/(2*k - 1). Borwein et al. gave an asymptotic expansion for the tails of this series: Pi/2 - 2*Sum_{k = 1..N/2} (-1)^(k-1)/(2*k - 1) ~ 1/N - 1/N^3 + 5/N^5 - 61/N^7 + ..., where N is an integer divisible by 4 and the sequence of unsigned coefficients [1, 1, 5, 61,...] is the sequence of Euler numbers A000364.
Similarly, we have the series representation Pi/8 = Sum_{k = 1..inf} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)): using the approach of Borwein et al. we can show the associated asymptotic expansion for the tails of the series is Pi/4 - 2*Sum_{k = 1..N/2} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)) ~ -1/N^3 + 2/N^5 - 31/N^7 + 692/N^9 - ..., where N is divisible by 4 and where the sequence of unsigned coefficients [1, 2, 31, 692,...] forms the present sequence. A numerical example is given below. Cf. A278080 and A278195. (End)

			tan(x)*sin(x)/2 = 1/2*x^2 + 1/12*x^4 + 31/720*x^6 + 173/10080*x^8 + ...
From _Peter Bala_, Nov 10 2016: (Start)
Asymptotic expansion at N = 100000.
The truncated series 2*Sum_{k = 1..N/2} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)) = 0.78539816339744(9)309615660(6)4581987(603) 104929(1657)84377... to 50 digits. The bracketed digits show where this decimal expansion differs from that of Pi/4. The numbers -1, 2, -31, 692 must be added to the bracketed numbers to give the correct decimal expansion to 50 digits: Pi/4 = 0.78539816339744(8)309615660(8)4581987(572)104929(2349)84377.... (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..50
J. M. Borwein, P. B. Borwein, K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
Eric Weisstein's World of Mathematics, Euler Polynomial.

Cf. A009744, A000364, A004174, A019675, A278080, A278195.

A000364 := proc(n)
   abs(euler(2*n));
end proc:
seq(1/2*(A000364(n) - (-1)^n), n = 0..20); # Peter Bala, Nov 10 2016

With[{nn=30},Take[CoefficientList[Series[Tan[x]*Sin[x]/2,{x,0,nn}], x]Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Apr 27 2012 *)

G.f.: 1/2*(G(0) - 1/(1+x)) where G(k) = 1 - x*(2*k+1)^2/(1 - x*(2*k+2)^2/G(k+1) );  (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 09 2013
a(n) ~ (2*n)! * (2/Pi)^(2*n+1). - Vaclav Kotesovec, Jan 23 2015
From Peter Bala, Nov 10 2016: (Start)
a(n) = 1/2*(A000364(n) - (-1)^n).
a(n) = 1/8*(-4)^n*( -E(2*n,3/2) + 2*E(2*n,1/2) - E(2*n,-1/2) ), where E(n,x) is the Euler polynomial of order n.
G.f. 1/2!*sin^2(x)/cos(x) = x^2/2! + 2*x^4/4! + 31*x^6/6! + 692*x^8/8! + ....
O.g.f. for a signed version of the sequence: Sum_{n >= 0} ( 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n, k)/((1 - (2*k - 1)*x)*(1 - (2*k + 1)*x)*(1 - (2*k + 3)*x)) ) = 1 - 2*x^2 + 31*x^4 - 692*x^6 + .... (End)

Extended and signs tested Mar 15 1997.

More terms from Harvey P. Dale, Apr 27 2012

A278080 Expansion of e.g.f. (1/4!)*sin^4(x)/cos(x) (coefficients of even powers only).

Original entry on oeis.org

0, 0, 1, -5, 126, 1490, 118151, 8256885, 808428076, 100199284180, 15432169163901, 2889536106161375, 646438926423519626, 170294687860735726470, 52177485058722877649251, 18397662218707151323777465, 7396641315814156362154666776

Offset: 0

Peter Bala, Nov 10 2016

This sequence gives the coefficients in an asymptotic expansion of a series related to the constant Pi. It can be shown that (1/4!)*Pi/4 = Sum_{k >= 1} (-1)^(k-1)/((2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)). Using Proposition 1 of Borwein et al. it can be shown that the following asymptotic expansion holds for the tails of the series: for N divisible by 4, 2*( (1/4!)*Pi/4 - Sum_{k = 1..N/2} (-1)^(k-1)/((2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)) ) ~ 1/N^5 - (-5)/N^7 + 126/N^9 - 1490/N^11 + 118151/N^13 - .... An example is given below. Cf. A024235 and A278195.

			Let N = 100000. The truncated series 2*Sum_{k = 1..N/2} (-1)^(k-1)/((2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)) = 0.065449846949787359134638(3)038183229(2)6754107(569)820314(8536)0364.... The bracketed digits show where this decimal expansion differs from that of Pi/48. The numbers 1, 5, 126, -1490 must be added to the bracketed numbers to give the correct decimal expansion to 60 digits: Pi/48 = 0.065449846949787359134638(4) 038183229(7)6754107(695)820314(7046)0364.. ..

J. M. Borwein, P. B. Borwein, K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
Eric Weisstein's World of Mathematics, Euler Polynomial.

Cf. A000364, A004174, A024235, A166984, A278079, A278194, A278195.

A000364 := n -> abs(euler(2*n)):
seq(1/4!*(A000364(n) + (-1)^n*(9^n - 5)/4), n = 0..20);

With[{nn=40},Take[CoefficientList[Series[1/4! Sin[x]^4/Cos[x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Feb 09 2025 *)

a(n) = [x^(2*n)/(2*n)!] ( (1/4!)*sin^4(x)/cos(x) ).
a(n) = (1/4!)*( A000364(n) + (-1)^n*(9^(n) - 5)/4 ).
a(n) = (-1)^n/(2^4*4!) * 2^(2*n)*( E(2*n,5/2) - 4*E(2*n,3/2) + 6*E(2*n,1/2) - 4*E(2*n,-1/2) + E(2*n,-3/2) ), where E(n,x) is the Euler polynomial of order n.
E.g.f. (1/4!)*sin^4(x)/cos(x) = x^4/4! - 5*x^6/6! + 126*x^8/8! + 1490*x^10/10! + ....
O.g.f. for a signed version of the sequence: Sum_{n >= 0} ( (1/2^n) * Sum_{k = 0..n} (-1)^k*binomial(n, k)/((1 - (2*k - 3)*x)*(1 - (2*k - 1)*x)*(1 - (2*k + 1)*x)*(1 - (2*k + 3)*x)*(1 - (2*k + 5)*x)) ) = 1 + 5*x^2 + 126*x^4 - 1490*x^6 + 118151*x^8 - ....

A278195 Expansion of e.g.f. (1/6!)*sin^6(x)/cos(x) (coefficients of even powers only).

Original entry on oeis.org

0, 0, 0, 1, -28, 882, -17116, 803803, 13713336, 3671012164, 506128123928, 96524822605365, 21542790273363260, 5676618945053498806, 1739246268204447115932, 613255488134158250903887, 246554708506039690689322544, 112115693433705109495581088008

Offset: 0

Peter Bala, Nov 15 2016

This sequence gives the coefficients in an asymptotic expansion of a series related to the constant Pi. It can be shown that (1/6!)*Pi/4 = Sum_{k >= 1} (-1)^k/((2*k - 7)*(2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)*(2*k + 5)). Using Proposition 1 of Borwein et al. it can be shown that the following asymptotic expansion holds for the tails of this series: for N divisible by 4, 2*{ (1/6!)*Pi/4 - Sum_{k = 1..N/2} (-1)^k/((2*k - 7)*(2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)*(2*k + 5)) } ~ -1/N^7 + (-28)/N^9 - 882/N^11 + (-17116)/N^13 - 803803/N^15 + .... An example is given below.

			Expansion of (1/6!)*sin^6(x)/cos(x) starts x^6/6! - 28*x^8/8! + 882*x^10/10! - 17116*x^12/12! + ....
Let N = 100000. The truncated series 2*Sum_{k = 1..N/2} (-1)^k/( (2*k - 7)*(2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)*(2*k + 5) ) = 0.0021816615649929119711546134606107(7)5891803(617)860677(2450)20121.... The bracketed digits show where this decimal expansion differs from that of Pi/1440. The numbers -1, -28, -882 must be added to the bracketed numbers to give the correct decimal expansion: Pi/1440 = 0.0021816615649929119711546134606107(6)5891803(589)860677(1568)20121....

J. M. Borwein, P. B. Borwein, K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
Eric Weisstein's World of Mathematics, Euler Polynomial.

Cf. A000364, A004174, A166984, A278079, A278080, A278194.

A000364 := n -> abs(euler(2*n)):
seq((1/6!)*(A000364(n) - (1/16)*((-25)^n - 7*(-9)^n + 22*(-1)^n) ), n = 0..20);

With[{nn=40},Take[CoefficientList[Series[Sin[x]^6/Cos[x] 1/6!,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Sep 12 2019 *)

a(n) = [x^(2*n)/(2*n)!] ( 1/6!*sin^6(x)/cos(x) ).
a(n) = (1/6!)*( A000364(n) - 1/16*((-25)^n - 7*(-9)^n + 22*(-1)^n) ).
a(n) = (-1)^(n+1)/(2^6*6!) * 2^(2*n)*( E(2*n,7/2) - 6*E(2*n,5/2) + 15*E(2*n,3/2) - 20*E(2*n,1/2) + 15*E(2*n,-1/2) - 6*E(2*n,-3/2) + E(2*n,-5/2) ), where E(n,x) is the Euler polynomial of order n.

A326480 T(n, k) = 2^n * n! * [x^k] [z^n] (4exp(xz))/(exp(z) + 1)^2, triangle read by rows, for 0 <= k <= n. Coefficients of Euler polynomials of order 2.

Original entry on oeis.org

1, -2, 2, 2, -8, 4, 4, 12, -24, 8, -16, 32, 48, -64, 16, -32, -160, 160, 160, -160, 32, 272, -384, -960, 640, 480, -384, 64, 544, 3808, -2688, -4480, 2240, 1344, -896, 128, -7936, 8704, 30464, -14336, -17920, 7168, 3584, -2048, 256

Offset: 0

Peter Luschny, Jul 11 2019

T(m, n, k) = 2^n * n! * [x^k] [z^n] (2^m*exp(x*z))/(exp(z) + 1)^m are the coefficients of the generalized Euler polynomials (or Euler polynomials of higher order).
The classical case (m=1) is in A004174, this sequence is case m=2. A different normalization for m=1 is given in A058940 and for m=2 in A326485.
Generalized Euler numbers are 2^n*Sum_{k=0..n} T(m, n, k)*(1/2)^k. The classical Euler numbers are in A122045 and for m=2 in A326483.

			Triangle starts:
[0] [     1]
[1] [    -2,       2]
[2] [     2,      -8,     4]
[3] [     4,      12,   -24,      8]
[4] [   -16,      32,    48,    -64,     16]
[5] [   -32,    -160,   160,    160,   -160,     32]
[6] [   272,    -384,  -960,    640,    480,   -384,    64]
[7] [   544,    3808, -2688,  -4480,   2240,   1344,  -896,   128]
[8] [ -7936,    8704, 30464, -14336, -17920,   7168,  3584, -2048,   256]
[9] [-15872, -142848, 78336, 182784, -64512, -64512, 21504,  9216, -4608, 512]

NIST Digital Library of Mathematical Functions, §24.16(i), Higher-Order Analogs (of Bernoulli and Euler Polynomials), Release 1.0.23 of 2019-06-15.

Let E2_{n}(x) = Sum_{k=0..n} T(n,k) x^k. Then E2_{n}(1) = A155585(n+1),
E2_{n}(0) = A326481(n), E2_{n}(-1) = A326482(n), 2^n*E2_{n}(1/2) = A326483(n),
2^n*E2_{n}(-1/2) = A326484(n), [x^n] E2_{n}(x) = A000079(n).
Cf. A004174, A058940, A326485, A122045, A081733.

E2 := proc(n) (4*exp(x*z))/(exp(z) + 1)^2;
series(%, z, 48); 2^n*n!*coeff(%, z, n) end:
ListTools:-Flatten([seq(PolynomialTools:-CoefficientList(E2(n), x), n=0..9)]);

T[n_, k_] := 2^n n! SeriesCoefficient[4 Exp[x z]/(Exp[z]+1)^2, {z, 0, n}, {x, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 15 2019 *)

A058940 Triangle of coefficients of Euler polynomials rescaled to integers by multiplication with 2^(binary carry sequence (A007814)).

Original entry on oeis.org

1, -1, 2, 0, -1, 1, 1, 0, -6, 4, 0, 1, 0, -2, 1, -1, 0, 5, 0, -5, 2, 0, -3, 0, 5, 0, -3, 1, 17, 0, -84, 0, 70, 0, -28, 8, 0, 17, 0, -28, 0, 14, 0, -4, 1, -31, 0, 153, 0, -126, 0, 42, 0, -9, 2, 0, -155, 0, 255, 0, -126, 0, 30, 0, -5, 1, 691, 0, -3410, 0, 2805, 0, -924, 0, 165, 0, -22, 4, 0, 2073, 0, -3410, 0, 1683, 0, -396, 0, 55, 0, -6

Offset: 0

Wouter Meeussen, Jan 12 2001

Sums of even rows are A002425, sums of odd rows are 0, first element of even rows is -row sum, first element of row(2^p) is second element of row(1+2^p), LCM of numerators of Euler polynomial coefficients is A007814.

Cf. A004174, A002425, A007814.

A058940_row := proc(n) local i; seq(coeff(euler(n,x)*2^padic[ordp](n+1,2),x,i), i=0..n) end: # Peter Luschny, Nov 26 2010

Flatten[ Table[ CoefficientList[ EulerE[n, x]*2^IntegerExponent[n+1, 2], x], {n, 0, 12}]] (* Jean-François Alcover, Nov 18 2011, after Wouter Meeussen *)

T(n, k) = [x^k] E(n, x)*2^A007814(n+1).

A278194 E.g.f. (1/5!)*sin^5(x)/cos(x) (coefficients of odd powers only).

Original entry on oeis.org

0, 0, 1, -14, 336, -1408, 256256, 14746368, 1766772736, 242121048064, 41267065061376, 8461792420167680, 2057680174397259776, 585429994601202057216, 192659868531986620481536, 72616356304572571212316672, 31078397531081274526066016256

Offset: 0

Peter Bala, Nov 15 2016

Andrew Howroyd, Table of n, a(n) for n = 0..100

Cf. A000364, A004174, A024235, A278079, A278080, A278195.

seq((-1)^n*( 4^(n-2)*(4^n - 3) + 4^(n-1)*(4^(n+1) - 1)*bernoulli(2*n + 2)/(n + 1) )/15, n = 0..20);

a(n)={my(m=2*n+1, A=O(x*x^m)); m!*polcoef(sin(x + A)^5/cos(x + A), m)/120} \\ Andrew Howroyd, May 04 2020

a(n) = [x^(2*n+1)/(2*n+1)!] ( 1/5!*sin^5(x)/cos(x) ).
a(n) = (-1)^n*( 4^(n-2)*(4^n - 3) + 4^(n-1)*(4^(n+1) - 1)*Bernoulli(2*n + 2)/(n + 1) )/15.
a(n) = (-1)^n/(3!*2^6) * Sum_{k = 0..n} ( 25^(n-k) - 3*9^(n-k) + 2 )*binomial(2*n+1, 2*k)*2^(2*k)*E(2*k, 1/2), where E(n,x) is the Euler polynomial of order n.
a(n) = (-1)^n/(2^5*5!) * 2^(2*n+1)*( E(2*n+1, 3) - 5*E(2*n+1, 2) + 10*E(2*n+1, 1) - 10*E(2*n+1, 0) + 5*E(2*n+1, -1) - E(2*n+1, -2) ).
G.f. 1/5!*sin^5(x)/cos(x) = x^5/5! - 14*x^7/7! + 336*x^9/9! - 1408*x^11/11! + ....

Terms a(15) and beyond from Andrew Howroyd, May 04 2020

A349003 Decimal expansion of lim_{n->infinity} E(2n, n)/n^(2n), where E(n, x) is the n-th Euler polynomial.

Original entry on oeis.org

2, 3, 8, 4, 0, 5, 8, 4, 4, 0, 4, 4, 2, 3, 5, 1, 1, 1, 8, 8, 0, 5, 4, 1, 7, 1, 7, 3, 9, 5, 2, 0, 6, 4, 0, 9, 5, 8, 7, 2, 3, 1, 4, 0, 2, 7, 4, 2, 0, 6, 3, 4, 4, 8, 4, 0, 3, 1, 8, 9, 4, 9, 9, 8, 7, 8, 0, 4, 6, 7, 5, 5, 4, 2, 3, 3, 6, 1, 5, 1, 6, 5, 4, 1, 0, 5, 2, 4, 7, 8, 3, 2, 6, 3, 2, 3, 2, 8, 5, 5, 7, 8, 0, 9, 7, 2

Offset: 0

Vaclav Kotesovec, Nov 05 2021

Asymptotic expansion: E(2*n,n) / n^(2*n) ~ c0 + c1/n + c2/n^2 + ..., where
c0 = A349003
c1 = -0.15992500211230612504712294232596098830480284076519978623574964079...
c2 = -0.07258631854606119935476518617230181507488028047324715883939525404...
In general, for k>=1, E(k*n,n) / n^(k*n) ~ 2/(1 + exp(k)).

			0.238405844044235111880541717395206409587231402742063448403189499878046...

Eric Weisstein's World of Mathematics, Euler Polynomial.

Cf. A004174, A292782, A349004.

$MaxExtraPrecision = 1000; funs[n_] := EulerE[2 n, n]/n^(2 n); Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[1000/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 110]], {m, 10, 100, 10}]
RealDigits[2/(1 + E^2), 10, 110][[1]]

Equals 2/(1 + exp(2)).

Equals lim_{n->infinity} (HurwitzZeta(-2*n, n/2) - HurwitzZeta(-2*n, (n+1)/2)) * 2^(2*n+1) / n^(2*n).

A278079 Expansion of e.g.f. (1/3!)*sin^3(x)/cos(x) (coefficients of odd powers only).

Original entry on oeis.org

0, 1, 0, 56, 1280, 59136, 3727360, 317295616, 34977546240, 4848147562496, 825249675345920, 169237314418507776, 41153580031698534400, 11708600267324004499456, 3853197364634932928839680, 1452327126187528216207425536, 621567950620088261848869109760

Offset: 0

Peter Bala, Nov 10 2016

Eric Weisstein's World of Mathematics, Euler Polynomial.

Cf. A000182, A000364, A004174, A024235, A278080, A278194, A278195.

seq((-1)^n*( 2/3*4^n*(4^(n+1) - 1)*bernoulli(2*n+2)/(2*n + 2) - 4^n/6 ), n = 0..20);

a(n) = [x^(2*n+1)/(2*n+1)!] ( 1/3!*sin^3(x)/cos(x) ).
a(n) = (-1)^n*( 2/3*4^n*(4^(n+1) - 1)*Bernoulli(2*n+2)/(2*n + 2) - 4^n/6 ).
a(n) = (-1)^(n+1)/(2^3*3!) * 2^(2*n+1)*( E(2*n+1,2) - 3*E(2*n+1,1) + 3*E(2*n+1,0) - E(2*n+1,-1) ), where E(n,x) is the Euler polynomial of order n.
a(n) = (-1)^(n+1)/8 * Sum_{k = 0..n} (9^(n-k) - 1)*binomial(2*n+1,2*k)*2^(2*k)* E(2*k, 1/2).
G.f. 1/3!*sin^3(x)/cos(x) = x^3/3! + 56*x^7/7! + 1280*x^9/9! + 59136*x^11/11! + ....

A059341 Triangle giving numerators of coefficients of Euler polynomials, highest powers first.

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, -3, 0, 1, 1, -2, 0, 1, 0, 1, -5, 0, 5, 0, -1, 1, -3, 0, 5, 0, -3, 0, 1, -7, 0, 35, 0, -21, 0, 17, 1, -4, 0, 14, 0, -28, 0, 17, 0, 1, -9, 0, 21, 0, -63, 0, 153, 0, -31, 1, -5, 0, 30, 0, -126, 0, 255, 0, -155, 0, 1, -11, 0, 165, 0, -231, 0, 2805, 0, -1705, 0, 691, 1, -6, 0, 55, 0, -396, 0, 1683, 0, -3410, 0, 2073, 0, 1, -13

Offset: 0

N. J. A. Sloane, Jan 27 2001

			1; x-1/2; x^2-x; x^3-3*x^2/2+1/4; ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 48, [14b].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A059342. See also A004172, A004173, A004174, A004175, A011934, A020523, A020524, A020525, A020526, A020547, A020548, A058940.

for n from 0 to 30 do for k from n to 0 by -1 do printf(`%d,`,numer(coeff(euler(n,x), x, k))) od:od:

Numerator[Table[Reverse[CoefficientList[Series[EulerE[n, x], {x, 0, 20}], x]], {n, 0, 10}]]//Flatten (* G. C. Greubel, Jan 07 2017 *)

More terms from James Sellers, Jan 29 2001

A059342 Triangle giving denominators of coefficients of Euler polynomials, highest powers first.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 4, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 0

N. J. A. Sloane, Jan 27 2001

			1; x-1/2; x^2-x; x^3-3*x^2/2+1/4; ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 48, [14b].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A059341. See also A004172 A004173 A004174 A004175 A011934 A020523 A020524 A020525 A020526 A020547 A020548 A058940.

for n from 0 to 30 do for k from n to 0 by -1 do printf(`%d,`,denom(coeff(euler(n,x), x, k))) od:od:

Denominator[Table[Reverse[CoefficientList[Series[EulerE[n, x], {x, 0, 20}], x]], {n, 0, 10}]] (* G. C. Greubel, Jan 07 2017 *)

More terms from James Sellers, Jan 29 2001

cp's OEIS Frontend

A024235 Expansion of e.g.f. tan(x)*sin(x)/2 (even powers only).

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A278080 Expansion of e.g.f. (1/4!)*sin^4(x)/cos(x) (coefficients of even powers only).

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A278195 Expansion of e.g.f. (1/6!)*sin^6(x)/cos(x) (coefficients of even powers only).

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A326480 T(n, k) = 2^n * n! * [x^k] [z^n] (4*exp(x*z))/(exp(z) + 1)^2, triangle read by rows, for 0 <= k <= n. Coefficients of Euler polynomials of order 2.

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A058940 Triangle of coefficients of Euler polynomials rescaled to integers by multiplication with 2^(binary carry sequence (A007814)).

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A278194 E.g.f. (1/5!)*sin^5(x)/cos(x) (coefficients of odd powers only).

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A349003 Decimal expansion of lim_{n->infinity} E(2*n, n)/n^(2*n), where E(n, x) is the n-th Euler polynomial.

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A326480 T(n, k) = 2^n * n! * [x^k] [z^n] (4exp(xz))/(exp(z) + 1)^2, triangle read by rows, for 0 <= k <= n. Coefficients of Euler polynomials of order 2.

A349003 Decimal expansion of lim_{n->infinity} E(2n, n)/n^(2n), where E(n, x) is the n-th Euler polynomial.