A020525 -id:A020525 - cp's OEIS frontend

A020523 a(n) = 3rd Euler polynomial evaluated at 2^n and multiplied by 4.

-1, 9, 161, 1665, 14849, 124929, 1024001, 8290305, 66715649, 535298049, 4288675841, 34334572545, 274777243649, 2198620602369, 17590575431681, 140731045904385, 1125874137038849, 9007096175525889, 72057181721067521, 576459103035981825, 4611679421357621249, 36893461759140036609

Offset: 0

Simon Plouffe

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-44,32).

Cf. A020524, A020525, A020526.

Maple
```
seq(euler(3,2^i),i=0..24);
```

Table[EulerE[3,2^n],{n,0,40}]*4 (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)

Vec(-(22*x-1)/((x-1)*(4*x-1)*(8*x-1)) + O(x^100)) \\ Colin Barker, May 04 2015

a(n) = 4*8^n - 6*4^n + 1.
From Colin Barker, May 04 2015: (Start)
a(n) = 13*a(n-1) - 44*a(n-2) + 32*a(n-3) for n > 2.
G.f.: -(22*x-1)/((x-1)*(4*x-1)*(8*x-1)). (End)
E.g.f.: exp(x)*(4*exp(7*x) - 6*exp(3*x) + 1). - Elmo R. Oliveira, Feb 23 2025

A020526 a(n) = 6th Euler polynomial evaluated at 2^n.

Original entry on oeis.org

0, 2, 1332, 166376, 13651920, 973242272, 65499561792, 4294977781376, 278176525712640, 17908846064302592, 1149543810255025152, 73678889946730981376, 4718907718699422044160, 302120774441963815411712, 19339271338993904793894912, 1237826702489967325274341376

Offset: 0

Simon Plouffe

Colin Barker, Table of n, a(n) for n = 0..553
Index entries for linear recurrences with constant coefficients, signature (106,-3024,22016,-32768).

Cf. A020523 - A020525.

Maple
```
seq(euler(6,2**i),i=0..24);
```

Table[EulerE[6,2^n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)

concat(0, Vec(2*x*(15616*x^2+560*x+1)/((2*x-1)*(8*x-1)*(32*x-1)*(64*x-1)) + O(x^100))) \\ Colin Barker, May 04 2015

a(n) = 106*a(n-1)-3024*a(n-2)+22016*a(n-3)-32768*a(n-4) for n>3. - Colin Barker, May 04 2015

G.f.: 2*x*(15616*x^2+560*x+1) / ((2*x-1)*(8*x-1)*(32*x-1)*(64*x-1)). - Colin Barker, May 04 2015

A020524 a(n) = 4th Euler polynomial evaluated at 2^n.

Original entry on oeis.org

0, 2, 132, 3080, 57360, 983072, 16252992, 264241280, 4261413120, 68451041792, 1097364145152, 17575006177280, 281337537761280, 4502500115750912, 72048797944922112, 1152851135862702080, 18446181123756195840, 295143401579725586432, 4722330454072626511872, 75557575495538172231680

Offset: 0

Simon Plouffe

Colin Barker, Table of n, a(n) for n = 0..830
Index entries for linear recurrences with constant coefficients, signature (26,-176,256).

Cf. A020523, A020525, A020526.

Maple
```
seq(euler(4, 2^n), n=0..24);
```

Table[EulerE[4,2^n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)

concat(0, Vec(-2*x*(40*x+1)/((2*x-1)*(8*x-1)*(16*x-1)) + O(x^100))) \\ Colin Barker, May 04 2015

From Colin Barker, May 04 2015: (Start)
a(n) = 2^n - 2^(1+3*n) + 16^n.
a(n) = 26*a(n-1) - 176*a(n-2) + 256*a(n-3) for n > 2.
G.f.: -2*x*(40*x+1)/((2*x-1)*(8*x-1)*(16*x-1)). (End)
E.g.f.: exp(2*x)*(exp(14*x) - 2*exp(6*x) + 1). - Elmo R. Oliveira, Feb 22 2025

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

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A020523 a(n) = 3rd Euler polynomial evaluated at 2^n and multiplied by 4.

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A020526 a(n) = 6th Euler polynomial evaluated at 2^n.

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A020524 a(n) = 4th Euler polynomial evaluated at 2^n.

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

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Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions