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 1-6 of 6 results.

A329498 Denominator of the rational number alpha_n involved in the calculation of the second moment of the n-th term of Ulam's "history-dependent random sequence".

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 1680, 40320, 362880, 725760, 7983360, 2128896, 113218560, 387459072, 261534873600, 836911595520, 71137485619200, 75322043596800, 1621934672117760, 37429261664256000, 10218188434341888000, 224800145555521536000
Offset: 1

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Author

N. J. A. Sloane, Nov 17 2019

Keywords

Examples

			0, 2, 21/2, 223/6, 2603/24, 33623/120, 477977/720, 2474153/1680, ...

Links

Crossrefs

Cf. A329495, A329496, A327497.

Programs

  • Maple
    s:=[1]; a:=[0];
for N from 2 to 40 do # N = n+1
n:=N-1;
t1:=s[n]+(1/n)*add(s[k],k=1..n)+2*s[n]/n+(2/n)*a[n];
t2:=s[n]+(1/n)*add(s[k],k=1..n)+a[n]+(2/n)*add(a[k],k=1..n);
s:=[op(s),t1];
a:=[op(a),t2];
od:
s; # sigma_n
a; # alpha_n
sn:=map(numer,s); # A329495
sd:=map(denom,s); # A329496
an:=map(numer,a); # A329497
ad:=map(denom,a); # A329498

A335264 a(n) = Numerator(-4*n^2*Zeta(1 - n)^2*(1 - 2^n)) for n >= 1, a(0) = 0.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 17, 0, 775, 0, 477481, 0, 267589, 0, 3362251073, 0, 421424697891, 0, 38751520678991, 0, 44386209501802003, 0, 228891128457907983257, 0, 1636462395711601387189, 0, 348063222218272291910609213, 0, 3710225622968600411572814809525
Offset: 0

Views

Author

Peter Luschny, Jun 13 2020

Keywords

Examples

			Rational sequence starts: 0, 1, 1/3, 0, 1/15, 0, 1/7, 0, 17/15, 0, 775/33, 0, 477481/455, ...

Crossrefs

Cf. A335265 (denominators), A164555/A027642 (Bernoulli numbers).
Cf. A335538, A335539, A327497.

Programs

  • Maple
    a := s -> `if`(s = 0, 0, -4*s^2*Zeta(1 - s)^2*(1 - 2^s)):
seq(numer(a(s)), s = 0..24);

Formula

a(n) = numerator(Bernoulli(n)^2*(2^(n+2) - 4)).

A335265 a(n) = Denominator(-4*n^2*Zeta(1 - n)^2*(1 - 2^n)) for n >= 1, a(0) = 1.

Original entry on oeis.org

1, 1, 3, 1, 15, 1, 7, 1, 15, 1, 33, 1, 455, 1, 3, 1, 255, 1, 133, 1, 33, 1, 69, 1, 455, 1, 3, 1, 435, 1, 2387, 1, 255, 1, 3, 1, 319865, 1, 3, 1, 1353, 1, 43, 1, 345, 1, 141, 1, 7735
Offset: 0

Views

Author

Peter Luschny, Jun 13 2020

Keywords

Examples

			Rational sequence starts: 0, 1, 1/3, 0, 1/15, 0, 1/7, 0, 17/15, 0, 775/33, 0, 477481/455, ...

Crossrefs

Cf. A335264 (numerators), A164555/A027642 (Bernoulli numbers).
Cf. A335538, A335539, A327497.

Programs

  • Maple
    a := s -> `if`(s = 0, 0, -4*s^2*Zeta(1 - s)^2*(1 - 2^s)):
seq(denom(a(s)), s = 0..24);

Formula

a(n) = denominator(Bernoulli(n)^2*(2^(n+2) - 4)).

A335538 a(n) = numerator(-4*n^2*zeta(1 - n)*zeta(n)*(1 - 2^(1 - n)) / Pi^n) for n >= 2, a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 1, 0, -7, 0, 31, 0, -127, 0, 365, 0, -977403607, 0, 57337, 0, -61240067209, 0, 252221719530919, 0, -15984987035583127, 0, 2841046127487821, 0, -468654557583574838590567, 0, 188822581306893585883, 0, -220710643004244238794643249, 0, 1594135539680034434970146279285311
Offset: 0

Views

Author

Peter Luschny, Jun 13 2020

Keywords

Examples

			Rational sequence starts: 0, 1, 1/9, 0, -7/1350, 0, 31/52920, 0, -127/1134000, 0, 365/11290752, ...

Crossrefs

Cf. A335539 (denominators), A164555/A027642 (Bernoulli numbers).
Cf. A335264, A335265, A327497.

Programs

  • Maple
    a := s -> `if`(s=1 or s=0, s, -4*s^2*Zeta(1 - s)*Zeta(s)*(1 - 2^(1 - s)) / Pi^s):
seq(numer(a(s)), s = 0..34);

Formula

a(n) = numerator(n*Bernoulli(n)*zeta(n)*(4-2^(3-n))/Pi^n) for n >= 2.

A335539 a(n) = denominator(-4*n^2*zeta(1 - n)*zeta(n)*(1 - 2^(1 - n)) / Pi^n) for n >= 2, a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, 9, 1, 1350, 1, 52920, 1, 1134000, 1, 11290752, 1, 74373979680000, 1, 8006169600, 1, 12147360825600000, 1, 56625794240311296000, 1, 3311787858630451200000, 1, 451287524451778560000, 1, 48168123888308960600064000000, 1, 10738530029998374912000000, 1
Offset: 0

Views

Author

Peter Luschny, Jun 13 2020

Keywords

Examples

			Rational sequence starts: 0, 1, 1/9, 0, -7/1350, 0, 31/52920, 0, -127/1134000, 0, 365/11290752, ...

Crossrefs

Cf. A335538 (numerators), A164555/A027642 (Bernoulli numbers).
Cf. A335264, A335265, A327497.

Programs

  • Maple
    a := s -> `if`(s = 1 or s = 0, s, -4*s^2*Zeta(1 - s)*Zeta(s)*(1 - 2^(1 - s))/Pi^s):
seq(denom(a(s)), s = 0..34);

Formula

a(n) = denominator(n*Bernoulli(n)*zeta(n)*(4-2^(3-n))/Pi^n) for n >= 2.

A327986 Denominators of the coefficients in the expansion of (4*sinh(sqrt(x)/2)^2*cosh(sqrt(x)))/x.

Original entry on oeis.org

1, 12, 360, 20160, 259200, 239500800, 43589145600, 1494484992000, 3201186852864000, 1216451004088320000, 11469395181404160000, 310224200866619719680000, 201645730563302817792000000, 21777738900836704321536000000, 132626429906095529318154240000000
Offset: 0

Views

Author

Peter Luschny, Oct 05 2019

Keywords

Examples

			r(n) = 1, 7/12, 31/360, 127/20160, 73/259200, 2047/239500800, 8191/43589145600, ...

Crossrefs

Numerators in A327497.

Programs

  • Maple
    gf := (4*sinh(sqrt(x)/2)^2*cosh(sqrt(x)))/x: ser := series(gf, x, 40):
seq(denom(coeff(ser, x, n)), n=0..14);

Formula

a(n) = denominator [x^n] (cosh(2*sqrt(x)) - 2*cosh(sqrt(x)) + 1)/x.
Showing 1-6 of 6 results.

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

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