Template:Sequence of the Day for October 16

Intended for: October 16, 2012

Timetable

• First draft entered by Daniel Forgues October 16, 2011 ✓
• Draft reviewed by Alonso del Arte on October 19, 2011 ✓
• Draft to be approved by September 16, 2012

Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.

---

A117972: Numerator of

ζ  ′( − 2n), n   ≥   0

.

{ 1, −1, 3, − 45, 315, −14175, 467775, − 42567525, 638512875, ... }

This sequence is related to the correlation function in Montgomery’s pair correlation conjecture for the nontrivial zeros of the Riemann zeta function

R2 (u)  =  1 − sincπ 2 u + δ (u)  =  1 − sin π  u π  u   2 + δ (u),

where

sincπ u

is the normalized sinc function.
Maclaurin series for

1  −  sin x x   2 = x 2 3  −  2 x 4 45 + x 6 315  −  2 x 8 14175 + 2 x 10 467775  −  ⋯

• Numerators of Maclaurin series for

  1  −  sin x x   2

  : A048896

  2 A000120 (n +1)  − 1, n   ≥   1

  . Also, maximal power of 2 dividing

  n

  th Catalan number.

{1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 2, 4, 4, 8, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, ...}

• Denominators of Maclaurin series for

  1  −  sin x x   2

  : A117972 Numerator of

  ζ  ′( − 2n), n   ≥   2

  .

{3, –45, 315, –14175, 467775, –42567525, 638512875, –97692469875, ...}

where A000120 is number of 1s in binary expansion of

n

(or the binary weight of

n, n   ≥   2

)

{1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, ...}

Why are the denominators of Maclaurin series for

1  −  sin x x   2

corresponding to the numerators of

ζ  ′( − 2n), n   ≥   2

?
Why are the numerators of Maclaurin series for

1  −  sin x x   2

corresponding to maximal power of 2 dividing

n

th,

n   ≥   1

, Catalan number?