Revision History for A009006
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#99 by N. J. A. Sloane at Fri Mar 08 16:13:59 EST 2024
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#98 by Peter Luschny at Fri Mar 08 03:43:53 EST 2024
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#97 by Peter Luschny at Fri Mar 08 03:43:15 EST 2024
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| COMMENTS
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If u(0) = 1 and u(n) = -Sum_{k = 0..n-1} u(k)*binomial(n,k)*2^(n-k-1) then a(n) = abs(u(n)) (in fact, u(n) = 1, -1, 0, 2, 0, -16, 0, 272, ...). - Robert FERREOL, Dec 30 2006
Sum_{k=0..n} A075263(n,k)*2^k = 1,-1,0,2,0,-16,0,272,0,-7936,0,... for n=0, 1, 2, 3, 4, ..., respectively. - Philippe Deléham, Aug 20 2007
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| FORMULA
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From Robert FERREOL, Dec 30 2006: (Start)
a(n) = 2^n*abs(Euler(n,0)) where Euler(n,x) is the n-th Eulerian polynomial. - _Robert FERREOL_, Dec 30 2006.
a(n) = abs(u(n)) where u(n) = -Sum_{k=0..n-1} u(k)*binomial(n, k)*2^(n-k-1) with u(0) = 1. (End)
Sum_{k=0..n} A075263(n, k)*2^k = 1,-1,0,2,0,-16,0,272,0,-7936,0,... for n = 0, 1, 2, 3, 4, ..., respectively. - Philippe Deléham, Aug 20 2007
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| STATUS
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proposed
editing
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Discussion
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| Fri Mar 08
| 03:43
| Peter Luschny: Moved as suggested.
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#96 by Peter Bala at Thu Mar 07 16:52:34 EST 2024
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Discussion
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| Thu Mar 07
| 17:05
| Michel Marcus: I wonder why not write: a(n) = abs(u(n)) where u(n) = -Sum_{k = 0..n-1} u(k)*binomial(n,k)*2^(n-k-1) with u(0)=1 ... and so this could go to the formula section?
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#95 by Peter Bala at Thu Mar 07 16:51:12 EST 2024
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| COMMENTS
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If bu(0)=) = 1 and bu(n+1) = -Sum_{k= = 0..n-1} u(k)*binomial(n,k)*2^(n-k-1) then a(n) = abs(bu(n)) (in fact, bu(n) = 1,, -1,, 0,-, 2,, 0,, -16,, 0,-, 272,...). - _, ...). - _Robert FERREOL_, Dec 30 2006
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| KEYWORD
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nonn,easy
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| STATUS
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approved
editing
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Discussion
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| Thu Mar 07
| 16:52
| Peter Bala: I think this is what the author intended.
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#94 by Charles R Greathouse IV at Thu Sep 08 08:44:37 EDT 2022
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| PROG
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(MAGMAMagma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1 + Tan(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 21 2018
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Discussion
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| Thu Sep 08
| 08:44
| OEIS Server: https://oeis.org/edit/global/2944
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#93 by Bruno Berselli at Mon Jun 14 11:52:45 EDT 2021
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#92 by Vaclav Kotesovec at Mon Jun 14 11:44:07 EDT 2021
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#91 by Peter Luschny at Mon Jun 14 10:38:01 EDT 2021
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#90 by Peter Luschny at Mon Jun 14 10:25:38 EDT 2021
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| MATHEMATICA
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A009006[n_] := Cos[Pi (n-1) / 2] (4^(n+1) - 2^(n+1)) * BBernoulliB[n+1] / (n+1); a[0] := 1; ; Table[A009006[n], {n, 0, 30}] (* _Peter Luschny_, Jun 14 2021 *)
Table[A009006[n], {n, 0, 30}] (* Peter Luschny, Jun 14 2021 *)
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Discussion
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| Mon Jun 14
| 10:27
| Peter Luschny: Yes, yes, thanks. Unfortunately, Mathematica has not yet adopted my conventions :)
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| 10:37
| Peter Luschny: With one serious disadvantage, by the way, as I have just seen: Mathematica cannot even plot this function with BernoulliB! With "my" 'B' you can do that and you can see a nice non-negative wave over the positive real numbers. Doesn't matter, for the present purpose it's enough.
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