The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A167584 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A167584

The ED4 array read by antidiagonals.
(history; published version)

#26 by Bruno Berselli at Fri Jan 20 02:52:25 EST 2017

STATUS

proposed

approved

#25 by Jon E. Schoenfield at Fri Jan 20 01:33:31 EST 2017

STATUS

editing

proposed

#24 by Jon E. Schoenfield at Fri Jan 20 01:33:29 EST 2017

COMMENTS

The Madhava-Gregory-Leibniz series representation for Pi/4 is the case m = 0 of the following more general result: for m = 0,1,2,... there holds 1/(2*m)! * Pi/4 = Sum_{k >= 0} ( (-1)^(m+k) * 1/Product_{j = -m .. m} (2*k + 1 + 2*j) ). The entries of this table are given by truncating these series to n-1 terms and then scaling by certain double factorials - - see the formula below. (End)

FORMULA

Sum(_{k=0..n-1} (-1)^k*binomial(n-1,k)*a(n,m-k),k=0..n-1) = 2^(n-1)*n!

STATUS

proposed

editing

#23 by G. C. Greubel at Fri Jan 20 01:24:55 EST 2017

STATUS

editing

proposed

#22 by G. C. Greubel at Fri Jan 20 01:24:23 EST 2017

LINKS

G. C. Greubel, <a href="/A167584/b167584.txt">Table of n, a(n) for the first 50 rows, flattened</a>

MATHEMATICA

T[0, k_] := 0; T[1, k_] := 1; T[n_, k_] := T[n, k] = (4*k - 2)*T[n - 1, k] + (2*n + 2*k - 5)*(2*n - 2*k - 1)*T[n - 2, k]; Table[T[n - k, k], {n, 2, 12}, {k, 1, n - 1}] (* G. C. Greubel, Jan 20 2017 *)

STATUS

approved

editing

#21 by Michael Somos at Wed Nov 09 12:59:44 EST 2016

STATUS

proposed

approved

#20 by Peter Bala at Tue Nov 08 05:18:38 EST 2016

STATUS

editing

proposed

#19 by Peter Bala at Tue Nov 08 05:16:55 EST 2016

FORMULA

Note the double factorial for a negative odd integer N is defined in terms of the gamma function as N!! = 2^((N+1)/2)* Gamma(N + 1/2 + 1)/sqrt(Pi).

STATUS

approved

editing

Discussion

Tue Nov 08

05:18

Peter Bala: Corrected one of my mistakes.

#18 by Bruno Berselli at Mon Nov 07 06:05:32 EST 2016

STATUS

editing

approved

#17 by Bruno Berselli at Mon Nov 07 06:05:28 EST 2016

MAPLE

- _# _Peter Bala_, Nov 06 2016

STATUS

proposed

editing