The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A160562

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A160562

Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).
(history; published version)

#40 by Michael De Vlieger at Mon Oct 30 09:52:50 EDT 2023

STATUS

proposed

approved

#39 by Jianing Song at Mon Oct 30 04:08:19 EDT 2023

STATUS

editing

proposed

#38 by Jianing Song at Mon Oct 30 04:08:15 EDT 2023

CROSSREFS

Cf. A002452 (column k=1), A002453 (column k=2), A000447 (right column k=n-1), A185375 (right column k=n-2).

#37 by Jianing Song at Mon Oct 30 04:07:34 EDT 2023

CROSSREFS

Cf. A002452 (column k=1), A002453 (column k=2), A000447 (right column k=n-1).

STATUS

approved

editing

#36 by Michael De Vlieger at Sun Oct 29 23:18:24 EDT 2023

STATUS

proposed

approved

#35 by Jianing Song at Sun Oct 29 23:02:18 EDT 2023

STATUS

editing

proposed

#34 by Jianing Song at Sun Oct 29 23:02:16 EDT 2023

FORMULA

T(n,k) = ((-1)^(n-k)*(2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sin(x)^(2*k+1) = ((2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sinh(x)^(2*k+1). Note that sin(x)^(2*k+1) = (Sum_{i=0..k} (-1)^i*binomial(2*k+1,k-i)*sin((2*i+1)*x))/(2^(2*k)). - Jianing Song, Oct 29 2023\

#33 by Jianing Song at Sun Oct 29 23:01:46 EDT 2023

FORMULA

T(n,k) = ((-1)^(n-k)*(2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sin(x)^(2*k+1) = ((2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sinh(x)^(2*k+1). Note that sin(x)^(2*k+1) = Sum_{i=0..k} (-1)^i*binomial(2*k+1,k-i)*sin((2*i+1)*x). - Jianing Song, Oct 29 2023\

#32 by Jianing Song at Sun Oct 29 23:00:22 EDT 2023

FORMULA

T(n,k) = ((-1)^(n-k)*(2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sin(x)^(2*k+1) = ((2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sinh(x)^(2*k+1). - Jianing Song, Oct 29 2023

STATUS

approved

editing

#31 by Joerg Arndt at Thu Mar 10 01:43:21 EST 2022

STATUS

editing

approved