The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A283151

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A283151

Triangle read by rows: Riordan array (1/(1-9x)^(2/3), x/(9x-1)).
(history; published version)

#44 by Joerg Arndt at Fri Sep 03 01:59:05 EDT 2021

STATUS

reviewed

approved

#43 by Michel Marcus at Fri Sep 03 00:20:04 EDT 2021

STATUS

proposed

reviewed

#42 by Jon E. Schoenfield at Thu Sep 02 22:03:47 EDT 2021

STATUS

editing

proposed

#41 by Jon E. Schoenfield at Thu Sep 02 22:03:45 EDT 2021

FORMULA

G(x) = (1/(1 - 9*b*x)^(2/3) ) * F(x/(1 - 9*b*x)) iff F(x) = (1/(1 + 9*b*x)^(2/3) ) * G(x/(1 + 9*b*x)).

STATUS

approved

editing

#40 by Bruno Berselli at Wed Sep 01 06:21:14 EDT 2021

STATUS

proposed

approved

#39 by Michel Marcus at Tue Aug 31 14:55:17 EDT 2021

STATUS

editing

proposed

#38 by Michel Marcus at Tue Aug 31 14:55:13 EDT 2021

FORMULA

Equivalently, if F(x) = Sum_{n >= 0} f(n)*x^n and G(x) = Sum_{n >= 0} g(n)*x^n are a pair of formal power series then

STATUS

proposed

editing

#37 by Peter Bala at Tue Aug 31 11:55:26 EDT 2021

STATUS

editing

proposed

#36 by Peter Bala at Tue Aug 31 11:55:22 EDT 2021

FORMULA

G(x) = 1/(1 - 9*b*x)^(2/3) * F(x/(1 - 9*b*a*x)) iff F(x) = 1/(1 + 9*b*x)^(2/3) * G(x/(1 + 9*b*x)).

#35 by Peter Bala at Tue Aug 31 11:49:50 EDT 2021

LINKS

H. Prodinger, <a href="https://www.fq.math.ca/Scanned/32-5/prodinger.pdf">Some information about the binomial transform</a>, The Fibonacci Quarterly, 32, 1994, 412-415.

FORMULA

T(n,k) = (-1)^k*binomial(n-1/3, n-k)*9^(n-k).

Analogous to the binomial transform we have the following sequence transformation formula: g(n) = Sum_{k = 0..n} T(n,k)*ab^(n-k)*f(k) iff f(n) = Sum_{k = 0..n} T(n,k)*ab^(n-k)*g(k). See Prodinger, bottom of p. 413, with b replaced with 9*b, c = -1 and d = 2/3.

G(x) = 1/(1 - 9*ab*x)^(2/3) * F(x/(1 - 9b*a*x)) iff F(x) = 1/(1 + 9*ab*x)^(2/3) * G(x/(1 + 9*ab*x)).