The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A340940 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A340940

Rectangular table of coefficients T(k,n) in row functions R(k,x) = sum_{n>=0} T(k,n)*x^n that satisfy the condition: Sum_{n>=0} x^n/(1 - x*R(k,x)^(n+k)) = Sum_{n>=0} x^n*R(k,x)^n/(1 - x*R(k,x)^(k*n+k-1)), for k >= 0, read here by antidiagonals.
(history; published version)

#17 by Paul D. Hanna at Fri Jun 18 16:38:49 EDT 2021

STATUS

editing

approved

#16 by Paul D. Hanna at Fri Jun 18 16:38:45 EDT 2021

FORMULA

(5) B(k,x) = Sum_{n>=0} x^(2*n) * AR(k,x)^(n^2+k*n) * (1 - x^2*AR(k,x)^(2*n+k)) / ((1 - x*AR(k,x)^n)*(1 - x*AR(k,x)^(n+k))),

(6) B(k,x) = Sum_{n>=0} x^(2*n) * AR(k,x)^(k*n^2+k*n) * (1 - x^2*AR(k,x)^(2*k*n+k)) / ((1 - x*AR(k,x)^(k*n+1))*(1 - x*AR(k,x)^(k*n+k-1)));

STATUS

approved

editing

#15 by Paul D. Hanna at Mon Mar 08 12:03:53 EST 2021

STATUS

editing

approved

#14 by Paul D. Hanna at Mon Mar 08 12:03:50 EST 2021

FORMULA

(6) B(k,x) = Sum_{n>=0} x^(2*n) * A(x)^(k*n^2+k*n) * (1 + - x^2*A(x)^(2*k*n+k)) / ((1 - x*A(x)^(k*n+1))*(1 - x*A(x)^(k*n+k-1)));

STATUS

approved

editing

#13 by Paul D. Hanna at Tue Feb 16 12:03:47 EST 2021

STATUS

editing

approved

#12 by Paul D. Hanna at Tue Feb 16 12:03:44 EST 2021

FORMULA

(4) B(k,x) = Sum_{n>=0} x^n*R(k,x)^(k*n) / (1 - x*R(k,x)^n);,

(5) B(k,x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+k*n) * (1 - x^2*A(x)^(2*n+k)) / ((1 - x*A(x)^n)*(1 - x*A(x)^(n+k))),

(6) B(k,x) = Sum_{n>=0} x^(2*n) * A(x)^(k*n^2+k*n) * (1 + x*A(x)^(2*k*n+k)) / ((1 - x*A(x)^(k*n+1))*(1 - x*A(x)^(k*n+k-1)));

CROSSREFS

Cf. A340941 (row 0), A340942 (row 1), A340894 (row 2), A340895 (row 3), A340943 (row 4), A341376 (row 5).

STATUS

approved

editing

#11 by Paul D. Hanna at Sat Feb 06 11:55:38 EST 2021

STATUS

editing

approved

#10 by Paul D. Hanna at Sat Feb 06 11:55:36 EST 2021

EXAMPLE

Note that this condition is satisfied by every row function in the table; however, the rows k=0 and k=1 cannot be uniquely determined by this condition alonerequire special handling to determine the coefficients; see A340941 (row 0) and A340942 (row 1) for further information.

STATUS

approved

editing

#9 by Paul D. Hanna at Thu Feb 04 11:23:51 EST 2021

STATUS

editing

approved

#8 by Paul D. Hanna at Thu Feb 04 11:23:49 EST 2021

EXAMPLE

where the row functions R(k,x) satisfy the condition:

Sum_{n>=0} x^n/(1 - x*RB(k,x)^(n+k)) ) = Sum_{n>=0} x^n*R(k,x)^n/(1 - x*R(k,x)^(k*mn+k-1)). ) and

B(k,x) = Sum_{n>=0} x^n*R(k,x)^n/(1 - x*R(k,x)^(k*m+k-1)) are equal.

STATUS

approved

editing