The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A324440 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A324440

a(n) = Product_{i=1..n, j=1..n} (i^7 + j^7).
(history; published version)

#21 by Vaclav Kotesovec at Sat Dec 02 06:02:59 EST 2023

STATUS

editing

approved

#20 by Vaclav Kotesovec at Sat Dec 02 05:49:26 EST 2023

CROSSREFS

Cf. A079478, A324403, A324426, A324437, A324438, A324439, A367834.

STATUS

approved

editing

#19 by Vaclav Kotesovec at Fri Dec 01 15:30:08 EST 2023

STATUS

editing

approved

#18 by Vaclav Kotesovec at Fri Dec 01 15:30:01 EST 2023

FORMULA

c = 0.06805650384668932892961265220725107128262312556515094156663626480587814403851617068056503846689328929612652207251071282623125565150941566636264805878144...

STATUS

approved

editing

#17 by Vaclav Kotesovec at Fri Dec 01 15:28:59 EST 2023

STATUS

editing

approved

#16 by Vaclav Kotesovec at Fri Dec 01 15:28:04 EST 2023

FORMULA

From Vaclav Kotesovec, Dec 01 2023: (Start)

a(n) ~ c * exp(n*(n+1)*s - 7*n*(n-2)/2) * n^(7*(n^2 - 1/4)), where

s = Sum_{j>=1} (-1)^(j+1)/(j*(1 + 7*j)) = Pi/(2*sin(Pi/7)) + 3*log(2)/2 - 7 - cos(Pi/7) * log(2*sin(Pi/14)^2) - log(2*sin(3*Pi/14)^2) * sin(Pi/14) + log(cos(3*Pi/14)*cos(Pi/7) / sin(Pi/7)) * sin(3*Pi/14) = 0.10150386842315637912206687298894641634315636548242136512503... and

c = 0.06805650384668932892961265220725107128262312556515094156663626480587814403851617...

Equivalently, s = log(2) - HurwitzLerchPhi(-1, 1, 1 + 1/7). (End)

#15 by Vaclav Kotesovec at Fri Dec 01 15:12:49 EST 2023

COMMENTS

For m>1, Product_{j=1..n, k=1..n} (j^m + k^m) ~ c(m) * exp(n*(n+1)*s(m) - m*n*(n-2)/2) * n^(m*(n^2 - 1/4)), where s(m) = Sum_{j>=1} (-1)^(j+1)/(j*(1 + m*j)) and c(m) is a constant (dependent only on m). Equivalently, s(m) = log(2) - HurwitzLerchPhi(-1, 1, 1 + 1/m). - Vaclav Kotesovec, Dec 01 2023

STATUS

approved

editing

#14 by Alois P. Heinz at Sun Nov 26 17:25:54 EST 2023

STATUS

proposed

approved

#13 by Michel Marcus at Sun Nov 26 15:55:39 EST 2023

STATUS

editing

proposed

#12 by Michel Marcus at Sun Nov 26 15:55:36 EST 2023

FORMULA

Limit_{n->infinityoo} (a(n)^(1/n^2))/n^7 = 2^(3/2) * (cos(3*Pi/14) / tan(Pi/7))^sin(3*Pi/14) / ((cos(Pi/14)*tan(3*Pi/14))^sin(Pi/14) * (sin(Pi/7)*tan(Pi/14))^cos(Pi/7)) * exp((Pi/sin(Pi/7) - 21)/2) = 0.0334234967249533921390751418772468470887965377...

STATUS

proposed

editing