A336239 Triangle read by rows, coefficients of polynomials in Pi^2, given by trigonometric double integrals over the unit square.

0, 16, -144, 36, 2304, -960, 96, -57600, 24000, -3600, 200, 2073600, -864000, 146880, -11520, 360, -101606400, 42336000, -7197120, 635040, -29400, 588, 6502809600, -2709504000, 460615680, -41932800, 2204160, -64512, 896

Offset

1,2

Comments

Formulas given by Srinivasa Raghava, see the picture in the links section.

Links

Table of n, a(n) for n=1..29.

Srinivasa Raghava, Trigonometric double integrals.

Formula

T(n, k) = Pi^(2*n-1) * [Pi^(2*k)] Integral_{x=0..1, y=0..1} sin(-Pi*(x^(1/n) + y^(1/n)) for n >= 2 and k = 0..n-2.

Example

Triangle starts:
                             [1] 0
                            [2] 16
                         [3] -144, 36
                      [4] 2304, -960, 96
                 [5] -57600, 24000, -3600, 200
           [6] 2073600, -864000, 146880, -11520, 360
    [7] -101606400, 42336000, -7197120, 635040, -29400, 588
[8] 6502809600, -2709504000, 460615680, -41932800, 2204160, -64512, 896
.
The first few polynomials are:
p_1(Pi) = 0;
p_2(Pi) = 16;
p_3(Pi) = 36*(Pi^2 - 4);
p_4(Pi) = 96*(Pi^4 - 10*Pi^2 + 24);
p_5(Pi) = 200*(Pi^6 - 18*Pi^4 + 120*Pi^2 - 288);
p_6(Pi) = 360*(Pi^8 - 32*Pi^6 + 408*Pi^4 - 2400*Pi^2 + 5760);

Maple

dblint := n -> Pi^(2*n-1)*int(sin(-Pi*(x^(1/n) + y^(1/n))), [x=0..1, y=0..1]):
Trow := proc(n) if n = 1 then return 0 fi; dblint(n);
seq(coeff(%, Pi, 2*k), k=0..n-2) end: seq(Trow(n), n=1..8);

Crossrefs

Sequence in context: A048533 A303145 A213349 * A332105 A032444 A358263

Adjacent sequences: A336236 A336237 A336238 * A336240 A336241 A336242

Keyword

sign,tabf

Author

Peter Luschny, Jul 13 2020

Status

approved