1,5

Compare to the law of sines of a spherical triangle: sin(A)/sin(a) = k.

The series reversion of e.g.f. A(x,k) wrt x equals A(x, 1/k).

Paul D. Hanna, Table of n, a(n) for n = 1..1035 for rows 1..45 of this triangle in flattened form.

E.g.f. A(x,k) = Sum_{n>=1, r=1..n} T(n,r) * x^(2*n-1) * k^(2*r-1)/(2*n-1)!, satisfies:

(1) sin(A(x,k)) = k * sin(x).

(2) A(x,k) = asin(k * sin(x)).

(3) A( A(x,k), 1/k) = x.

(4) sin( A^r(x,k) ) = k^r * sin(x) where A^r(x,k) = A(x,k^r) is the r-th iteration of A(x,k) wrt x, with A^0(x,k) = x.

(5) A(x,1) = x.

Row sums of n-th row equals zero for n>1.

T(n+1,1) = (-1)^n for n>=0.

T(n+1,2) = (-1)^(n-1) * (9^n - 1)/8 for n>=1.

T(n+1,n+1) = ( (2*n)! / (n!*2^n) )^2 = A001818(n) for n>=0.

T(n, r) = (-1)^n / ((2*r - 1)^2 * 4^(2*r - 1)) * ((2*r)! / r!)^2 * Sum_{i=1..n} (-1)^i * (2*i - 1)^(2*n - 1) / ((r - i)! * (r + i - 1)!). - Vjekoslav-Leonard Prcic, Oct 10 2018

This triangle of coefficients T(n,r) in e.g.f. A(x,k) begins:

[1],

[-1, 1],

[1, -10, 9],

[-1, 91, -315, 225],

[1, -820, 8694, -18900, 11025],

[-1, 7381, -224730, 1143450, -1819125, 893025],

[1, -66430, 5684679, -61647300, 203378175, -255405150, 108056025],

[-1, 597871, -142714845, 3162834675, -19494349875, 47377655325, -49165491375, 18261468225],

[1, -5380840, 3573251964, -158546770200, 1734021238950, -7311738634200, 14041664336700, -12417798393000, 4108830350625],

[-1, 48427561, -89379726660, 7858123038900, -148224512094750, 1025176095093150, -3257761647640500, 5167045911327300, -3981456609755625, 1187451971330625],

[1, -435848050, 2234929014549, -387282522072600, 12391233508580850, -136052492985945900, 674608025957515650, -1713147048499887000, 2313226290268018125, -1579311121869731250, 428670161650355625], ...

where e.g.f. A(x,k) = Sum_{n>=1, r=1..n} T(n,r) * x^(2*n-1) * k^(2*r-1) / (2*n-1)!.

E.g.f.: A(x,k) = k*x + (k^3 - k)*x^3/3! + (9*k^5 - 10*k^3 + k)*x^5/5! + (225*k^7 - 315*k^5 + 91*k^3 - k)*x^7/7! + (11025*k^9 - 18900*k^7 + 8694*k^5 - 820*k^3 + k)*x^9/9! + (893025*k^11 - 1819125*k^9 + 1143450*k^7 - 224730*k^5 + 7381*k^3 - k)*x^11/11! + (108056025*k^13 - 255405150*k^11 + 203378175*k^9 - 61647300*k^7 + 5684679*k^5 - 66430*k^3 + k)*x^13/13! + (18261468225*k^15 - 49165491375*k^13 + 47377655325*k^11 - 19494349875*k^9 + 3162834675*k^7 - 142714845*k^5 + 597871*k^3 - k)*x^15/15! + (4108830350625*k^17 - 12417798393000*k^15 + 14041664336700*k^13 - 7311738634200*k^11 + 1734021238950*k^9 - 158546770200*k^7 + 3573251964*k^5 - 5380840*k^3 + k)*x^17/17! + (1187451971330625*k^19 - 3981456609755625*k^17 + 5167045911327300*k^15 - 3257761647640500*k^13 + 1025176095093150*k^11 - 148224512094750*k^9 + 7858123038900*k^7 - 89379726660*k^5 + 48427561*k^3 - k)*x^19/19! +...

such that sin(A(x,k)) = k * sin(x).

T[n_, k_] := If[ n < 1, 0, (2 n - 1)! Coefficient[ SeriesCoefficient[ ArcSin[y Sin[x]], {x, 0, 2 n - 1}], y, 2 k - 1]]; (* Michael Somos, Jul 03 2018 *)

T[n_, k_] := ((-1)^n/((2*k - 1)^2*4^(2*k - 1)))*((2*k)!/k!)^2 * Sum[((-1)^i*(2*i - 1)^(2*n - 1))/((k - i)!*(k + i - 1)!), {i, 1, n}]; (* Vjekoslav-Leonard Prcic, Oct 10 2018 *)

(PARI) {T(n, r) = (2*n-1)! * polcoeff( polcoeff( asin( k*sin(x + O(x^(2*n)))), 2*n-1, x), 2*r-1, k)}

for(n=1, 10, for(r=1, n, print1(T(n, r), ", ")); print(""))

Cf. A002452 (column 1), A001818 (diagonal), A291561 (diagonal), A291562 (central terms).

Cf. A291527 (variant).

Sequence in context: A118768 A318255 A008956 * A382714 A259567 A022966

Adjacent sequences: A291557 A291558 A291559 * A291561 A291562 A291563

sign,tabl

Paul D. Hanna, Aug 26 2017

approved