A291561 -id:A291561 - cp's OEIS frontend

A292119 O.g.f. equals the square of the e.g.f. of A291561.

1, 10, 130, 2100, 40950, 943740, 25269300, 774635400, 26836251750, 1038607069500, 44448725821500, 2084869401615000, 106355178306877500, 5861473946222895000, 346999395775257225000, 21956626245257906202000, 1478562610889805715023750, 105561794005139231136877500, 7963731010308915234880987500, 632966979266333111428303275000, 52862553418201438508049805852500

Offset: 2

Paul D. Hanna, Sep 18 2017

nonn

A291561 is a diagonal in triangle A291560: a(n) = -A291560(n+1, n) for n >= 1; the e.g.f. of triangle A291560 equals arcsin( k*sin(x) ).

			O.g.f.: A(x) = x^2 + 10*x^3 + 130*x^4 + 2100*x^5 + 40950*x^6 + 943740*x^7 + 25269300*x^8 + 774635400*x^9 + 26836251750*x^10 + 1038607069500*x^11 + 44448725821500*x^12 + 2084869401615000*x^13 + 106355178306877500*x^14 + 5861473946222895000*x^15 + 346999395775257225000*x^16 + 21956626245257906202000*x^17 + 1478562610889805715023750*x^18 + ...
such that the square root of the g.f. equals the e.g.f. of A291561, which begins:
A(x)^(1/2) = x + 10*x^2/2! + 315*x^3/3! + 18900*x^4/4! + 1819125*x^5/5! + 255405150*x^6/6! + 49165491375*x^7/7! + 12417798393000*x^8/8! + 3981456609755625*x^9/9! + 1579311121869731250*x^10/10! + ... + A291561(n)*x^n/n! + ...

Cf. A291561, A291560.

{A291560(n, r) = (2*n-1)! * polcoeff( polcoeff( asin( k*sin(x + O(x^(2*n)))), 2*n-1, x), 2*r-1, k)}
{a(n) = polcoeff( sum(m=1,n,-A291560(m+1, m) * x^m / m! +x*O(x^n) )^2, n)}
for(n=2, 25, print1(a(n), ", "))

A291560 E.g.f. A(x,k) satisfies: sin(A(x,k)) = k * sin(x).

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 26 2017

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).

			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).

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

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

Cf. A291527 (variant).

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 *)

{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(""))

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

cp's OEIS Frontend

A292119 O.g.f. equals the square of the e.g.f. of A291561.

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