A291561 Diagonal in triangle A291560: a(n) = -A291560(n+1, n) for n>=1.

1, 10, 315, 18900, 1819125, 255405150, 49165491375, 12417798393000, 3981456609755625, 1579311121869731250, 759174856282779811875, 434800144961955710437500, 292511797523155704196828125, 228384211143079261353677343750, 204811697921525723306815646484375, 209071781238293458351597411931250000, 241020562808770177455950891441994140625, 311597054671244174125111099536008660156250

Offset: 1

Author

Paul D. Hanna, Sep 03 2017

Keywords

• nonn

Comments

The e.g.f. G(x,k) of triangle A291560 satisfies: sin(G(x,k)) = k * sin(x).

Examples

			E.g.f.: A(x) = 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! +...
Notice that the square of the e.g.f is an integer series:
A(x)^2 = 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 +...+ A292119(n)*x^n +...
		

Crossrefs

Cf. A291560, A291562, A292119.

Programs

PARI

{A291560(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, 20, print1(-A291560(n+1, n), ", "))

Formula

Conjecture: a(n) = 4^n*gamma(-1/2 + n)*gamma(3/2 + n)*n/(3*Pi). - Thomas Scheuerle, Jan 27 2025