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

%I A291561 #14 Jan 29 2025 08:12:56
%S A291561 1,10,315,18900,1819125,255405150,49165491375,12417798393000,
%T A291561 3981456609755625,1579311121869731250,759174856282779811875,
%U A291561 434800144961955710437500,292511797523155704196828125,228384211143079261353677343750,204811697921525723306815646484375,209071781238293458351597411931250000,241020562808770177455950891441994140625,311597054671244174125111099536008660156250
%N A291561 Diagonal in triangle A291560: a(n) = -A291560(n+1, n) for n>=1.
%C A291561 The e.g.f. G(x,k) of triangle A291560 satisfies: sin(G(x,k)) = k * sin(x).
%F A291561 Conjecture: a(n) = 4^n*gamma(-1/2 + n)*gamma(3/2 + n)*n/(3*Pi). - _Thomas Scheuerle_, Jan 27 2025
%e A291561 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! +...
%e A291561 Notice that the square of the e.g.f is an integer series:
%e A291561 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 +...
%o A291561 (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)}
%o A291561 for(n=1, 20, print1(-A291560(n+1, n), ", "))
%Y A291561 Cf. A291560, A291562, A292119.
%K A291561 nonn
%O A291561 1,2
%A A291561 _Paul D. Hanna_, Sep 03 2017