A322733 Row sums of triangle A322730.

1, 4, 56, 1856, 103936, 8893952, 1080485888, 176673603584, 37417114009600, 9963927777050624, 3258468236083331072, 1283800576046355447808, 599766792781193235398656, 327833344685493011492110336, 207273007354010852132258840576, 150092604050210703675500571656192, 123411833893721992920958314267803648, 114350314285874476606422143072852770816, 118597403963878781240540181376561622024192, 136853838009919098121511181384398866938331136

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..49

Formula

E.g.f. A(x) = S(x,y=x) given S(x,y) = Integral C(x,y)*C(y,x) dx such that C(x,y)^2 - S(x,y)^2 = 1 and C(y,x) = 1 + Integral S(y,x)*C(x,y) dy, where A(x) = Sum_{n>=0} a(n) * x^(2*n+1)/(2*n+1)!.

Example

E.g.f.: A(x) = x + 4*x^3/3! + 56*x^5/5! + 1856*x^7/7! + 103936*x^9/9! + 8893952*x^11/11! + 1080485888*x^13/13! + 176673603584*x^15/15! + 37417114009600*x^17/17! + 9963927777050624*x^19/19! + ...
RELATED SERIES.
sqrt(1 + A(x)^2) = 1 + x^2/2! + 13*x^4/4! + 301*x^6/6! + 13049*x^8/8! + 916441*x^10/10! + 94195333*x^12/12! + 13347584069*x^14/14! + 2494336502897*x^16/16! + 594306468307633*x^18/18! + ... + A322734(n)*x^(2*n)/(2*n)! + ...
A(x) = sinh( Integral D(x) dx ) where D(x) = A'(x)/sqrt(1 + A(x)^2) begins
D(x) = 1 + 3*x^2/2! + 25*x^4/4! + 595*x^6/6! + 26193*x^8/8! + 1832611*x^10/10! + 188365801*x^12/12! + 26696014003*x^14/14! + 4988672502305*x^16/16! + 1188611267890243*x^18/18! + ...

Prog

(PARI) {A322730(n, k) = my(Sx=x, Sy=y, Cx=1, Cy=1); for(i=1, 2*n,
Sx = intformal( Cx*Cy +x*O(x^(2*n)), x);
Cx = 1 + intformal( Sx*Cy, x);
Sy = intformal( Cy*Cx +y*O(y^(2*k)), y);
Cy = 1 + intformal( Sy*Cx, y));
(2*n+1)! *polcoeff(polcoeff(Sx, 2*n+1-2*k, x), 2*k, y)}
a(n) = sum(k=0, n, A322730(n, k))
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A322730, A322734.

Sequence in context: A009159 A013055 A243486 * A012983 A012959 A013113

Adjacent sequences: A322730 A322731 A322732 * A322734 A322735 A322736

Keyword

nonn

Author

Paul D. Hanna, Jan 01 2019

Status

approved