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A322733 Row sums of triangle A322730. 3
1, 4, 56, 1856, 103936, 8893952, 1080485888, 176673603584, 37417114009600, 9963927777050624, 3258468236083331072, 1283800576046355447808, 599766792781193235398656, 327833344685493011492110336, 207273007354010852132258840576, 150092604050210703675500571656192, 123411833893721992920958314267803648, 114350314285874476606422143072852770816, 118597403963878781240540181376561622024192, 136853838009919098121511181384398866938331136 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 5 03:05 EDT 2020. Contains 335457 sequences. (Running on oeis4.)