A322734 Row sums of triangle A322731.

1, 1, 13, 301, 13049, 916441, 94195333, 13347584069, 2494336502897, 594306468307633, 175843898741580413, 63256176039805178717, 27187896853252573010537, 13760130793027073955444361, 8099868218813257097451686389, 5486947453030516571774107669621, 4238147510641905386674640667028193, 3702563543334459672543167863851104609, 3632508681950057312629014573578376827629, 3976725158932698015861683248077453527809421

Offset

0,3

Links

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

Formula

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

Example

E.g.f.: A(x) = 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! + ...
RELATED SERIES.
sqrt(A(x)^2 - 1) = 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! + ... + A322733(n)*x^(2*n+1)/(2*n+1)! + ...
A(x) = cosh( Integral D(x) dx ) where D(x) = A'(x)/sqrt(A(x)^2 - 1) 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) {A322731(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)! *polcoeff(polcoeff(Cx, 2*n-2*k, x), 2*k, y)}
a(n) = sum(k=0, n, A322731(n, k))
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A322731, A322733.

Sequence in context: A035272 A296319 A009119 * A331657 A142425 A234236

Adjacent sequences: A322731 A322732 A322733 * A322735 A322736 A322737

Keyword

nonn

Author

Paul D. Hanna, Jan 01 2019

Status

approved