login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Row sums of triangle A322731.
3

%I #11 Jan 10 2019 23:02:47

%S 1,1,13,301,13049,916441,94195333,13347584069,2494336502897,

%T 594306468307633,175843898741580413,63256176039805178717,

%U 27187896853252573010537,13760130793027073955444361,8099868218813257097451686389,5486947453030516571774107669621,4238147510641905386674640667028193,3702563543334459672543167863851104609,3632508681950057312629014573578376827629,3976725158932698015861683248077453527809421

%N Row sums of triangle A322731.

%H Paul D. Hanna, <a href="/A322734/b322734.txt">Table of n, a(n) for n = 0..49</a>

%F 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)!.

%e 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! + ...

%e RELATED SERIES.

%e 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)! + ...

%e A(x) = cosh( Integral D(x) dx ) where D(x) = A'(x)/sqrt(A(x)^2 - 1) begins

%e 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! + ...

%o (PARI) {A322731(n, k) = my(Sx=x, Sy=y, Cx=1, Cy=1); for(i=1, 2*n,

%o Sx = intformal( Cx*Cy +x*O(x^(2*n)), x);

%o Cx = 1 + intformal( Sx*Cy, x);

%o Sy = intformal( Cy*Cx +y*O(y^(2*k)), y);

%o Cy = 1 + intformal( Sy*Cx, y));

%o (2*n)! *polcoeff(polcoeff(Cx, 2*n-2*k, x), 2*k, y)}

%o a(n) = sum(k=0, n, A322731(n, k))

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A322731, A322733.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 01 2019