A059837 Diagonal T(s,s) of triangle A059836.

1, 1, 4, 18, 144, 1200, 14400, 176400, 2822400, 45722880, 914457600, 18441561600, 442597478400, 10685567692800, 299195895398400, 8414884558080000, 269276305858560000, 8646761377013760000, 311283409572495360000

Offset: 1

Author

N. J. A. Sloane, Feb 25 2001

Keywords

• nonn
• easy

References

• S. G. Mikhlin, Constants in Some Inequalities of Analysis, Wiley, NY, 1986, see p. 59.

Crossrefs

Cf. A059836.

Programs

Maple

T := proc(s,t) option remember: if s=1 or t=1 then RETURN(1) fi: if t>1 and t mod 2 = 1 then RETURN(product((s-i)^2, i=1..(t-1)/2)) else RETURN((s-t/2)*product((s-i)^2, i=1..t/2-1)) fi: end: for s from 1 to 50 do printf(`%d,`, T(s,s)) od:

Mathematica

a[n_] := (n-1)! Binomial[n-1, Quotient[n-1, 2]];
Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Apr 29 2023 *)

Formula

T(s, s) = (s-1)^2 * T(s-1, s-1) / floor(s/2) - Larry Reeves.

a(n) = Sum_{k=0..n} (-1)^(n+k)*C(n, k)*Sum_{i=0..n} C(n, floor(i/2))*k^i. - Paul Barry, Aug 05 2004

a(n) = (n-1)!*binomial(n-1,floor(n-1,2)), n>=1.

E.g.f. is the integral of the o.g.f. of A001405. With offset 0: e.g.f. is o.g.f. of A001405.

Conjecture: +(n+1)*a(n) -2*n*a(n-1) -4*n*(n-1)^2*a(n-2)=0. - R. J. Mathar, Nov 24 2012

Extensions

More terms from James Sellers, Feb 26 2001 and from Larry Reeves (larryr(AT)acm.org), Feb 26 2001