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

1,3

References

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

Links

Table of n, a(n) for n=1..19.

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

From Amiram Eldar, Dec 04 2025: (Start)

Sum_{n>=1} 1/a(n) = 3/2 + (Pi/4)*(2*StruveL(0, 1/2) + StruveL(1, 1/2)), where StruveL is the modified Struve function.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - (Pi/4)*(StruveL(0, 1/2) + StruveL(1, 1/2)). (End)

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 *)

Crossrefs

Cf. A001405, A059836.

Sequence in context: A382813 A370774 A060841 * A220266 A218917 A054759

Adjacent sequences: A059834 A059835 A059836 * A059838 A059839 A059840

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 25 2001

Extensions

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

Status

approved