A228143 Determinant of the (n+1) X (n+1) Hankel-type matrix with (i,j)-entry equal to A005259(i+j) for all i,j = 0,...,n.

1, 48, 161856, 39002646528, 674708032182398976, 839431510934341028210638848, 75178263784150214825106859877233852416, 484905075185415831301477770434885768003422223597568, 225327830550164300895512117291590826401931052058453494726924435456, 7544971365077550026405694467600069733983243666195122776655161969325034606646263808

Offset: 0

Author

Zhi-Wei Sun, Aug 14 2013

Keywords

• nonn
• easy

Comments

Conjecture: a(n)/24^n is always a positive integer. Similarly, if b(n) denotes the (n+1) X (n+1) Hankel-type determinant with (i,j)-entry equal to A005258(i+j) for all i,j = 0,...,n, then b(n)/10^n is always a positive integer; also, if p is a prime with floor(p/10) odd and p is not congruent to 31 or 39 modulo 40, then p divides b((p-1)/2).

Conjecture: if A(x) = 1 + 48*x + 161856*x^2 + ... denotes the o.g.f. then A(x/3)^(1/8) has integer coefficients (checked up to x^30). - Peter Bala, Apr 22 2018

Examples

			a(0) = 1 since A005259(0+0) = 1.
A(x/3)^(1/8) = 1 + 2*x + 2234*x^2 + 180536476*x^3 + 1041213553880806*x^4 + 431806318205326490858140*x^5 + 12890648790962619413782473229673892*x^6 + 27715196341006992690056202634389754569453086008*x^7 + 4292939920556011562306504817069205738464230629574745210785030*x^8 + 47915532217380103151430239883031701095737468980424637791531495548671526291244*x^9 + .... - _Peter Bala_, Apr 22 2018
		

Crossrefs

Cf. A005258, A005259, A225776.

Programs

Mathematica

A[n_]:=Sum[Binomial[n,k]^2*Binomial[n+k,k]^2,{k,0,n}]; a[n_]:=Det[Table[A[i+j],{i,0,n},{j,0,n}]]; Table[a[n],{n,0,10}]