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

0,2

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

This was proved by an autonomous AI agent, see the Tsoukalas paper and the Lean file. The proof uses two divisibility facts about the Apéry-like Hankel determinant a(n): working mod 3 and mod 4, row-reducing the matrix by unitriangular P matrices peels off diagonal factors, giving 3^n | a(n) and 4^n | a(n) (with an extra factor 16 from a(1)=48). Hence B(n) = a(n)/3^n is 1 + 16*(integer series). Writing B = 1 + 16Y, it constructs an eighth root coefficient-by-coefficient: solving (1+2X)^8 = 1 + 16*(X + P(X)) recursively via a valuation argument shows Y is realized, so B, and thus the scaled generating function, is a perfect eighth power (Summary by Opus 4.7). - Ralf Stephan, May 25 2026

Links

Table of n, a(n) for n=0..9.

Google Deepmind, AlphaProof Nexus: A228143 Lean file

George Tsoukalas et al., Advancing Mathematics Research with AI-Driven Formal Proof Search, arXiv:2605.22763 [cs.AI], 2026.

Example

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

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}]

Crossrefs

Cf. A005258, A005259, A225776.

Sequence in context: A081262 A381626 A238001 * A340186 A008704 A037947

Adjacent sequences: A228140 A228141 A228142 * A228144 A228145 A228146

Keyword

nonn,easy

Author

Zhi-Wei Sun, Aug 14 2013

Status

approved