A225776 -id:A225776 - cp's OEIS frontend

A228289 Determinant of the p_n X p_n matrix with (i,j)-entry equal to D(i+j) for all i,j = 0,...,p_n-1, where D(k) = A002895(k) is the k-th Domb number and p_n is the n-th prime.

12, 2448, 428587718400, 4994319435309277891448832, 191901511752240055024005979549622856313555581586068578283027431424, 637213222716753775758429677219909335140503764595701930312765250413280716374852064945052319744

Offset: 1

Zhi-Wei Sun, Aug 19 2013

nonn

Conjecture: If p_n == 1 (mod 3) and p_n = x^2 + 3*y^2 with x and y integers, then we have a(n) == (-1)^{(p_n-1)/2}*(4*x^2-2*p_n) (mod p_n^2). In the case p_n == 2 (mod 3), we have a(n) == 0 (mod p_n^2).
Zhi-Wei Sun also made the following similar conjecture:
If p is an odd prime and b(p) is the p X p determinant with (i,j)-entry equal to A053175(i+j) for all i,j = 0,...,p-1, then we have the congruence b(p) == (-1)^{(p-1)/2} (mod p^2).

Zhi-Wei Sun, Conjectures and results on x^2 mod p^2 with 4*p = x^2 + d*y^2, in: Number Theory and Related Area (eds., Y. Ouyang, C. Xing, F. Xu and P. Zhang), Higher Education Press & International Press, Beijing and Boston, 2013, pp. 147-195.

Cf. A002895, A053175, A225776, A228143.

d[n_]:=Sum[Binomial[n,k]^2*Binomial[2k,k]Binomial[2(n-k),n-k],{k,0,n}]
a[n_]:=Det[Table[d[i+j],{i,0,Prime[n]-1},{j,0,Prime[n]-1}]]
Table[a[n],{n,1,8}]

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.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Aug 14 2013

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

			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

Cf. A005258, A005259, A225776.

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}]

cp's OEIS Frontend

A228289 Determinant of the p_n X p_n matrix with (i,j)-entry equal to D(i+j) for all i,j = 0,...,p_n-1, where D(k) = A002895(k) is the k-th Domb number and p_n is the n-th prime.

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