A283028 Number of inequivalent 4 X 4 matrices with entries in {1,2,3,...,n} up to vertical and horizontal reflections.

0, 1, 16576, 10766601, 1073790976, 38147265625, 705278736576, 8308236966001, 70368756760576, 463255079498001, 2500000075000000, 11487432626662201, 46221065046245376, 166354152907593001, 544488335559184576, 1642102090850390625, 4611686021648613376

Offset

0,3

Comments

Cycle index of dihedral group D2 acting on the 16 entries is (3s(2)^8 + s(1)^16)/4.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Formula

a(n) = n^8 * (n^8 + 3)/4.

From Chai Wah Wu, Dec 07 2018: (Start)

a(n) = 17*a(n-1) - 136*a(n-2) + 680*a(n-3) - 2380*a(n-4) + 6188*a(n-5) - 12376*a(n-6) + 19448*a(n-7) - 24310*a(n-8) + 24310*a(n-9) - 19448*a(n-10) + 12376*a(n-11) - 6188*a(n-12) + 2380*a(n-13) - 680*a(n-14) + 136*a(n-15) - 17*a(n-16) + a(n-17) for n > 16.

G.f.: -x*(x + 1)*(x^14 + 16558*x^13 + 10468387*x^12 + 882544028*x^11 + 20463263441*x^10 + 175065686258*x^9 + 626804969739*x^8 + 968894839176*x^7 + 626804969739*x^6 + 175065686258*x^5 + 20463263441*x^4 + 882544028*x^3 + 10468387*x^2 + 16558*x + 1)/(x - 1)^17. (End)

E.g.f.: (1/4)*x*exp(x)*(x^15 + 120*x^14 + 6020*x^13 + 165620*x^12 + 2757118*x^11 + 28936908*x^10 + 193754990*x^9 + 820784250*x^8 + 2141764056*x^7 + 3281882688*x^6 + 2734927356*x^5 + 1096193700*x^4 + 171804004*x^3 + 7144584*x^2 + 33148*x + 4). - Stefano Spezia, Dec 07 2018

Example

For n=2 we get a(2)=16576 inequivalent 4 X 4 binary matrices up to vertical and horizontal reflections.

Maple

[n^8*(n^8+3)/4$n=0..18]; # Muniru A Asiru, Dec 07 2018

Mathematica

Table[n^8(n^8 + 3)/4, {n, 0, 30}]

Prog

(PARI) a(n) = n^8 * (n^8 + 3)/4; \\ Altug Alkan, Feb 27 2017
(Python) def A283028(n): return n**8*(n**8 + 3)/4 # Indranil Ghosh, Feb 27 2017
(Magma) [n^8*(n^8+3)/4: n in [0..20]]; // G. C. Greubel, Dec 07 2018
(Sage) [n^8*(n^8+3)/4 for n in range(20)] # G. C. Greubel, Dec 07 2018
(GAP) List([0..20], n -> n^8*(n^8+3)/4); # G. C. Greubel, Dec 07 2018

Crossrefs

Cf. A282612, A282613, A283026, A283027, A283029, A283030, A283031, A283033.

Cf. A283032 (5 X 5 version), A282614 (3 X 3 version), A039623 (2 X 2 version).

Sequence in context: A193244 A240901 A234894 * A344241 A344242 A234374

Adjacent sequences: A283025 A283026 A283027 * A283029 A283030 A283031

Keyword

nonn,easy

Author

David Nacin, Feb 27 2017

Status

approved