A214595 -id:A214595 - cp's OEIS frontend

A214589 Number of nXnXn triangular 0..2 arrays with every horizontal row having the same average value.

3, 5, 23, 401, 20351, 2869211, 1127599139, 1248252244661, 3918263795984603, 35080215765450132755, 899912775031092255512711, 66403663756769266442027284403, 14140062564030204365431731967633343

Offset: 1

R. H. Hardin, Jul 22 2012

Column 2 of A214595.

a(n) appears to equal A386649(n) + 2 for n >= 1, where A386649(n) = product of first n central trinomial coefficients (A002426). - Paul D. Hanna, Aug 20 2025

			Some solutions for n=4
.....1........1........1........1........1........1........1........1
....1.1......2.0......0.2......0.2......0.2......2.0......0.2......1.1
...2.0.1....2.0.1....1.2.0....2.1.0....1.1.1....2.1.0....1.2.0....0.2.1
..0.2.1.1..1.0.1.2..0.2.2.0..1.2.0.1..2.1.1.0..0.1.1.2..0.2.1.1..0.0.2.2

R. H. Hardin, Table of n, a(n) for n = 1..66

Cf. A214595, A386649.

A214590 Number of nXnXn triangular 0..3 arrays with every horizontal row having the same average value.

Original entry on oeis.org

4, 8, 62, 1862, 187862, 63120962, 71200442882, 272199293130242, 3552472974642762242, 159161446682919676638722, 24593308418551382594861736962, 13156534644821921838477614251069442

Offset: 1

R. H. Hardin, Jul 22 2012

Column 3 of A214595.

a(n) appears to equal 2*A386650(n) + 2 for n >= 1, where A386650(n) = product of first n quadrinomial coefficients (A005725). - Paul D. Hanna, Aug 20 2025

			Some solutions for n=4
.....2........2........2........2........2........1........2........2
....1.3......1.3......3.1......2.2......3.1......0.2......3.1......2.2
...1.3.2....3.2.1....2.3.1....3.0.3....0.3.3....1.0.2....2.3.1....3.0.3
..1.2.2.3..3.0.3.2..3.2.3.0..0.3.3.2..3.2.1.2..0.0.3.1..3.0.3.2..3.2.0.3

R. H. Hardin, Table of n, a(n) for n = 1..61

Cf. A214595, A386650.

A214591 Number of nXnXn triangular 0..4 arrays with every horizontal row having the same average value.

Original entry on oeis.org

5, 13, 157, 10177, 3330677, 5495329427, 43988453613377, 1673440688496376877, 301618636251020715984377, 258456549967842122873898234377, 1057468726916674176186809546298140627

Offset: 1

R. H. Hardin, Jul 22 2012

			Some solutions for n=4
.....1........1........2........2........2........2........2........2
....2.0......2.0......3.1......4.0......2.2......0.4......4.0......2.2
...2.0.1....0.1.2....2.4.0....2.2.2....4.1.1....1.3.2....1.2.3....0.3.3
..0.1.1.2..2.0.0.2..1.4.2.1..4.1.2.1..2.3.3.0..2.0.3.3..2.3.1.2..4.1.3.0

R. H. Hardin, Table of n, a(n) for n = 1..54

Column 4 of A214595.

A214592 Number of nXnXn triangular 0..5 arrays with every horizontal row having the same average value.

Original entry on oeis.org

6, 18, 312, 33352, 20608352, 69920063852, 1279414854562952, 126404181535040391152, 67823930097039015417498752, 198754894778868023542228903871252, 3195893205534165606040279967148673282502, 283070945017969156158537463029538134823210477502

Offset: 1

R. H. Hardin, Jul 22 2012

			Some solutions for n=4
.....2........3........1........2........2........2........2........2
....1.3......3.3......2.0......1.3......0.4......3.1......1.3......1.3
...2.4.0....3.1.5....2.1.0....1.5.0....4.2.0....0.2.4....3.1.2....1.4.1
..3.3.2.0..4.3.0.5..0.2.1.1..1.2.5.0..3.1.3.1..2.1.2.3..0.2.5.1..1.5.1.1

R. H. Hardin, Table of n, a(n) for n = 1..52

Column 5 of A214595.

apply( {A214592(n)=A214595(n,5)}, [1..12]) \\ M. F. Hasler, Aug 21 2025

From M. F. Hasler, Aug 21 2025: (Start)
a(n) == 2 (mod 10) for n > 2, (mod 50) for n > 3, (mod 2500) for n > 10, (mod 25000) for n > 21, (mod 125000000000) for n > 26.
a(n) = Sum_{k=0..5} Product_{L=2..n} NC(k*L, L, 5), where NC(s, n, m) is the number of compositions with sum s and n parts between 0 and m. (End)

A214593 Number of nXnXn triangular 0..6 arrays with every horizontal row having the same average value.

Original entry on oeis.org

7, 25, 601, 103651, 121537201, 971302214871, 53475958137232241, 20261509828593742587755, 52256372938247946481045323041, 909113988784102810475556803145162725

Offset: 1

R. H. Hardin, Jul 22 2012

			Some solutions for n=4
.....2........3........2........4........4........3........2........3
....0.4......2.4......0.4......2.6......4.4......6.0......3.1......6.0
...0.3.3....6.0.3....1.0.5....2.4.6....2.4.6....3.0.6....4.2.0....4.1.4
..0.6.2.0..3.4.3.2..3.5.0.0..5.6.4.1..2.5.6.3..5.0.1.6..1.6.1.0..2.4.1.5

R. H. Hardin, Table of n, a(n) for n = 1..49

Column 6 of A214595.

A214594 Number of nXnXn triangular 0..7 arrays with every horizontal row having the same average value.

Original entry on oeis.org

8, 32, 986, 250042, 493575042, 7427553264322, 841074553776240002, 708809429969357784359042, 4431510042384136382251663178882, 205943724467315535144140586838561052162, 71400892546992940004221531659013363645000504322

Offset: 1

R. H. Hardin, Jul 22 2012

			Some solutions for n=4
.....4........4........4........2........2........4........4........3
....6.2......6.2......6.2......0.4......2.2......2.6......6.2......2.4
...7.3.2....0.7.5....5.2.5....0.4.2....3.3.0....5.6.1....2.5.5....2.6.1
..4.5.4.3..1.3.5.7..7.2.1.6..3.4.1.0..0.0.2.6..5.3.3.5..7.1.4.4..3.5.0.4

R. H. Hardin, Table of n, a(n) for n = 1..48

Column 7 of A214595.

apply( {A214594(n)=A214595(n, 7)}, [1..11]) \\ M. F. Hasler, Aug 21 2025

See A214595. - M. F. Hasler, Aug 21 2025

A214596 Number of 3X3X3 triangular 0..n arrays with every horizontal row having the same average value.

Original entry on oeis.org

2, 23, 62, 157, 312, 601, 986, 1619, 2426, 3589, 5028, 6997, 9314, 12371, 15914, 20377, 25488, 31789, 38846, 47399, 56906, 68161, 80592, 95137, 111002, 129383, 149342, 172141, 196800, 224761, 254762, 288563, 324722, 365077, 408132, 455941, 506666

Offset: 1

R. H. Hardin, Jul 22 2012

			Some solutions for n=4
....3......2......1......1......2......2......3......2......3......2......2
...2.4....1.3....1.1....2.0....3.1....0.4....3.3....1.3....4.2....4.0....0.4
..2.4.3..3.2.1..1.0.2..1.1.1..3.1.2..2.0.4..3.2.4..3.3.0..2.3.4..1.2.3..0.3.3

R. H. Hardin, Table of n, a(n) for n = 1..210

Row 3 of A214595.

Empirical: a(n) = 3*a(n-2) +2*a(n-3) -3*a(n-4) -6*a(n-5) +6*a(n-7) +3*a(n-8) -2*a(n-9) -3*a(n-10) +a(n-12).

Empirical: G.f. ( -x*(2+23*x+56*x^2+84*x^3+86*x^4+87*x^5+60*x^6+23*x^7-2*x^9+x^11) ) / ( (1+x+x^2)^2*(1+x)^3*(x-1)^5 ). - R. J. Mathar, Jul 23 2012

A214597 Number of 4X4X4 triangular 0..n arrays with every horizontal row having the same average value.

Original entry on oeis.org

2, 401, 1862, 10177, 33352, 103651, 250042, 589763, 1199614, 2377843, 4299560, 7631589, 12644114, 20709813, 32235854, 49618485, 73512984, 107991735, 153683050, 217344807, 299596702, 410511063, 550861288, 735611705, 964611698

Offset: 1

R. H. Hardin, Jul 22 2012

			Some solutions for n=4
.....2........2........2........2........2........2........1........2
....3.1......2.2......1.3......2.2......4.0......1.3......2.0......4.0
...2.2.2....1.2.3....3.2.1....0.2.4....1.1.4....4.2.0....0.3.0....4.2.0
..2.3.3.0..3.3.2.0..3.3.1.1..3.0.4.1..2.3.2.1..0.4.0.4..1.0.3.0..3.4.0.1

R. H. Hardin, Table of n, a(n) for n = 1..210

Row 4 of A214595.

Empirical: a(n) = -3*a(n-1) -4*a(n-2) +a(n-3) +15*a(n-4) +28*a(n-5) +22*a(n-6) -18*a(n-7) -74*a(n-8) -96*a(n-9) -42*a(n-10) +78*a(n-11) +177*a(n-12) +165*a(n-13) +24*a(n-14) -155*a(n-15) -234*a(n-16) -155*a(n-17) +24*a(n-18) +165*a(n-19) +177*a(n-20) +78*a(n-21) -42*a(n-22) -96*a(n-23) -74*a(n-24) -18*a(n-25) +22*a(n-26) +28*a(n-27) +15*a(n-28) +a(n-29) -4*a(n-30) -3*a(n-31) -a(n-32)

A214598 Number of 5X5X5 triangular 0..n arrays with every horizontal row having the same average value.

Original entry on oeis.org

2, 20351, 187862, 3330677, 20608352, 121537201, 493575042, 1877543213, 5767190924, 16841357893, 42868858260, 105147415789, 233567627514, 505276944133, 1015252318954, 1995846081585, 3708725498234, 6772671546085

Offset: 1

R. H. Hardin, Jul 22 2012

			Some solutions for n=3
......2..........1..........2..........1..........1..........2..........2
.....1.3........0.2........2.2........0.2........1.1........2.2........2.2
....3.0.3......0.1.2......1.3.2......1.2.0......0.2.1......3.2.1......1.3.2
...0.3.3.2....1.0.3.0....2.3.3.0....1.1.0.2....0.0.1.3....1.3.1.3....2.3.1.2
..3.1.2.1.3..1.1.3.0.0..1.2.3.3.1..2.1.0.0.2..1.0.2.2.0..3.1.0.3.3..1.3.2.3.1

R. H. Hardin, Table of n, a(n) for n = 1..210

Row 5 of A214595

Empirical: a(n) = -14*a(n-1) -103*a(n-2) -523*a(n-3) -2037*a(n-4) -6416*a(n-5) -16803*a(n-6) -37004*a(n-7) -68163*a(n-8) -101576*a(n-9) -108846*a(n-10) -35709*a(n-11) +188220*a(n-12) +610219*a(n-13) +1182816*a(n-14) +1683142*a(n-15) +1681232*a(n-16) +634962*a(n-17) -1837434*a(n-18) -5548937*a(n-19) -9379421*a(n-20) -11246005*a(n-21) -8675945*a(n-22) -19085*a(n-23) +14068354*a(n-24) +29634368*a(n-25) +39812706*a(n-26) +37051642*a(n-27) +16770832*a(n-28) -19008463*a(n-29) -60093479*a(n-30) -90231081*a(n-31) -93014760*a(n-32) -59648029*a(n-33) +4933489*a(n-34) +81105797*a(n-35) +140396619*a(n-36) +156579200*a(n-37) +117529811*a(n-38) +32282085*a(n-39) -70640875*a(n-40) -153698573*a(n-41) -185452424*a(n-42) -153698573*a(n-43) -70640875*a(n-44) +32282085*a(n-45) +117529811*a(n-46) +156579200*a(n-47) +140396619*a(n-48) +81105797*a(n-49) +4933489*a(n-50) -59648029*a(n-51) -93014760*a(n-52) -90231081*a(n-53) -60093479*a(n-54) -19008463*a(n-55) +16770832*a(n-56) +37051642*a(n-57) +39812706*a(n-58) +29634368*a(n-59) +14068354*a(n-60) -19085*a(n-61) -8675945*a(n-62) -11246005*a(n-63) -9379421*a(n-64) -5548937*a(n-65) -1837434*a(n-66) +634962*a(n-67) +1681232*a(n-68) +1683142*a(n-69) +1182816*a(n-70) +610219*a(n-71) +188220*a(n-72) -35709*a(n-73) -108846*a(n-74) -101576*a(n-75) -68163*a(n-76) -37004*a(n-77) -16803*a(n-78) -6416*a(n-79) -2037*a(n-80) -523*a(n-81) -103*a(n-82) -14*a(n-83) -a(n-84)

A214599 Number of 6X6X6 triangular 0..n arrays with every horizontal row having the same average value.

Original entry on oeis.org

2, 2869211, 63120962, 5495329427, 69920063852, 971302214871, 7427553264322, 51611647496273, 265462242006596, 1257264229573585, 4911963280790040, 18044741780879451, 57852506775385206, 177133944592970795

Offset: 1

R. H. Hardin, Jul 22 2012

			Some solutions for n=3
.......1............1............1............1............1............1
......0.2..........0.2..........0.2..........0.2..........0.2..........0.2
.....1.1.1........1.1.1........1.1.1........1.1.1........1.1.1........1.1.1
....0.1.3.0......1.0.1.2......0.1.0.3......0.0.1.3......0.1.1.2......1.1.1.1
...2.0.0.1.2....1.1.1.2.0....0.2.2.0.1....1.1.1.2.0....0.1.1.3.0....2.0.0.1.2
..1.3.0.2.0.0..1.0.0.3.0.2..1.2.2.0.0.1..1.1.1.3.0.0..1.3.0.0.0.2..1.1.0.0.2.2

R. H. Hardin, Table of n, a(n) for n = 1..210

Row 6 of A214595.

cp's OEIS Frontend

A214589 Number of nXnXn triangular 0..2 arrays with every horizontal row having the same average value.

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A214590 Number of nXnXn triangular 0..3 arrays with every horizontal row having the same average value.

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A214591 Number of nXnXn triangular 0..4 arrays with every horizontal row having the same average value.

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A214592 Number of nXnXn triangular 0..5 arrays with every horizontal row having the same average value.

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A214593 Number of nXnXn triangular 0..6 arrays with every horizontal row having the same average value.

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A214594 Number of nXnXn triangular 0..7 arrays with every horizontal row having the same average value.

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A214596 Number of 3X3X3 triangular 0..n arrays with every horizontal row having the same average value.

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A214597 Number of 4X4X4 triangular 0..n arrays with every horizontal row having the same average value.

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A214598 Number of 5X5X5 triangular 0..n arrays with every horizontal row having the same average value.

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A214599 Number of 6X6X6 triangular 0..n arrays with every horizontal row having the same average value.

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