A127670 -id:A127670 - cp's OEIS frontend

A251695 a(n) = (3n+1) (4n+1)^(n-2) 5^n.

1, 4, 175, 16250, 2348125, 463050000, 115966796875, 35253537343750, 12611991884765625, 5191587030710937500, 2417311348659677734375, 1256208098030090332031250, 720779749270420907470703125, 452589644988876542822265625000, 308707218248583408960223388671875

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 4*x + 175*x^2/2! + 16250*x^3/3! + 2348125*x^4/4! + 463050000*x^5/5! +...
such that A(x) = exp( 5*x*A(x)^4 * G(x*A(x)^4)^4 ) / G(x*A(x)^4),
where G(x) = 1 + x*G(x)^5 is the g.f. A002294:
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^4) where
F(x) = 1 + 4*x + 47*x^2/2! + 1034*x^3/3! + 34349*x^4/4! + 1540480*x^5/5! +...
F(x) = exp( 5*x*G(x)^4 ) / G(x) is the e.g.f. of A251665.

G. C. Greubel, Table of n, a(n) for n = 0..268

Cf. A251665, A002294.

Cf. Variants: A127670, A251693, A251694, A251696, A251697, A251698, A251699, A251700.

[(3*n + 1)*(4*n + 1)^(n - 2)*5^n: n in [0..50]]; // G. C. Greubel, Nov 13 2017

Table[(3*n + 1)*(4*n + 1)^(n - 2)*5^n, {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

{a(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^5 +x*O(x^n));
A = ( serreverse( x*G^4 / exp(20*x*G^4) )/x )^(1/4); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^5 be the g.f. of A002294, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 5*x*A(x)^4 * G(x*A(x)^4)^4 ) / G(x*A(x)^4).
(2) A(x) = F(x*A(x)^4) where F(x) = exp(5*x*G(x)^4)/G(x) is the e.g.f. of A251665.
(3) A(x) = ( Series_Reversion( x*G(x)^4 / exp(20*x*G(x)^4) )/x )^(1/4).
E.g.f.: (-LambertW(-20*x)/(20*x))^(1/4) * (1 + LambertW(-20*x)/20). - Vaclav Kotesovec, Dec 07 2014

A251696 a(n) = (4n+1) (5n+1)^(n-2) 6^n.

Original entry on oeis.org

1, 5, 324, 44928, 9716112, 2870090496, 1077194894400, 490873123897344, 263285585800098048, 162505400851637010432, 113463916253636561519616, 88423664876285081860177920, 76086820231309990402228260864, 71651521268311905104861664903168, 73298071049899905319337719679434752

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 5*x + 324*x^2/2! + 44928*x^3/3! + 9716112*x^4/4! + 2870090496*x^5/5! +...
such that A(x) = exp( 6*x*A(x)^5 * G(x*A(x)^5)^5 ) / G(x*A(x)^5),
where G(x) = 1 + x*G(x)^6 is the g.f. A002295:
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^5) where
F(x) = 1 + 5*x + 74*x^2/2! + 2028*x^3/3! + 83352*x^4/4! + 4607496*x^5/5! +...
F(x) = exp( 6*x*G(x)^5 ) / G(x) is the e.g.f. of A251666.

G. C. Greubel, Table of n, a(n) for n = 0..256

Cf. A251666, A002295.

Cf. Variants: A127670, A251693, A251694, A251695, A251697, A251698, A251699, A251700.

[(4*n + 1)*(5*n + 1)^(n - 2)*6^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Table[(4*n + 1)*(5*n + 1)^(n - 2)*6^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^6 +x*O(x^n));
A = ( serreverse( x*G^5 / exp(30*x*G^5) )/x )^(1/5); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^6 be the g.f. of A002295, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 6*x*A(x)^5 * G(x*A(x)^5)^5 ) / G(x*A(x)^5).
(2) A(x) = F(x*A(x)^5) where F(x) = exp(6*x*G(x)^5)/G(x) is the e.g.f. of A251666.
(3) A(x) = ( Series_Reversion( x*G(x)^5 / exp(30*x*G(x)^5) )/x )^(1/5).
E.g.f.: (-LambertW(-30*x)/(30*x))^(1/5) * (1 + LambertW(-30*x)/30). - Vaclav Kotesovec, Dec 07 2014

A251697 a(n) = (5n+1) (6n+1)^(n-2) 7^n.

Original entry on oeis.org

1, 6, 539, 104272, 31513125, 13018130762, 6835288192159, 4358439870247764, 3271482918202092041, 2826044644022395468750, 2761781119675422226696419, 3012587650584028093856586776, 3628565076873134344787430377389, 4783177086109789054912470697687698, 6849486554475843842876951982177734375

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 6*x + 539*x^2/2! + 104272*x^3/3! + 31513125*x^4/4! + 13018130762*x^5/5! +...
such that A(x) = exp( 7*x*A(x)^6 * G(x*A(x)^6)^6 ) / G(x*A(x)^6),
where G(x) = 1 + x*G(x)^7 is the g.f. A002296:
G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^6) where
F(x) = 1 + 6*x + 107*x^2/2! + 3508*x^3/3! + 171741*x^4/4! + 11280842*x^5/5! +...
F(x) = exp( 7*x*G(x)^6 ) / G(x) is the e.g.f. of A251667.

G. C. Greubel, Table of n, a(n) for n = 0..248

Cf. A251667, A002296.

Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251698, A251699, A251700.

[(5*n + 1)*(6*n + 1)^(n - 2)*7^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Table[(5*n + 1)*(6*n + 1)^(n - 2)*7^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^7 +x*O(x^n));
A = ( serreverse( x*G^6 / exp(42*x*G^6) )/x )^(1/6); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 7*x*A(x)^6 * G(x*A(x)^6)^6 ) / G(x*A(x)^6).
(2) A(x) = F(x*A(x)^6) where F(x) = exp(7*x*G(x)^6)/G(x) is the e.g.f. of A251667.
(3) A(x) = ( Series_Reversion( x*G(x)^6 / exp(42*x*G(x)^6) )/x )^(1/6).
E.g.f.: (-LambertW(-42*x)/(42*x))^(1/6) * (1 + LambertW(-42*x)/42). - Vaclav Kotesovec, Dec 07 2014

A251698 a(n) = (6n+1) (7n+1)^(n-2) 8^n.

Original entry on oeis.org

1, 7, 832, 214016, 86118400, 47393538048, 33160072265728, 28180480000000000, 28194546272924860416, 32466269569728810844160, 42295727044150128912891904, 61505801717703291002224115712, 98762474157744880353280000000000, 173565347832317233669371533581090816, 331360760866451564310212841997955235840

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 7*x + 832*x^2/2! + 214016*x^3/3! + 86118400*x^4/4! + 47393538048*x^5/5! +...
such that A(x) = exp( 8*x*A(x)^7 * G(x*A(x)^7)^7 ) / G(x*A(x)^7),
where G(x) = 1 + x*G(x)^8 is the g.f. A007556:
G(x) = 1 + x + 8*x^2 + 92*x^3 + 1240*x^4 + 18278*x^5 + 285384*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^7) where
F(x) = 1 + 7*x + 146*x^2/2! + 5570*x^3/3! + 316376*x^4/4! + 24070168*x^5/5! +...
F(x) = exp( 8*x*G(x)^7 ) / G(x) is the e.g.f. of A251668.

G. C. Greubel, Table of n, a(n) for n = 0..240

Cf. A251668, A007556.

Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251697, A251699, A251700.

[(6*n + 1)*(7*n + 1)^(n - 2)*8^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Table[(6*n + 1)*(7*n + 1)^(n - 2)*8^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (6*n+1) * (7*n+1)^(n-2) * 8^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^8 +x*O(x^n));
A = ( serreverse( x*G^7 / exp(56*x*G^7) )/x )^(1/7); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^8 be the g.f. of A007556, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 8*x*A(x)^7 * G(x*A(x)^7)^7 ) / G(x*A(x)^7).
(2) A(x) = F(x*A(x)^7) where F(x) = exp(8*x*G(x)^7)/G(x) is the e.g.f. of A251668.
(3) A(x) = ( Series_Reversion( x*G(x)^7 / exp(56*x*G(x)^7) )/x )^(1/7).
E.g.f.: (-LambertW(-56*x)/(56*x))^(1/7) * (1 + LambertW(-56*x)/56). - Vaclav Kotesovec, Dec 07 2014

A251699 a(n) = (7n+1) (8n+1)^(n-2) 9^n.

Original entry on oeis.org

1, 8, 1215, 400950, 207202941, 146509780644, 131737019154363, 143893722808861650, 185052548743241765625, 273919266332571877019712, 458736814135093804224189111, 857575304752878031562956215918, 1770298011965146072399475770453365, 3999656915702652258291935606835937500

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 8*x + 1215*x^2/2! + 400950*x^3/3! + 207202941*x^4/4! + 146509780644*x^5/5! +...
such that A(x) = exp( 9*x*A(x)^8 * G(x*A(x)^8)^8 ) / G(x*A(x)^8),
where G(x) = 1 + x*G(x)^9 is the g.f. A062994:
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^8) where
F(x) = 1 + 8*x + 191*x^2/2! + 8310*x^3/3! + 537117*x^4/4! + 46444164*x^5/5! +...
F(x) = exp( 9*x*G(x)^8 ) / G(x) is the e.g.f. of A251669.

G. C. Greubel, Table of n, a(n) for n = 0..235

Cf. A251669, A062994.

Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251697, A251698, A251700.

[(7*n + 1)*(8*n + 1)^(n - 2)*9^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Table[(7*n + 1)*(8*n + 1)^(n - 2)*9^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^9 +x*O(x^n));
A = ( serreverse( x*G^8 / exp(72*x*G^8) )/x )^(1/8); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 9*x*A(x)^8 * G(x*A(x)^8)^8 ) / G(x*A(x)^8).
(2) A(x) = F(x*A(x)^8) where F(x) = exp(9*x*G(x)^8)/G(x) is the e.g.f. of A251669.
(3) A(x) = ( Series_Reversion( x*G(x)^8 / exp(72*x*G(x)^8) )/x )^(1/8).
E.g.f.: (-LambertW(-72*x)/(72*x))^(1/8) * (1 + LambertW(-72*x)/72). - Vaclav Kotesovec, Dec 07 2014

A251700 a(n) = (8n+1) (9n+1)^(n-2) 10^n.

Original entry on oeis.org

1, 9, 1700, 700000, 451770000, 399077600000, 448380625000000, 612032839680000000, 983672470878500000000, 1819783935144064000000000, 3809045473682732010000000000, 8900000000000000000000000000000, 22963427643543537169897000000000000, 64847222511564256853913600000000000000

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

In general, sequence a(n) = (b*n+1)*(c*n+1)^(n-2)*d^n, b > 0, c > 0, d > 0, has e.g.f. (1 + (1-b/c)*LambertW(-c*d*x)/(c+1)) * (-LambertW(-c*d*x)/(c*d*x))^(1/c). - Vaclav Kotesovec, Dec 07 2014

			E.g.f.: A(x) = 1 + 9*x + 1700*x^2/2! + 700000*x^3/3! + 451770000*x^4/4! + 399077600000*x^5/5! +...
such that A(x) = exp( 10*x*A(x)^9 * G(x*A(x)^9)^9 ) / G(x*A(x)^9),
where G(x) = 1 + x*G(x)^10 is the g.f. A059968:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^9) where
F(x) = 1 + 9*x + 242*x^2/2! + 11824*x^3/3! + 856824*x^4/4! + 82986080*x^5/5! +...
F(x) = exp( 10*x*G(x)^9 ) / G(x) is the e.g.f. of A251670.

G. C. Greubel, Table of n, a(n) for n = 0..230

Cf. A251670, A059968.

Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251697, A251698, A251699.

[(8*n + 1)*(9*n + 1)^(n - 2)*10^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Table[(8*n + 1)*(9*n + 1)^(n - 2)*10^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (8*n+1) * (9*n+1)^(n-2) * 10^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^10 +x*O(x^n));
A = ( serreverse( x*G^9 / exp(90*x*G^9) )/x )^(1/9); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^10 be the g.f. of A059968, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 10*x*A(x)^9 * G(x*A(x)^9)^9 ) / G(x*A(x)^9).
(2) A(x) = F(x*A(x)^9) where F(x) = exp(10*x*G(x)^9)/G(x) is the e.g.f. of A251670.
(3) A(x) = ( Series_Reversion( x*G(x)^9 / exp(90*x*G(x)^9) )/x )^(1/9).
E.g.f.: (-LambertW(-90*x)/(90*x))^(1/9) * (1 + LambertW(-90*x)/90). - Vaclav Kotesovec, Dec 07 2014

A171860 Number of n-cell fixed polycubes that are proper in n-2 dimensions.

Original entry on oeis.org

0, 1, 17, 348, 8640, 254800, 8749056, 343901376, 15257600000, 755110160640, 41278242816000, 2471677136321536, 160961785787056128, 11330322120000000000, 857485369051342438400, 69444841895469240729600, 5993559601317659925282816, 549242871950650346384195584

Offset: 2

N. J. A. Sloane, Oct 16 2010

nonn

Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016, http://www.csun.edu/~ctoth/Handbook/chap14.pdf
G. Barequet, M. Shalah, Automatic Proofs for Formulae Enumerating Proper Polycubes, 31st International Symposium on Computational Geometry (SoCG'15). Editors: Lars Arge and János Pach; pp. 19-22, 2015.
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica, 30 (2010), 257-275. See Th. 6.

Vincenzo Librandi, Table of n, a(n) for n = 2..100
A. Asinowski, G. Barequet, R. Barequet, and G. Rote, Proper n-Cell Polycubes in n-3 Dimensions, J. Int. Seq. 15 (2012) #12.8.4.
M. Shalah, Formulae and growth rates of animals on cubical and triangular lattices, PhD Thesis, Israel Inst. Techn. (2017).

Cf. A127670, A191092, A036364 (free).

Diagonal 2 of A195739.

[2^(n-3)*n^(n-5)*(n-2)*(2*n^2-6*n+9): n in [2..20]]; // Vincenzo Librandi, May 26 2011

Table[2^(n-3)n^(n-5)(n-2)(2n^2-6n+9),{n,2,30}] (* Harvey P. Dale, Nov 27 2024 *)

a(n) = 2^(n-3)*n^(n-5)*(n-2)*(2*n^2 - 6*n + 9).

Slightly edited by Gill Barequet, May 25 2011

A385582 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional polysticks of size n.

Original entry on oeis.org

1, 1, 4, 1, 20, 32, 1, 86, 420, 400, 1, 370, 4164, 10368, 6912, 1, 1626, 38205, 186552, 301840, 153664, 1, 7310, 343380, 2934560, 8637760, 10223616, 4194304, 1, 33464, 3086049, 43517697, 207353960, 427708848, 396809280, 136048896

Offset: 1

Pontus von Brömssen, Jul 04 2025

			Triangle begins:
  n\d| 1     2       3        4         5         6         7         8
  ---+-----------------------------------------------------------------
  1  | 1
  2  | 1     4
  3  | 1    20      32
  4  | 1    86     420      400
  5  | 1   370    4164    10368      6912
  6  | 1  1626   38205   186552    301840    153664
  7  | 1  7310  343380  2934560   8637760  10223616   4194304
  8  | 1 33464 3086049 43517697 207353960 427708848 396809280 136048896

Pontus von Brömssen, Table of n, a(n) for n = 1..78 (first 12 rows)
Stephan Mertens and Cristopher Moore, Series expansion of the percolation threshold on hypercubic lattices, J. Phys. A: Math. Theor., 51 (2018), 475001; arXiv:1805.02701 [cond-mat.stat-mech], 2018.
Index entries for sequences related to polyominoes.

Cf. A127670 (main diagonal), A195739 (polyominoes), A365566 (free), A385581.

T(n,d) = Sum_{k=1..d} (-1)^(d-k)*binomial(d,k)*A385581(n,k).

A086804 a(0)=0; for n > 0, a(n) = (n+1)^(n-2)*2^(n^2).

Original entry on oeis.org

0, 1, 16, 2048, 1638400, 7247757312, 164995463643136, 18446744073709551616, 9803356117276277820358656, 24178516392292583494123520000000, 271732164163901599116133024293512544256

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 05 2003

Discriminant of Chebyshev polynomial U_n (x) of second kind.
Chebyshev second kind polynomials are defined by U(0)=0, U(1)=1 and U(n) = 2xU(n-1) - U(n-2) for n > 1.
The absolute value of the discriminant of Pell polynomials is a(n-1).
Pell polynomials are defined by P(0)=0, P(1)=1 and P(n) = 2x P(n-1) + P(n-2) if n > 1. - Rigoberto Florez, Sep 01 2018

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219, 5.1.2.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind
Eric Weisstein's World of Mathematics, Pell Polynomial
Index entries for sequences related to Chebyshev polynomials.

Cf. A007701, A127670 (discriminant for S-polynomials), A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

[0] cat [(n+1)^(n-2)*2^(n^2): n in [1..10]]; // G. C. Greubel, Nov 11 2018

Join[{0},Table[(n+1)^(n-2) 2^n^2,{n,10}]] (* Harvey P. Dale, May 01 2015 *)

PARI
```
a(n)=if(n<1,0,(n+1)^(n-2)*2^(n^2))
```

a(n)=if(n<1,0,n++; poldisc(poltchebi(n)'/n))

a(n) = ((n+1)^(n-2))*2^(n^2), n >= 1, a(0):=0.
a(n) = ((2^(2*(n-1)))*Det(Vn(xn[1],...,xn[n])))^2, n >= 1, with the determinant of the Vandermonde matrix Vn with elements (Vn)i,j:= xn[i]^j, i=1..n, j=0..n-1 and xn[i]:=cos(Pi*i/(n+1)), i=1..n, are the zeros of the Chebyshev U(n,x) polynomials.
a(n) = ((-1)^(n*(n-1)/2))*(2^(n*(n-2)))*Product_{i=1..n}((d/dx)U(n,x)|_{x=xn[i]}), n >= 1, with the zeros xn[i], i=1..n, given above.

Formula and more terms from Vladeta Jovovic, Aug 07 2003

A355997 Triangle read by rows: T(n,d) is the number of fixed orthoplex n-ominoes with cell centers determining d-space.

Original entry on oeis.org

1, 0, 4, 0, 1, 32, 0, 0, 48, 400, 0, 0, 28, 1728, 6912, 0, 0, 8, 4240, 62720, 153664, 0, 0, 1, 7272, 344320, 2457600, 4194304, 0, 0, 0, 8720, 1465600, 23872320, 105815808, 136048896, 0, 0, 0, 7136, 5254576, 182691200, 1603840000, 5017600000, 512000000

Offset: 2

Robert A. Russell, Jul 22 2022

Orthoplex polyominoes are connected sets of cells of regular tilings with Schläfli symbols {}, {4}, {3,4}, {3,3,4}, {3,3,3,4}, etc. These are tilings of regular orthoplexes projected on their circumspheres. Orthoplex polyominoes are equivalent to multidimensional polyominoes that do not extend more than two units along any axis, i.e., fit within a 2^d cube. Two fixed polyominoes are identical only if one is a translation of the other.

Conjecture: T(n,n-4) = 2^(n-7) * n^(n-9) * (n-4) * (n-5) * (n-6) * (n^6-14*n^5+65*n^4-189*n^3+594*n^2-1305*n+6832) / 6 ~ A259015(n) / 8.

			Triangle begins with T(2,1):
n\d 1 2  3    4       5         6          7          8         9
2   1
3   0 4
4   0 1 32
5   0 0 48  400
6   0 0 28 1728    6912
7   0 0  8 4240   62720    153664
8   0 0  1 7272  344320   2457600    4194304
9   0 0  0 8720 1465600  23872320  105815808  136048896
10  0 0  0 7136 5254576 182691200 1603840000 5017600000 512000000

Robert A. Russell, Table of n, a(n) for n = 2..73

Cf. A195739 (multidimensional).

Diagonals (with formulas) are A127670, A355998, A355999.

cp's OEIS Frontend

A251695 a(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n.

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A251696 a(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n.

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A251697 a(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n.

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A251698 a(n) = (6*n+1) * (7*n+1)^(n-2) * 8^n.

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A251699 a(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n.

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A251700 a(n) = (8*n+1) * (9*n+1)^(n-2) * 10^n.

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Formula

A251695 a(n) = (3n+1) (4n+1)^(n-2) 5^n.

A251696 a(n) = (4n+1) (5n+1)^(n-2) 6^n.

A251697 a(n) = (5n+1) (6n+1)^(n-2) 7^n.

A251698 a(n) = (6n+1) (7n+1)^(n-2) 8^n.

A251699 a(n) = (7n+1) (8n+1)^(n-2) 9^n.

A251700 a(n) = (8n+1) (9n+1)^(n-2) 10^n.