A061993 - OEIS

%I #14 Apr 29 2022 03:34:04

%S 0,0,0,0,0,0,0,40,312,2038,9632,37248,120104,335010,835056,1897702,

%T 3998456,7907094,14818300,26512942,45562852,75580634,121520020,

%U 190031678,289879092,432420154,632159540,907376502,1280833348

%N Number of ways to place 7 nonattacking queens on a 7 X n board.

%H Vincenzo Librandi, <a href="/A061993/b061993.txt">Table of n, a(n) for n = 0..1000</a>

%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>, part of V. Kotesovec, Between chessboard and computer, 1996, pp. 204 - 206.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F Explicit formula (V. Kotesovec, 1992): a(n) = n^7 - 63*n^6 + 1879*n^5 - 34411*n^4 + 417178*n^3 - 3336014*n^2 + 16209916*n - 36693996, n >= 23.

%F Recurrence: a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), n >= 31.

%F G.f.: 2*x^7*(20 - 4*x + 331*x^2 - 88*x^3 + 1292*x^4 - 1356*x^5 + 2019*x^6 + 264*x^7 - 2857*x^8 + 6472*x^9 - 7616*x^10 + 7462*x^11 - 7831*x^12 + 8326*x^13 - 5672*x^14 + 1998*x^15 - 308*x^16 - 142*x^17 + 510*x^18 - 284*x^19 - 220*x^20 + 320*x^21 - 140*x^22 + 24*x^23)/(1 - x)^8.

%t CoefficientList[Series[2*x^7*(20-4*x+331*x^2-88*x^3+1292*x^4-1356*x^5+2019*x^6 +264*x^7-2857*x^8+6472*x^9-7616*x^10+7462*x^11-7831*x^12+8326*x^13-5672*x^14 +1998*x^15-308*x^16-142*x^17+510*x^18-284*x^19-220*x^20+320*x^21-140*x^22 +24*x^23)/(1-x)^8, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 12 2013 *)

%o (SageMath)

%o def p(x): return 20-4*x+331*x^2-88*x^3+1292*x^4-1356*x^5+2019*x^6 +264*x^7-2857*x^8+6472*x^9-7616*x^10+7462*x^11-7831*x^12+8326*x^13-5672*x^14 +1998*x^15-308*x^16-142*x^17+510*x^18-284*x^19-220*x^20+320*x^21-140*x^22 +24*x^23

%o [( 2*x^7*p(x)/(1-x)^8 ).series(x,n+1).list()[n] for n in (0..40)] # _G. C. Greubel_, Apr 29 2022

%Y Cf. A061989, A061990, A061991, A061992.

%K nonn,easy

%O 0,8

%A Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 10 2001