This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A008419 #44 Jul 08 2025 01:03:47
%S A008419 1,19,181,1159,5641,22363,75517,224143,598417,1462563,3317445,7059735,
%T A008419 14218905,27298155,50250765,89129247,152951073,254831667,413442773,
%U A008419 654862247,1014889769,1541911931,2300409629,3375210671,4876601009,6946419011,9765268709
%N A008419 Crystal ball sequence for 9-dimensional cubic lattice.
%C A008419 This is row/column 9 of the Delannoy numbers array, A008288, which is the main entry for these numbers, listing many more properties. - _Shel Kaphan_, Jan 07 2023
%H A008419 T. D. Noe, <a href="/A008419/b008419.txt">Table of n, a(n) for n = 0..1000</a>
%H A008419 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%H A008419 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%H A008419 <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>
%F A008419 G.f.: (1+x)^9/(1-x)^10.
%F A008419 a(n) = (4*n^9+18*n^8+240*n^7+756*n^6+3612*n^5+7182*n^4+14360*n^3+14724*n^2+ 10134*n+2835)/2835. - _Johannes W. Meijer_, Jul 14 2013
%F A008419 a(0)=1, a(1)=19, a(2)=181, a(3)=1159, a(4)=5641, a(5)=22363, a(6)=75517, a(7)=224143, a(8)=598417, a(9)=1462563, a(n)=10*a(n-1)-45*a(n-2)+ 120*a(n-3)- 210*a(n-4)+252*a(n-5)-210*a(n-6)+120*a(n-7)-45*a(n-8)+ 10*a(n-9)- a(n-10). - _Harvey P. Dale_, Jul 25 2013
%F A008419 Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = (1 - 1/2 + 1/3 - ... + 1/9) - log(2). - _Peter Bala_, Mar 23 2024
%t A008419 CoefficientList[Series[(z + 1)^9/(z - 1)^10, {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 19 2011 *)
%t A008419 LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,19,181,1159,5641,22363,75517,224143,598417,1462563},40] (* _Harvey P. Dale_, Jul 25 2013 *)
%Y A008419 Cf. A001847, A001848, A001849.
%Y A008419 Cf. A240876.
%Y A008419 Row/column 9 of A008288.
%K A008419 nonn,easy
%O A008419 0,2
%A A008419 _N. J. A. Sloane_