A008419 Crystal ball sequence for 9-dimensional cubic lattice.

1, 19, 181, 1159, 5641, 22363, 75517, 224143, 598417, 1462563, 3317445, 7059735, 14218905, 27298155, 50250765, 89129247, 152951073, 254831667, 413442773, 654862247, 1014889769, 1541911931, 2300409629, 3375210671, 4876601009, 6946419011, 9765268709

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
• Index entries for crystal ball sequences

Crossrefs

Cf. A001847, A001848, A001849.

Cf. A240876.

Row/column 9 of A008288.

Programs

Mathematica

CoefficientList[Series[(z + 1)^9/(z - 1)^10, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
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 *)

Formula

G.f.: (1+x)^9/(1-x)^10.

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

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

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