A053805 Expansion of (1 + x)^12 / (1 - x)^13.

1, 25, 313, 2625, 16641, 85305, 369305, 1392065, 4673345, 14218905, 39753273, 103274625, 251595969, 579168825, 1267854873, 2653649025, 5334940545, 10343052825, 19403906105, 35330137025, 62596382081, 108167252025, 182668423833, 302016962625, 489658242241

Offset: 0

Author

R. K. Guy, Apr 07 2000

Keywords

• nonn
• easy

Comments

This is row/column 12 of the Delannoy numbers array, A008288, which is the main entry for these numbers, listing many more properties. - Peter Munn, Jan 05 2023

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Crossrefs

Cf. A008288, A240876.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)^12/(1-x)^13)); // Bruno Berselli, Apr 17 2014

Mathematica

CoefficientList[Series[(1 + x)^12/(1 - x)^13, {x, 0, 30}], x] (* Bruno Berselli, Apr 17 2014 *)

PARI

Vec((1+x)^12/(1-x)^13+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

Formula

G.f.: (1+x)^12/(1-x)^13.

a(n) = A240876(n) + 2*Sum_{i=0..n-1} A240876(i) for n>0, a(0)=1. - Bruno Berselli, Apr 17 2014

a(n) = 13*a(n-1) - 78*a(n-2) + 286*a(n-3) - 715*a(n-4) + 1287*a(n-5) - 1716*a(n-6) + 1716*a(n-7) - 1287*a(n-8) + 715*a(n-9) - 286*a(n-10) + 78*a(n-11) - 13*a(n-12) + a(n-13). - Wesley Ivan Hurt, Jul 09 2025