A157059 Number of integer sequences of length n+1 with sum zero and sum of absolute values 20.

2, 60, 1002, 11750, 106752, 794598, 5025692, 27717948, 135916002, 601585512, 2432878866, 9079799742, 31534801116, 102644594262, 315029394792, 916470530808, 2538818182782, 6724224543708, 17088309885542, 41800229045610, 98698280879352, 225524301678170

Offset: 1

Author

R. H. Hardin, Feb 22 2009

Keywords

• nonn
• easy

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (21,-210,1330,-5985,20349,-54264, 116280,-203490,293930,-352716, 352716,-293930,203490,-116280,54264,-20349,5985, -1330,210,-21,1).

Crossrefs

Cf. A103881, A156554.

Programs

Mathematica

A103881[n_, k_]:= (n+1)*Binomial[n+k-1, k]*HypergeometricPFQ[{1-n,-n,1-k}, {2,1-n-k}, 1];
A157059[n_]:= A103881[n,10];
Table[A157059[n], {n, 50}] (* G. C. Greubel, Jan 24 2022 *)

Sage

def A103881(n,k): return sum( binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k) for i in (0..n) )
def A157059(n): return A103881(n, 10)
[A157059(n,10) for n in (1..50)] # G. C. Greubel, Jan 24 2022

Formula

a(n) = T(n,10); T(n,k) = Sum_{i=1..n} binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k).

G.f.: 2*x*(x^18 +9*x^17 +81*x^16 +324*x^15 +1296*x^14 +3024*x^13 +7056*x^12 +10584*x^11 +15876*x^10 +15876*x^9 +15876*x^8 +10584*x^7 +7056*x^6 +3024*x^5 +1296*x^4 +324*x^3 +81*x^2 +9*x +1)/(1-x)^21. - Colin Barker, Jan 25 2013

From G. C. Greubel, Jan 24 2022: (Start)

a(n) = (184756/20!)*n*(n+1)*(1316818944000 [phone]*n [phone]*n^2 [phone]*n^3 [phone]*n^4 [phone]*n^5 [phone]*n^6 [phone]*n^7 [phone]*n^8 [phone]*n^9 [phone]*n^10 [phone]*n^11 [phone]*n^12 +873558*n^13 +136302*n^14 +4644*n^15 +606*n^16 +9*n^17 +n^18).

a(n) = (n+1)*binomial(n+9, 10)*Hypergeometric3F2([-9, -n, 1-n], [2, -n-9], 1). (End)