A157060 Number of integer sequences of length n+1 with sum zero and sum of absolute values 22.

2, 66, 1212, 15620, 155850, 1272810, 8823080, 53265960, 285510150, 1379301990, 6078578508, 24680519604, 93093230958, 328512273390, 1091144804400, 3429182092560, 10244035242630, 29206656395910, 79759293448100

Offset: 1

Author

R. H. Hardin, Feb 22 2009

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (23,-253,1771,-8855,33649,-100947,245157,-490314, 817190,-1144066,1352078, -1352078,1144066,-817190,490314,-245157,100947,-33649, 8855,-1771,253,-23,1).

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];
A157060[n_] := A103881[n, 11];
Table[A157060[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 A157060(n): return A103881(n, 11)
[A157060(n) for n in (1..50)] # G. C. Greubel, Jan 24 2022

Formula

a(n) = T(n,11); T(n,k) = Sum_{i=1..n} binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k).

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

a(n) = (n+1)*binomial(n+10, 11)*Hypergeometric3F2([-10, -n, 1-n], [2, -n-10], 1).

a(n) = (705432/22!)*n*(n+1)*(144850083840000 [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 [phone]*n^13 [phone]*n^14 +1873860*n^15 +254046*n^16 +7050*n^17 +815*n^18 +10*n^19 +n^20).

G.f.: 2*x*(1 +10*x +100*x^2 +450*x^3 +2025*x^4 +5400*x^5 +14400*x^6 +25200*x^7 +44100*x^8 +52920*x^9 +63504*x^10 +52920*x^11 +44100*x^12 +25200*x^13 +14400*x^14 +5400*x^15 +2025*x^16 +450*x^17 +100*x^18 +10*x^19 +x^20)/(1-x)^23. (End)