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

1,1

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).

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 +292579402752000*n +440986525516800*n^2 +325146872079360*n^3 +235868591146176*n^4 +94960596391200*n^5 +43658519177360*n^6 +10953312870160*n^7 +3585704220196*n^8 +593523073650*n^9 +147783744195*n^10 +16467776610*n^11 +3255909581*n^12 +242376100*n^13 +39230830*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)

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 *)

Prog

(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

Crossrefs

Cf. A103881, A156554.

Sequence in context: A131472 A098532 A159716 * A154637 A069865 A218433

Adjacent sequences: A157057 A157058 A157059 * A157061 A157062 A157063

Keyword

nonn

Author

R. H. Hardin, Feb 22 2009

Status

approved