A157062 Number of integer sequences of length n+1 with sum zero and sum of absolute values 26.

2, 78, 1692, 25740, 302850, 2912910, 23744840, 168278760, 1056789450, 5968878630, 30684132468, 144977296932, 634756203018, 2593322651430, 9946019437200, 35995371261360, 123490242018990, 403237594259010, 1257743358034100, 3759426449644740, 10799525727846702

Offset

1,1

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (27,-351,2925,-17550,80730,-296010,888030,-2220075, 4686825,-8436285,13037895,-17383860,20058300,-20058300,17383860,-13037895, 8436285,-4686825,2220075,-888030,296010,-80730,17550,-2925,351,-27,1).

Formula

a(n) = T(n,13); 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+12, 13)*Hypergeometric3F2([-12, -n, 1-n], [2, -n-12], 1).

a(n) = (10400600/26!)*n*(n+1)*(2982752926433280000 + 6502800338141184000*n + 10192999816651161600*n^2 + 8194549559065989120*n^3 + 6217354001317404672*n^4 + 2785907939555600640*n^5 + 1345736958526293696*n^6 + 386128480881709632*n^7 + 133329525393692848*n^8 + 26155830342678960*n^9 + 6893260441243396*n^10 + 955286585044572*n^11 + 200534847420673*n^12 + 19880275030680*n^13 + 3426180791086*n^14 + 242021337492*n^15 + 35027635423*n^16 + 1724131200*n^17 + 213288856*n^18 + 6959172*n^19 + 746383*n^20 + 14520*n^21 + 1366*n^22 + 12*n^23 + n^24).

G.f.: 2*x*(1 + 12*x + 144*x^2 + 792*x^3 + 4356*x^4 + 14520*x^5 + 48400*x^6 + 108900*x^7 + 245025*x^8 + 392040*x^9 + 627264*x^10 + 731808*x^11 + 853776*x^12 + 731808*x^13 + 627264*x^14 + 392040*x^15 + 245025*x^16 + 108900*x^17 + 48400*x^18 + 14520*x^19 + 4356*x^20 + 792*x^21 + 144*x^22 + 12*x^23 + x^24)/(1-x)^27. (End)

Mathematica

A103881[n_, k_]:= (n+1)*Binomial[n+k-1, k]*HypergeometricPFQ[{1-n, -n, 1-k}, {2, 1-n - k}, 1];
A157062[n_]:= A103881[n, 13];
Table[A157062[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 A157062(n): return A103881(n, 13)
[A157062(n) for n in (1..50)] # G. C. Greubel, Jan 24 2022

Crossrefs

Cf. A103881, A156554.

Sequence in context: A308373 A183578 A184965 * A008273 A231240 A197101

Adjacent sequences: A157059 A157060 A157061 * A157063 A157064 A157065

Keyword

nonn

Author

R. H. Hardin, Feb 22 2009

Status

approved