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A157062
Number of integer sequences of length n+1 with sum zero and sum of absolute values 26.
1
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
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
Sequence in context: A308373 A183578 A184965 * A008273 A231240 A197101
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 22 2009
STATUS
approved