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

%I A157062 #11 Jan 25 2022 08:49:40
%S A157062 2,78,1692,25740,302850,2912910,23744840,168278760,1056789450,
%T A157062 5968878630,30684132468,144977296932,634756203018,2593322651430,
%U A157062 9946019437200,35995371261360,123490242018990,403237594259010,1257743358034100,3759426449644740,10799525727846702
%N A157062 Number of integer sequences of length n+1 with sum zero and sum of absolute values 26.
%H A157062 T. D. Noe, <a href="/A157062/b157062.txt">Table of n, a(n) for n = 1..1000</a>
%H A157062 <a href="/index/Rec#order_27">Index entries for linear recurrences with constant coefficients</a>, 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).
%F A157062 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).
%F A157062 From _G. C. Greubel_, Jan 24 2022: (Start)
%F A157062 a(n) = (n+1)*binomial(n+12, 13)*Hypergeometric3F2([-12, -n, 1-n], [2, -n-12], 1).
%F A157062 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).
%F A157062 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)
%t A157062 A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
%t A157062 A157062[n_]:= A103881[n, 13];
%t A157062 Table[A157062[n], {n, 50}] (* _G. C. Greubel_, Jan 24 2022 *)
%o A157062 (Sage)
%o A157062 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) )
%o A157062 def A157062(n): return A103881(n, 13)
%o A157062 [A157062(n) for n in (1..50)] # _G. C. Greubel_, Jan 24 2022
%Y A157062 Cf. A103881, A156554.
%K A157062 nonn
%O A157062 1,1
%A A157062 _R. H. Hardin_, Feb 22 2009