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

%I A157060 #11 Sep 28 2022 04:11:31
%S A157060 2,66,1212,15620,155850,1272810,8823080,53265960,285510150,1379301990,
%T A157060 6078578508,24680519604,93093230958,328512273390,1091144804400,
%U A157060 3429182092560,10244035242630,29206656395910,79759293448100
%N A157060 Number of integer sequences of length n+1 with sum zero and sum of absolute values 22.
%H A157060 G. C. Greubel, <a href="/A157060/b157060.txt">Table of n, a(n) for n = 1..1000</a>
%H A157060 <a href="/index/Rec#order_23">Index entries for linear recurrences with constant coefficients</a>, signature (23,-253,1771,-8855,33649,-100947,245157,-490314, 817190,-1144066,1352078, -1352078,1144066,-817190,490314,-245157,100947,-33649, 8855,-1771,253,-23,1).
%F A157060 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).
%F A157060 From _G. C. Greubel_, Jan 24 2022: (Start)
%F A157060 a(n) = (n+1)*binomial(n+10, 11)*Hypergeometric3F2([-10, -n, 1-n], [2, -n-10], 1).
%F A157060 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).
%F A157060 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)
%t A157060 A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2,1-n - k}, 1];
%t A157060 A157060[n_] := A103881[n, 11];
%t A157060 Table[A157060[n], {n, 50}] (* _G. C. Greubel_, Jan 24 2022 *)
%o A157060 (Sage)
%o A157060 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 A157060 def A157060(n): return A103881(n, 11)
%o A157060 [A157060(n) for n in (1..50)] # _G. C. Greubel_, Jan 24 2022
%Y A157060 Cf. A103881, A156554.
%K A157060 nonn
%O A157060 1,1
%A A157060 _R. H. Hardin_, Feb 22 2009