cp's OEIS Frontend

A157064 Number of integer sequences of length n+1 with sum zero and sum of absolute values 30.

%I A157064 #9 Jan 25 2022 08:49:56
%S A157064 2,90,2252,39500,535502,5930022,55599992,452715672,3262336002,
%T A157064 21114177018,124188986196,670283877588,3346707628446,15564971674518,
%U A157064 67830161708592,278406848295312,1081149205136382,3988232552194662,14025412751733092,47171740235162340
%N A157064 Number of integer sequences of length n+1 with sum zero and sum of absolute values 30.
%H A157064 T. D. Noe, <a href="/A157064/b157064.txt">Table of n, a(n) for n = 1..1000</a>
%H A157064 <a href="/index/Rec#order_31">Index entries for linear recurrences with constant coefficients</a>, signature (31,-465,4495,-31465,169911,-736281,2629575, -7888725,20160075,-44352165,84672315,-141120525,206253075,-265182525,300540195, -300540195,265182525,-206253075,141120525,-84672315,44352165,-20160075,7888725, -2629575,736281,-169911,31465,-4495,465,-31,1).
%F A157064 a(n) = T(n,15); T(n,k) = Sum_{i=1..n} binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k).
%F A157064 From _G. C. Greubel_, Jan 24 2022: (Start)
%F A157064 a(n) = (n+1)*binomial(n+14, 15)*Hypergeometric3F2([-14, -n, 1-n], [2, -n-14], 1).
%F A157064 a(n) = (155117520/30!)*n*(n+1)*(114000816848279961600000 + 264279998869470904320000*n + 428198206877484244992000*n^2 + 368310644587032673075200*n^3 + 290167678780290006589440*n^4 + 141041429579778368449536*n^5 + 71004668064572092241664*n^6 + 22493711118572061653376*n^7 + 8120370606956264477184*n^8 + 1797910570397283902560*n^9 + 496779939204280228640*n^10 + 79886837991962961960*n^11 + 17626771834821917040*n^12 + 2101988853205045350*n^13 + 381651017327064975*n^14 + 34037459504198850*n^15 + 5201044031664375*n^16 + 346174867450230*n^17 + 45303425489595*n^18 + 2220034746930*n^19 + 252351294195*n^20 + 8844405570*n^21 + 883381005*n^22 + 20963670*n^23 + 1857765*n^24 + 26754*n^25 + 2121*n^26 + 14*n^27 + n^28).
%F A157064 G.f.: 2*x*(1 + 14*x + 196*x^2 + 1274*x^3 + 8281*x^4 + 33124*x^5 + 132496*x^6 + 364364*x^7 + 1002001*x^8 + 2004002*x^9 + 4008004*x^10 + 6012006*x^11 + 9018009*x^12 + 10306296*x^13 + 11778624*x^14 + 10306296*x^15 + 9018009*x^16 + 6012006*x^17 + 4008004*x^18 + 2004002 x^19 + 1002001*x^20 + 364364*x^21 + 132496*x^22 + 33124*x^23 + 8281*x^24 + 1274*x^25 + 196*x^26 + 14*x^27 + x^28)/(1-x)^31. (End)
%t A157064 A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
%t A157064 A157064[n_]:= A103881[n, 15];
%t A157064 Table[A157064[n], {n, 50}] (* _G. C. Greubel_, Jan 24 2022 *)
%o A157064 (Sage)
%o A157064 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 A157064 def A157064(n): return A103881(n, 15)
%o A157064 [A157064(n) for n in (1..50)] # _G. C. Greubel_, Jan 24 2022
%Y A157064 Cf. A103881, A156554.
%K A157064 nonn
%O A157064 1,1
%A A157064 _R. H. Hardin_, Feb 22 2009