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Number of integer sequences of length n+1 with sum zero and sum of absolute values 32.
1

%I #9 Jan 25 2022 08:49:23

%S 2,96,2562,47920,692610,8174544,81659522,708113304,5431848930,

%T 37403270520,233931828834,1341750437352,7114703302434,35117045235720,

%U 162298598439330,705951252118284,2903050518427962,11331495633292524,42132555868774010,149703679118108220

%N Number of integer sequences of length n+1 with sum zero and sum of absolute values 32.

%H T. D. Noe, <a href="/A157065/b157065.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_33">Index entries for linear recurrences with constant coefficients</a>, signature (33,-528,5456,-40920,237336,-1107568,4272048, -13884156,38567100,-92561040,193536720,-354817320,573166440,-818809200, 1037158320,-1166803110,1166803110,-1037158320,818809200,-573166440,354817320, -193536720,92561040,-38567100,13884156,-4272048,1107568,-237336,40920,-5456, 528,-33,1).

%F a(n) = T(n,16); T(n,k) = Sum_{i=1..n} binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k).

%F From _G. C. Greubel_, Jan 25 2022: (Start)

%F a(n) = (n+1)*binomial(n+15, 16)*Hypergeometric3F2([-15, -n, 1-n], [2, -n-15], 1).

%F a(n) = (601080390/32!)*n*(n+1)*(27360196043587190784000000 + 65137211981397216460800000*n + 107110050449356033228800000*n^2 + 94817527804050105212928000*n^3 + 75961411427539608595660800*n^4 + 38202458280851158526730240*n^5 + 19587950887554046039781376*n^6 + 6463560689425876180435200*n^7 + 2379792991631228553219840*n^8 + 553304095999692103772160*n^9 + 156114125142340061791744*n^10 + 26624540206135314300000*n^11 + 6005394587432947709600*n^12 + 768878902291539639600*n^13 + 142854837644598236640*n^14 + 13893755540913698625*n^15 + 2174500936993696575*n^16 + 161097628663020825*n^17 + 21612664028370855*n^18 + 1212359721607125*n^19 + 141388292047275*n^20 + 5907926749725*n^21 + 605873224515*n^22 + 18281995875*n^23 + 1664663325*n^24 + 34287435*n^25 + 2794869*n^26 + 35175*n^27 + 2585*n^28 + 15*n^29 + n^30).

%F G.f.: 2*x*(1 + 15*x + 225*x^2 + 1575*x^3 + 11025*x^4 + 47775*x^5 + 207025*x^6 + 621075*x^7 + 1863225*x^8 + 4099095*x^9 + 9018009*x^10 + 15030015*x^11 + 25050025*x^12 + 32207175*x^13 + 41409225*x^14 + 41409225*x^15 + 41409225*x^16 + 32207175*x^17 + 25050025*x^18 + 15030015*x^19 + 9018009*x^20 + 4099095*x^21 + 1863225*x^22 + 621075*x^23 + 207025*x^24 + 47775*x^25 + 11025*x^26 + 1575*x^27 + 225*x^28 + 15*x^29 + x^30)/(1-x)^33. (End)

%t A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];

%t A157065[n_]:= A103881[n, 16];

%t Table[A157065[n], {n, 50}] (* _G. C. Greubel_, Jan 25 2022 *)

%o (Sage)

%o 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 def A157065(n): return A103881(n, 16)

%o [A157065(n) for n in (1..50)] # _G. C. Greubel_, Jan 25 2022

%Y Cf. A103881, A156554.

%K nonn

%O 1,1

%A _R. H. Hardin_, Feb 22 2009