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A157068 Number of integer sequences of length n+1 with sum zero and sum of absolute values 38.

%I A157068 #8 Jan 27 2022 21:05:31
%S A157068 2,114,3612,80180,1374690,19234194,227605448,2335932504,21186110970,
%T A157068 172295622730,1271112537684,8588601364668,53573492643034,
%U A157068 310601807143530,1683493452034320,8573748834211984,41210997268585158,187693442844729174,812839595630249540
%N A157068 Number of integer sequences of length n+1 with sum zero and sum of absolute values 38.
%H A157068 T. D. Noe, <a href="/A157068/b157068.txt">Table of n, a(n) for n = 1..1000</a>
%H A157068 <a href="/index/Rec#order_39">Index entries for linear recurrences with constant coefficients</a>, signature (39,-741,9139,-82251,575757,-3262623,15380937, -61523748,211915132,-635745396,1676056044,-3910797436,8122425444,-15084504396, 25140840660,-37711260990,51021117810,-62359143990,68923264410,-68923264410, 62359143990,-51021117810,37711260990,-25140840660,15084504396,-8122425444, 3910797436,-1676056044,635745396,-211915132,61523748,-15380937,3262623,-575757, 82251,-9139,741,-39,1).
%F A157068 a(n) = T(n,19); T(n,k) = Sum_{i=1..n} binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k).
%F A157068 From _G. C. Greubel_, Jan 25 2022: (Start)
%F A157068 a(n) = (n+1)*binomial(n+18, 19)*Hypergeometric3F2([-18, -n, 1-n], [2, -n-18], 1).
%F A157068 a(n) = (35345263800/38!)*n*(n+1)*(778817392288148379660189696000000 + 1984223956005743569581323059200000*n + 3392214823876583668626122342400000*n^2 + 3227079634641025484578928197632000*n^3 + 2701114821085872776574503662387200*n^4 + 1477486663626167257723210367631360*n^5 + 794678697494482855499280703586304*n^6 + 289485264886342590944226501328896*n^7 + 112195641614805001937808853208064*n^8 + 29309532027252333838411983247872*n^9 + 8732100429652853130168723017472*n^10 + 1708566742801697011435174735872*n^11 + 408081704870580048838437092992*n^12 + 61460345467484307832839519168*n^13 + 12123027157132710911533327584*n^14 + 14298582910205269163512480328n^15 + 238150505845545646647030204*n^16 + 222226805381963345901159308n^17 + 3179819458407554816818235*n^18 + 235823049245552968253250*n^19 + 29394217444775030780985*n^20 + 17315150592375085755608n^21 + 190160234133314656140*n^22 + 8844512620448927880*n^23 + 864030358357843740*n^24 + 31339517913669420*n^25 + 2745580274521866*n^26 + 76036376515644*n^27 + 6015727425006*n^28 + 122857968168*n^29 + 8831668028*n^30 + 125358408*n^31 + 8231808*n^32 + 72522*n^33 + 4371*n^34 + 18*n^35 + n^36).
%F A157068 G.f.: 2*x*(1 + 18*x + 324*x^2 + 2754*x^3 + 23409*x^4 + 124848*x^5 + 665856*x^6 + 2496960*x^7 + 9363600*x^8 + 26218080*x^9 + 73410624*x^10 + 159056352*x^11 + 344622096*x^12 + 590780736*x^13 + 1012766976*x^14 + 1392554592*x^15 + 1914762564*x^16 + 2127513960*x^17 + 2363904400*x^18 + 2127513960*x^19 + 1914762564*x^20 + 1392554592*x^21 + 1012766976*x^22 + 590780736*x^23 + 344622096*x^24 + 159056352*x^25 + 73410624*x^26 + 26218080*x^27 + 9363600*x^28 + 2496960*x^29 + 665856*x^30 + 124848*x^31 + 23409*x^32 + 2754*x^33 + 324*x^34 + 18*x^35 + x^36)/(1-x)^39. (End)
%t A157068 A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
%t A157068 A157068[n_]:= A103881[n, 19];
%t A157068 Table[A157068[n], {n, 50}] (* _G. C. Greubel_, Jan 25 2022 *)
%o A157068 (Sage)
%o A157068 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 A157068 def A157068(n): return A103881(n, 19)
%o A157068 [A157068(n) for n in (1..50)] # _G. C. Greubel_, Jan 25 2022
%Y A157068 Cf. A103881, A156554.
%K A157068 nonn
%O A157068 1,1
%A A157068 _R. H. Hardin_, Feb 22 2009