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%I #11 Sep 26 2022 13:58:34
%S 2,84,1962,32130,406800,4208610,36881420,280819260,1893408750,
%T 11472968760,63221641758,319917948246,1498750896708,6545498596110,
%U 26808012135000,103501142484360,378407481456870,1315394383751460,4363052456797550,13853429338548630
%N Number of integer sequences of length n+1 with sum zero and sum of absolute values 28.
%H T. D. Noe, <a href="/A157063/b157063.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_29">Index entries for linear recurrences with constant coefficients</a>, signature (29, -406, 3654, -23751, 118755, -475020, 1560780, -4292145, 10015005, -20030010, 34597290, -51895935, 67863915, -77558760, 77558760, -67863915, 51895935, -34597290, 20030010, -10015005, 4292145, -1560780, 475020, -118755, 23751, -3654, 406, -29, 1).
%F a(n) = T(n,14); 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 24 2022: (Start)
%F a(n) = (n+1)*binomial(n+13, 14)*Hypergeometric3F2([-13, -n, 1-n], [2, -n-13], 1).
%F a(n) = (40116600/28!)*n*(n+1)*(542861032610856960000 + 1222285449585328128000*n + 1949147924290921267200*n^2 + 1623917017366475120640*n^3 + 1256475121883342659584*n^4 + 587860847016245577216*n^5 + 290144191606881266304*n^6 + 87769963981312971072*n^7 + 31017509312522326880*n^8 + 6493644952485577744*n^9 + 1754084550497496360*n^10 + 263474544214276252*n^11 + 56764614862429890*n^12 + 6225163072052509*n^13 + 1102423601827845*n^14 + 88588233707662*n^15 + 13189509162960*n^16 + 769151138899*n^17 + 97984044015*n^18 + 4039324432*n^19 + 446558970*n^20 + 12345619*n^21 + 1198275*n^22 + 19942*n^23 + 1716*n^24 + 13*n^25 + n^26).
%F G.f.: 2*x*(1 + 13*x + 169*x^2 + 1014*x^3 + 6084*x^4 + 22308*x^5 + 81796*x^6 + 204490*x^7 + 511225*x^8 + 920205*x^9 + 1656369*x^10 + 2208492*x^11 + 2944656*x^12 + 2944656*x^13 + 2944656*x^14 + 2208492*x^15 + 1656369*x^16 + 920205*x^17 + 511225*x^18 + 204490*x^19 + 81796*x^20 + 22308*x^21 + 6084*x^22 + 1014*x^23 + 169*x^24 + 13*x^25 + x^26)/(1-x)^29. (End)
%t A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
%t A157063[n_]:= A103881[n, 14];
%t Table[A157063[n], {n, 50}] (* _G. C. Greubel_, Jan 24 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 A157063(n): return A103881(n, 14)
%o [A157063(n) for n in (1..50)] # _G. C. Greubel_, Jan 24 2022
%Y Cf. A103881, A156554.
%K nonn
%O 1,1
%A _R. H. Hardin_, Feb 22 2009
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