%I #11 Jan 25 2022 06:19:58
%S 2,72,1442,20260,220250,1958460,14768810,96900810,563873400,
%T 2953859370,14097919968,61908797418,252208679268,959882556570,
%U 3433533723900,11603837100660,37221177046410,113779617228060,332648955112250,933146517188760,2518877938240202
%N Number of integer sequences of length n+1 with sum zero and sum of absolute values 24.
%H T. D. Noe, <a href="/A157061/b157061.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_25">Index entries for linear recurrences with constant coefficients</a>, signature (25,-300,2300,-12650,53130,-177100,480700,-1081575, 2042975,-3268760,4457400,-5200300,5200300,-4457400,3268760,-2042975,1081575, -480700,177100,-53130,12650,-2300,300,-25,1).
%F a(n) = T(n,12); 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+11, 12)*Hypergeometric3F2([-11, -n, 1-n], [2, -n-11], 1).
%F a(n) = (2704156/24!)*n*(n+1)*(19120211066880000 + 40213832085504000*n + 61866024285081600*n^2 + 47770238895160320*n^3 + 35477403021764352*n^4 + 15129353226246336*n^5 + 7138320279252096*n^6 + 1926081009812080*n^7 + 648411230685152*n^8 + 117787792143956*n^9 + 30215435337736*n^10 + 3799367698665*n^11 + 775177128207*n^12 + 67808650591*n^13 + 11342892341*n^14 + 678888650*n^15 + 95251222*n^16 + 3725106*n^17 + 446226*n^18 + 10285*n^19 + 1067*n^20 + 11*n^21 + n^22).
%F G.f.: 2*x*(1 + 11*x + 121*x^2 + 605*x^3 + 3025*x^4 + 9075*x^5 + 27225*x^6 + 54450*x^7 + 108900*x^8 + 152460*x^9 + 213444*x^10 + 213444*x^11 + 213444*x^12 + 152460*x^13 + 108900*x^14 + 54450*x^15 + 27225*x^16 + 9075*x^17 + 3025*x^18 + 605*x^19 + 121*x^20 + 11*x^21 + x^22)/(1-x)^25. (End)
%t A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2,1-n - k}, 1];
%t A157061[n_]:= A103881[n, 12];
%t Table[A157061[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 A157061(n): return A103881(n, 12)
%o [A157061(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