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%I #12 Nov 02 2021 07:37:02
%S 1,2,7,29,128,587,2759,13190,63844,311948,1535488,7602971,37829455,
%T 188989166,947399951,4763280965,24009574400,121291129748,613939110308,
%U 3112989719080,15809048927000,80397234851080,409378690617344,2086928493438299,10649867701045871
%N a(n) = Sum_{k=0..n} binomial(n, k)^2 * hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).
%C Binomial convolution of the Motzkin numbers.
%F a(n) ~ sqrt((76 + 5*(38*(247 - 27*sqrt(57)))^(1/3) + 5*(38*(247 + 27*sqrt(57)))^(1/3))/57)/4 * ((1261 + 57*sqrt(57))^(1/3)/6 + 56/(3*(1261 + 57*sqrt(57))^(1/3)) + 5/3)^n / sqrt(Pi*n). - _Vaclav Kotesovec_, May 24 2021
%F Conjecture D-finite with recurrence -4*(3035*n-11997) *(2*n+1) *(n+1) *a(n) +2*(153546*n^3-490325*n^2-62942*n+71982) *a(n-1) +2*(-452090*n^3+1745622*n^2-1405285*n-226200) *a(n-2) -2 *(n-2)*(83741*n^2-200458*n-19650) *a(n-3) -3*(n-2) *(n-3) *(46423*n+6938) *a(n-4)=0. - _R. J. Mathar_, Nov 02 2021
%p a := n -> add(binomial(n, k)^2 * hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4), k = 0..n); seq(simplify(a(n)), n = 0..24);
%Y Cf. A064189 (Motzkin numbers), A344503.
%Y Cf. A348840.
%K nonn
%O 0,2
%A _Peter Luschny_, May 23 2021
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