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A006442 Expansion of 1/sqrt(1 - 10*x + x^2).

%I A006442 #107 Aug 30 2025 10:25:29
%S A006442 1,5,37,305,2641,23525,213445,1961825,18205345,170195525,1600472677,
%T A006442 15122515985,143457011569,1365435096485,13033485491077,
%U A006442 124715953657025,1195966908404545,11490534389896325,110584004488276645,1065853221648055025
%N A006442 Expansion of 1/sqrt(1 - 10*x + x^2).
%C A006442 Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where H can choose from two colors. - _Paul Barry_, May 25 2005
%C A006442 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,1), and two kinds of steps (1,0). - _Joerg Arndt_, Jul 01 2011
%C A006442 The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - _Peter Bala_, Jan 09 2022
%H A006442 Seiichi Manyama, <a href="/A006442/b006442.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from T. D. Noe)
%H A006442 Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%H A006442 Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%H A006442 Ömür Deveci and Anthony G. Shannon, <a href="https://doi.org/10.20948/mathmontis-2021-50-4">Some aspects of Neyman triangles and Delannoy arrays</a>, Mathematica Montisnigri (2021) Vol. L, 36-43.
%H A006442 Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%H A006442 T. R. S. Walsh, <a href="/A007401/a007401.pdf">Number of sensed planar maps with n edges and m vertices</a>
%F A006442 Legendre polynomial evaluated at 5. - _Michael Somos_, Dec 04 2001
%F A006442 G.f.: 1/sqrt(1 - 10*x + x^2).
%F A006442 a(n) equals the central coefficient of (1 + 5*x + 6*x^2)^n. - _Paul D. Hanna_, Jun 03 2003
%F A006442 a(n) equals the (n+1)-th term of the binomial transform of 1/(1-2x)^(n+1). - _Paul D. Hanna_, Sep 29 2003
%F A006442 a(n) = Sum_{k=0..n} 2^k*binomial(n, k)*binomial(n+k, k). - _Benoit Cloitre_, Apr 13 2004
%F A006442 a(n) = Sum_{k=0..n} binomial(n,k)^2 * 2^k * 3^(n-k). - _Paul D. Hanna_, Feb 04 2012
%F A006442 E.g.f.: exp(5*x) * Bessel_I(0, 2*sqrt(6)*x). - _Paul Barry_, May 25 2005
%F A006442 D-finite with recurrence: n*a(n) - 5*(2n-1)*a(n-1) + (n-1)*a(n-2) = 0 [Eq (4) in the T. D. Noe article]. _R. J. Mathar_, Jun 26 2012
%F A006442 a(n) ~ (5 + 2*sqrt(6))^n/(2*sqrt(Pi*n)*sqrt(5*sqrt(6) - 12)). - _Vaclav Kotesovec_, Oct 05 2012
%F A006442 a(n) = hypergeom([-n, n+1], [1], -2). - _Peter Luschny_, May 23 2014
%F A006442 a(n) = Sum_{k=0..n} 2^k * C(2*k, k) * C(n+k, n-k). - _Paul D. Hanna_, Aug 17 2014
%F A006442 a(n) = Sum_{k=0..n} (k+1) * 3^k * (-1)^(n-k) * binomial(n,k) * binomial(n+k+1,n) / (n+k+1). - _Vladimir Kruchinin_, Nov 23 2014
%F A006442 From _Peter Bala_, Nov 28 2021: (Start)
%F A006442 a(n) = (1/3)*(1/2)^n*Sum_{k >= n} binomial(k,n)^2*(2/3)^k.
%F A006442 a(n) = (1/3)^(n+1)*hypergeom([n+1, n+1], [1], 2/3).
%F A006442 a(n) = (2^n)*hypergeom([-n, -n], [1], 3/2).
%F A006442 a(n) = [x^n] ((x - 1)*(3 - 2*x))^n
%F A006442 a(n) = (1/2)^n*A098270(n). (End)
%F A006442 a(n) = (-1)^n * Sum_{k=0..n} (1/10)^(n-2*k) * binomial(-1/2,k) * binomial(k,n-k). - _Seiichi Manyama_, Aug 28 2025
%F A006442 a(n) = Sum_{k=0..floor(n/2)} 6^k * 5^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - _Seiichi Manyama_, Aug 30 2025
%p A006442 seq(orthopoly[P](n,5), n = 0 .. 20); # _Robert Israel_, Aug 18 2014
%t A006442 Table[LegendreP[n, 5], {n, 0, 19}] (* _Arkadiusz Wesolowski_, Aug 13 2012 *)
%t A006442 CoefficientList[Series[1 / Sqrt[1 - 10 x + x^2], {x, 0, 20}], x] (* _Vincenzo Librandi_, Nov 23 2014 *)
%o A006442 (PARI) a(n)=subst(pollegendre(n),x,5)
%o A006442 (PARI) /* as lattice paths: same as in A092566 but use */
%o A006442 steps=[[1,0], [1,0], [0,1], [1,1]]; /* note the double [1,0] */
%o A006442 /* _Joerg Arndt_, Jul 01 2011 */
%o A006442 (PARI) {a(n)=sum(k=0,n,binomial(n,k)^2*2^k*3^(n-k))} /* _Paul D. Hanna_ */
%o A006442 (PARI) {a(n) = sum(k=0, n, 2^k * binomial(2*k, k) * binomial(n+k, n-k) )}
%o A006442 for(n=0, 25, print1(a(n), ", ")) \\ _Paul D. Hanna_, Aug 17 2014
%o A006442 (Magma) [Evaluate(LegendrePolynomial(n), 5): n in [0..40]]; // _G. C. Greubel_, May 21 2023
%o A006442 (SageMath) [gen_legendre_P(n,0,5) for n in range(41)] # _G. C. Greubel_, May 21 2023
%Y A006442 Column k=2 of A335333.
%Y A006442 Sequences of the form LegendreP(n, 2*m+1): A000012 (m=0), A001850 (m=1), this sequence (m=2), A084768 (m=3), A084769 (m=4).
%Y A006442 Cf. A098270, A243943 (a(n)^2).
%K A006442 nonn,easy,changed
%O A006442 0,2
%A A006442 _N. J. A. Sloane_