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A081671 Expansion of e.g.f. exp(4x) * I_0(2x).

%I A081671 #96 Aug 18 2025 09:34:42
%S A081671 1,4,18,88,454,2424,13236,73392,411462,2325976,13233628,75682512,
%T A081671 434662684,2505229744,14482673832,83940771168,487610895942,
%U A081671 2838118247064,16547996212044,96635257790352,565107853947444,3308820294176016,19395905063796312,113814537122646432
%N A081671 Expansion of e.g.f. exp(4x) * I_0(2x).
%C A081671 Binomial transform of A026375. Second binomial transform of A000984.
%C A081671 Largest coefficient of (1+4x+x^2)^n. - _Paul Barry_, Dec 15 2003
%C A081671 Row sums of triangle in A124574. - _Philippe Deléham_, Sep 27 2007
%C A081671 Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in 4 colors. - _N-E. Fahssi_, Feb 05 2008
%C A081671 Diagonal of rational function 1/(1 - (x^2 + 4*x*y + y^2)). - _Gheorghe Coserea_, Aug 01 2018
%H A081671 Vincenzo Librandi, <a href="/A081671/b081671.txt">Table of n, a(n) for n = 0..200</a>
%H A081671 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 A081671 Isaac DeJager, Madeleine Naquin and Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%H A081671 Nachum Dershowitz, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Dershowitz/dersh3.html">Touchard's Drunkard</a>, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
%H A081671 Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, <a href="https://arxiv.org/abs/1110.6638">Sato-Tate distributions and Galois endomorphism modules in genus 2</a>, arXiv preprint arXiv:1110.6638 [math.NT], 2011.
%H A081671 Francesc Fite and Andrew V. Sutherland, <a href="http://arxiv.org/abs/1203.1476">Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1</a>, arXiv preprint arXiv:1203.1476 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 14 2012
%H A081671 Rigoberto Flórez, Leandro Junes and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Florez/florez4.html">Further Results on Paths in an n-Dimensional Cubic Lattice</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
%H A081671 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 A081671 Paveł Szabłowski, <a href="https://cdm.ucalgary.ca/article/view/76214">Beta distributions whose moment sequences are related to integer sequences listed in the OEIS</a>, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 97.
%F A081671 a(n) = Sum_{m=0..n} Sum_{k=0..m} C(n, m)*C(m, k)*C(2k, k).
%F A081671 G.f.: 1/sqrt((1-2*x)*(1-6*x)). - _Vladeta Jovovic_, Oct 09 2003
%F A081671 a(n) = Sum_{k=0..n} 2^(n-k) * C(n, k) * C(2*k, k). - _Paul Barry_, Jan 27 2005
%F A081671 a(n) = Sum_{k=0..n} 6^(n-k) * (-1)^k * C(n,k) * C(2*k,k). - _Paul D. Hanna_, Dec 09 2018
%F A081671 D-finite with recurrence: n*a(n) = 4*(2*n-1)*a(n-1) - 12*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 13 2012
%F A081671 a(n) ~ sqrt(3/(2*Pi*n))*6^n. - _Vaclav Kotesovec_, Oct 13 2012
%F A081671 a(n) = 2^n*hypergeom([-n,1/2], [1], -2). - _Peter Luschny_, Apr 26 2016
%F A081671 a(n) = GegenbauerC(n, -n, -2). - _Peter Luschny_, May 09 2016
%F A081671 a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - _Seiichi Manyama_, May 04 2019
%F A081671 a(n) = (1/Pi) * Integral_{x = -1..1} (2 + 4*x^2)^n/sqrt(1 - x^2) dx  = (1/Pi) * Integral_{x = -1..1} (6 - 4*x^2)^n/sqrt(1 - x^2) dx . - _Peter Bala_, Jan 27 2020
%F A081671 From _Peter Bala_, Jan 10 2022: (Start)
%F A081671 exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + 4*x + 17*x^2 + 76*x^3 + 354*x^4 + ... is the o.g.f. of A005572.
%F A081671 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. (End)
%F A081671 a(n) = (1/2)^n * Sum_{k=0..n} 3^k * binomial(2*k,k) * binomial(2*(n-k),n-k). - _Seiichi Manyama_, Aug 18 2025
%p A081671 seq(simplify(2^n*hypergeom([-n,1/2], [1], -2)),n=0..23); # _Peter Luschny_, Apr 26 2016
%p A081671 seq(simplify(GegenbauerC(n,-n,-2)),n=0..23); # _Peter Luschny_, May 09 2016
%t A081671 Table[SeriesCoefficient[1/Sqrt[(1-2*x)*(1-6*x)],{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 13 2012 *)
%o A081671 (Maxima) a(n):=coeff(expand((1+4*x+x^2)^n),x^n);
%o A081671 makelist(a(n),n,0,30); /* _Emanuele Munarini_, Apr 27 2012 */
%o A081671 (PARI) x='x+O('x^66); Vec(1/sqrt((1-2*x)*(1-6*x))) \\ _Joerg Arndt_, May 07 2013
%o A081671 (PARI) {a(n) = sum(k=0, n\2, 4^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ _Seiichi Manyama_, May 04 2019
%Y A081671 Column 4 of A292627.
%Y A081671 m-th binomial transforms of A000984: A126869 (m = -2), A002426 (m = -1 and m = -3 for signed version), A000984 (m = 0 and m = -4 for signed version), A026375 (m = 1 and m = -5 for signed version), A081671 (m = 2 and m = -6 for signed version), A098409 (m = 3 and m = -7 for signed version), A098410 (m = 4 and m = -8 for signed version), A104454 (m = 5 and m = -9 for signed version).
%K A081671 nonn,easy
%O A081671 0,2
%A A081671 _Paul Barry_, Mar 28 2003