A084771 Coefficients of expansion of 1/sqrt(1 - 10*x + 9*x^2); also, a(n) is the central coefficient of (1 + 5*x + 4*x^2)^n.

1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525, 653808673, 5614995765, 48416454529, 418895174885, 3634723102113, 31616937184725, 275621102802945, 2407331941640325, 21061836725455905, 184550106298084725

Offset: 0

Author

Paul D. Hanna, Jun 10 2003

Keywords

• nonn
• easy

Comments

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps come in four colors and the H steps come in five colors. - N-E. Fahssi, Mar 30 2008

Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and three kinds of steps (1,1). - Joerg Arndt, Jul 01 2011

Sums of squares of coefficients of (1+2*x)^n. - Joerg Arndt, Jul 06 2011

The Hankel transform of this sequence gives A103488. - Philippe Deléham, Dec 02 2007

Partial sums of A085363. - J. M. Bergot, Jun 12 2013

Diagonal of rational functions 1/(1 - x - y - 3*x*y), 1/(1 - x - y*z - 3*x*y*z). - Gheorghe Coserea, Jul 06 2018

Examples

			G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
• Tewodros Amdeberhan, In search of multiple expressions for a sequence
• Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
• Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
• Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
• Curtis Greene, Posets of shuffles, Journal of Combinatorial Theory, Series A 47.2 (1988): 191-206. See Eq. (30).
• Christopher Huffaker, Nathan McCue, Cameron N. Miller, and Kayla S. Miller, The M&M Game: From Morsels to Modern Mathematics, arXiv:1508.06542 [math.HO], 2015.
• Greg Morrow, Some probability distributions and integer sequences related to rook paths, Univ. Colorado Springs (2024). See p. 3.
• Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
• Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
• Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 97.

Crossrefs

Cf. A001850, A059231, A059304, A246923 (a(n)^2).

Programs

GAP

List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*4^k)); # Muniru A Asiru, Jul 29 2018

Magma

[3^n*Evaluate(LegendrePolynomial(n), 5/3) : n in [0..40]]; // G. C. Greubel, May 30 2023

Maple

seq(simplify(hypergeom([-n,1/2], [1], -8)),n=0..19); # Peter Luschny, Apr 26 2016

Mathematica

Table[n! SeriesCoefficient[E^(5 x) BesselI[0, 4 x], {x, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, May 10 2013 *)
Table[Hypergeometric2F1[-n, -n, 1, 4], {n,0,30}] (* Vladimir Reshetnikov, Nov 29 2013 *)
CoefficientList[Series[1/Sqrt[1-10x+9x^2],{x,0,30}],x] (* Harvey P. Dale, Mar 08 2016 *)
Table[3^n*LegendreP[n, 5/3], {n, 0, 40}] (* G. C. Greubel, May 30 2023 *)

PARI

{a(n) = if( n<0, -3 * 9^n * a(-1-n), sum(k=0,n, binomial(n, k)^2 * 4^k))}; /* Michael Somos, Oct 08 2003 */

PARI

{a(n) = if( n<0, -3 * 9^n * a(-1-n), polcoeff((1 + 5*x + 4*x^2)^n, n))}; /* Michael Somos, Oct 08 2003 */

PARI

/* as lattice paths: same as in A092566 but use */
steps=[[1,0], [0,1], [1,1], [1,1], [1,1]]; /* note the triple [1,1] */
/* Joerg Arndt, Jul 01 2011 */

PARI

a(n)={local(v=Vec((1+2*x)^n));sum(k=1,#v,v[k]^2);} /* Joerg Arndt, Jul 06 2011 */

PARI

a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1,#v, real(v[k])^2+imag(v[k])^2);} /* Joerg Arndt, Jul 06 2011 */

SageMath

[3^n*gen_legendre_P(n, 0, 5/3) for n in range(41)] # G. C. Greubel, May 30 2023

Formula

G.f.: 1 / sqrt(1 - 10*x + 9*x^2).

From Vladeta Jovovic, Aug 20 2003: (Start)

Binomial transform of A059304.

G.f.: Sum_{k >= 0} binomial(2*k,k)*(2*x)^k/(1-x)^(k+1).

E.g.f.: exp(5*x)*BesselI(0, 4*x). (End)

a(n) = Sum_{k = 0..n} Sum_{j = 0..n-k} C(n,j)*C(n-j,k)*C(2*n-2*j,n-j). - Paul Barry, May 19 2006

a(n) = Sum_{k = 0..n} 4^k*C(n,k)^2. - heruneedollar (heruneedollar(AT)gmail.com), Mar 20 2010

a(n) ~ 3^(2*n+1)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Sep 11 2012

D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-1)*a(n-2). - R. J. Mathar, Nov 26 2012

a(n) = hypergeom([-n, -n], [1], 4). - Vladimir Reshetnikov, Nov 29 2013

a(n) = hypergeom([-n, 1/2], [1], -8). - Peter Luschny, Apr 26 2016

From Michael Somos, Jun 01 2017: (Start)

a(n) = -3 * 9^n * a(-1-n) for all n in Z.

0 = a(n)*(+81*a(n+1) -135*a(n+2) +18*a(n+3)) +a(n+1)*(-45*a(n+1) +100*a(n+2) -15*a(n+3)) +a(n+2)*(-5*a(n+2) +a(n+3)) for all n in Z. (End)

From Peter Bala, Nov 13 2022: (Start)

1 + x*exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 5*x^2 + 29*x^3 + 185*x^4 + 1257*x^5 + ... is the g.f. of A059231.

The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all positive integers n and r and all primes p. (End)

a(n) = 3^n * LegendreP(n, 5/3). - G. C. Greubel, May 30 2023

a(n) = (1/4)^n * Sum_{k=0..n} 9^k * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025