A098335 -id:A098335 - cp's OEIS frontend

A098331 Expansion of 1/sqrt(1 - 2x + 5x^2).

1, 1, -1, -5, -5, 11, 41, 29, -125, -365, -131, 1409, 3301, -155, -15625, -29485, 16115, 170035, 254525, -309775, -1813055, -2064655, 4617755, 18909175, 14903725, -61552739, -192390589, -81290561, 767919595, 1901796395, 28588201

Offset: 0

Paul Barry, Sep 03 2004

Central coefficients of (1+x-x^2)^n. Binomial transform of 1/sqrt(1+4x^2), or (1,0,-2,0,6,0,-20,...). Binomial transform is A098335. (-1)^nA098331(n) is the inverse binomial transform of (1,0,-2,0,6,0,-20,...).

Hankel transform is 2^n*(-1)^C(n+1,2). Hankel transform of 0,1,1,-1,-5,-5,... is F(n)*(-1)^C(n+2,2)*(2^n+0^n)/2. - Paul Barry, Jan 13 2009

Robert Israel, Table of n, a(n) for n = 0..2865
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Eric Weisstein's World of Mathematics, Trinomial Coefficient

Cf. A098332, A098333, A098334.

A098331 := n -> hypergeom([-n/2, 1/2-n/2], [1], -4);
seq(round(evalf(A098331(n),99)),n=0..30); # Peter Luschny, Sep 18 2014
f:= gfun:-rectoproc({(5*n+5)*a(n)+(-3-2*n)*a(n+1)+(n+2)*a(n+2), a(0) = 1, a(1) = 1},a(n),remember):
map(f, [$0..50]); # Robert Israel, Jan 30 2018

a=b=1; Join[{a, b}, Table[c=((2n-1)b-5(n-1)a)/n; a=b; b=c; c, {n, 2, 30}]] (Noe)
CoefficientList[Series[1/Sqrt[1-2x+5x^2],{x,0,40}],x] (* Harvey P. Dale, Aug 17 2015 *)

my(x='x+O('x^99)); Vec(1/(1-2*x+5*x^2)^(1/2)) \\ Altug Alkan, Mar 18 2018

E.g.f.: exp(x)*BesselI(0, 2*i*x), i=sqrt(-1);
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*(-1)^k;
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*(-1)^k;
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k, k/2)*cos(pi*k/2).
D-finite with recurrence: a(0)=a(1)=1, a(n) = ((2n-1)a(n-1)-5(n-1)a(n-2))/n. - T. D. Noe, Oct 19 2005
a(n) = hypergeom([-n/2, 1/2-n/2], [1], -4). - Peter Luschny, Sep 18 2014
a(n) ~ 5^(n/2 + 1/4) * cos((Pi*n - arctan(1/2) - n*arctan(4/3))/2) / sqrt(Pi*n). - Vaclav Kotesovec, Oct 31 2017
a(n) = (sqrt(5))^n*P(n,1/sqrt(5)), where P(n,x) is the Legendre polynomial of degree n. Note the general result (sqrt(4*m+1))^n*P(n, 1/sqrt(4*m+1)) = Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)(-m)^k due to Catalan. - Peter Bala, Mar 18 2018
G.f.: 1/(1 - x + 2*x^2/(1 - x + x^2/(1 - x + x^2/(1 - x + x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 19 2021
From Peter Bala, Feb 08 2022: (Start)
G.f.: A(x) = Sum_{n >= 0} (-1)^n*binomial(2*n,n)*x^(2*n)/(1 - x)^(2*n+1).
a(n)^2 = Sum_{k = 0..n} (-1)^k*5^(n-k)*binomial(2*k,k)*binomial(n,k)* binomial(n+k,k).
Sum_{n >= 0} (-1)^n*binomial(2*n,n)^2 * x^n/(1 - 5*x)^(2*n+1) = 1 + x + x^2 + 25*x^3 + 25*x^4 + 121*x^5 + ... is the g.f. of a(n)^2.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all prime p and positive integers n and k. (End)

Corrected by T. D. Noe, Oct 19 2005

A123199 Irregular triangle read by rows: row n is the expansion of (1 + 2*x - x^2)^n.

Original entry on oeis.org

1, 1, 2, -1, 1, 4, 2, -4, 1, 1, 6, 9, -4, -9, 6, -1, 1, 8, 20, 8, -26, -8, 20, -8, 1, 1, 10, 35, 40, -30, -68, 30, 40, -35, 10, -1, 1, 12, 54, 100, 15, -168, -76, 168, 15, -100, 54, -12, 1, 1, 14, 77, 196, 161, -238, -427, 184, 427, -238, -161, 196, -77, 14

Offset: 0

Roger L. Bagula, Oct 04 2006

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A007318(n, j)*(2*x)^j*(1 - x^2)^(n-j).

			Triangle begins:
    1;
    1,  2, -1;
    1,  4,  2, -4,   1;
    1,  6,  9, -4,  -9,   6, -1;
    1,  8, 20,  8, -26,  -8, 20, -8,   1;
    1, 10, 35, 40, -30, -68, 30, 40, -35, 10, -1;
    ...

Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, p. 164, figure 26.1.
Henry McKean and Victor Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic, Cambridge University Press, 1997, p. 106, figure 2.22.

G. C. Greubel, Rows n = 0..40 of the irregular triangle, flattened

Row sums: A000079 (powers of 2).

Cf. A122753, A123018, A123019, A123021, A123027, A123202, A123217, A123221.

Table[CoefficientList[(-x^2 + 2*x + 1)^n, x], {n, 0, 10}]//Flatten

create_list(ratcoef((-x^2 + 2*x + 1)^n, x, k), n, 0, 10, k, 0, 2*n); /* Franck Maminirina Ramaharo, Oct 13 2018 */

def T(n): return ( (1+2*x-x^2)^n ).full_simplify().coefficients(sparse=False)
[T(n) for n in (0..12)] # G. C. Greubel, Jul 15 2021

Row n is made of coefficients of: (1 + 2*x - x^2)^n. - Thomas Baruchel, Jan 15 2015
From Franck Maminirina Ramaharo, Oct 13 2018: (Start)
G.f.: 1/(1 - (1 +  2*x - x^2)*y).
E.g.f.: exp((1 +  2*x - x^2)*y).
T(n,1) = A005843(n).
T(n,2) = A014107(n).
T(n,n) = A098335(n). (End)

New name from Thomas Baruchel, Jan 15 2015

Edited, and offset corrected by Franck Maminirina Ramaharo, Oct 13 2018

A294035 a(n) = 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], -1).

Original entry on oeis.org

1, 3, 9, 33, 153, 783, 4059, 21087, 110889, 592899, 3214989, 17608077, 97150491, 539331237, 3010588317, 16887545793, 95134584969, 537942476907, 3051902823849, 17365639042449, 99076018204413, 566622950463099, 3247670747106927, 18651711493531539, 107315246617831179

Offset: 0

Peter Luschny, Nov 02 2017

nonn

Diagonal of rational function 1/(1 - (x^3 + y^3 + z^3 + 3*x*y*z)). - Gheorghe Coserea, Aug 04 2018

Robert Israel, Table of n, a(n) for n = 0..1288

H(1, n, 1) = A000007(n), H(2, n, 1) = A000984(n), H(3, n, 1) = A006077(n), H(4, n, 1) = A294036(n), H(1, n, -1) = A000079(n), H(2, n, -1) = A098335(n), H(3, n, -1) = this seq., H(4, n, -1) = A294037(n).

T := (m,n,x) -> m^n*hypergeom([seq((k-n)/m, k=0..m-1)], [seq(1,k=0..m-2)], x):
seq(simplify(T(3, n, -1)), n=0..39);

Table[3^n * HypergeometricPFQ[{-n/3, (1 - n)/3, (2 - n)/3}, {1, 1}, -1], {n, 0, 30}] (* Vaclav Kotesovec, Nov 02 2017 *)

Let H(m, n, x) = m^n*hypergeom([(k-n)/m for k=0..m-1], [1 for k=0..m-2], x) then a(n) = H(3, n, -1).
a(n) ~ sqrt(3) * 6^n / (Pi*n) . - Vaclav Kotesovec, Nov 02 2017
-(54*(n+2))*(n+1)*a(n)+27*(n+2)^2*a(n+1)-(3*(3*n^2+15*n+19))*a(n+2)+(n+3)^2*a(n+3) = 0. - Robert Israel, Nov 02 2017

A294036 a(n) = 4^n*hypergeom([-n/4, (1-n)/4, (2-n)/4, (3-n)/4], [1, 1, 1], 1).

Original entry on oeis.org

1, 4, 16, 64, 280, 1504, 9856, 70144, 498136, 3449440, 23506816, 160566784, 1115048896, 7905796864, 56994288640, 414928113664, 3034880623576, 22255957312864, 163667338903936, 1208070406612480, 8955840250934080, 66678657938510080

Offset: 0

Peter Luschny, Nov 02 2017

nonn

Diagonal of rational function 1/(1 - (x^4 + y^4 + z^4 + t^4 + 4*x*y*z*t)). - Gheorghe Coserea, Aug 04 2018

H(1, n, 1) = A000007(n), H(2, n, 1) = A000984(n), H(3, n, 1) = A006077(n), H(4, n, 1) = this seq., H(1, n, -1) = A000079(n), H(2, n, -1) = A098335(n), H(3, n, -1) = A294035(n), H(4, n, -1) = A294037(n).

T := (m,n,x) -> m^n*hypergeom([seq((k-n)/m, k=0..m-1)], [seq(1, k=0..m-2)], x):
lprint(seq(simplify(T(4,n,1)), n=0..39));

Table[4^n * HypergeometricPFQ[{-n/4, (1-n)/4, (2-n)/4, (3-n)/4}, {1, 1, 1}, 1], {n, 0, 30}] (* Vaclav Kotesovec, Nov 02 2017 *)

Let H(m, n, x) = m^n*hypergeom([(k-n)/m for k=0..m-1], [1 for k=0..m-2], x) then a(n) = H(4, n, 1).

a(n) ~ 2^(3*n + 2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Nov 02 2017

A294037 a(n) = 4^n*hypergeom([-n/4, (1-n)/4, (2-n)/4, (3-n)/4], [1, 1, 1], -1).

Original entry on oeis.org

1, 4, 16, 64, 232, 544, -1664, -37376, -362024, -2743712, -17780864, -98955776, -442825664, -1129423616, 5536033792, 118591811584, 1224814969816, 9905491019104, 68032143081856, 398051159254528, 1854461906222272, 4784426026102528

Offset: 0

Peter Luschny, Nov 02 2017

sign

Diagonal of rational function 1/(1 - (x^4 + y^4 + z^4 - t^4 + 4*x*y*z*t)). - Gheorghe Coserea, Aug 04 2018

H(1, n, 1) = A000007(n), H(2, n, 1) = A000984(n), H(3, n, 1) = A006077(n), H(4, n, 1) = A294036(n), H(1, n, -1) = A000079(n), H(2, n, -1) = A098335(n), H(3, n, -1) = A294035(n), H(4, n, -1) = this seq..

T := (m,n,x) -> m^n*hypergeom([seq((k-n)/m, k=0..m-1)], [seq(1, k=0..m-2)], x):
lprint(seq(simplify(T(4,n,-1)), n=0..39));

Table[4^n * HypergeometricPFQ[{-n/4, (1-n)/4, (2-n)/4, (3-n)/4}, {1, 1, 1}, -1], {n, 0, 20}] (* Vaclav Kotesovec, Nov 02 2017 *)

Let H(m, n, x) = m^n*hypergeom([(k-n)/m for k=0..m-1], [1 for k=0..m-2], x) then a(n) = H(4, n, -1).

A098338 Expansion of 1/sqrt(1-6x+13x^2).

Original entry on oeis.org

1, 3, 7, 9, -21, -207, -911, -2769, -5213, 2457, 74997, 400491, 1409109, 3323583, 2219343, -27453951, -186624333, -750905127, -2088947819, -2955863589, 8506703569, 86421384387, 401183114163, 1280139325101, 2522745571021

Offset: 0

Paul Barry, Sep 03 2004

Binomial transform of A098335. Second binomial transform of A098331.

Central coefficients of (1+3x-x^2)^n.

Robert Israel, Table of n, a(n) for n = 0..1796
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Robert Israel, Plot of a(n) sqrt(n)/13^(n/2) for 1<=n<=10000.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

f:= gfun:-rectoproc({(13*n+13)*a(n)+(-9-6*n)*a(n+1)+(n+2)*a(n+2), a(0)=1, a(1)=3},a(n),remember):
map(f, [$0..50]); # Robert Israel, Mar 02 2017

CoefficientList[Series[1/Sqrt[1-6*x+13*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 29 2013 *)

E.g.f.: exp(3*x)*BesselI(0, 2*I*x), I=sqrt(-1).
a(n) = Sum{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*3^n*(-9)^(-k).
a(n) = Sum{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*3^n*(-9)^(-k).
D-finite with recurrence: n*a(n) +3*(1-2*n)*a(n-1) +13*(n-1)*a(n-2)=0. - R. J. Mathar, Sep 26 2012
Recurrence follows from the differential equation (13x-3) g(x) + (13x^2-6x+1) g'(x) = 0 satisfied by the generating function. - Robert Israel, Mar 02 2017
Lim sup n->infinity |a(n)|^(1/n) = sqrt(13). - Vaclav Kotesovec, Sep 29 2013

cp's OEIS Frontend

A098331 Expansion of 1/sqrt(1 - 2*x + 5*x^2).

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A123199 Irregular triangle read by rows: row n is the expansion of (1 + 2*x - x^2)^n.

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A294035 a(n) = 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], -1).

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A294036 a(n) = 4^n*hypergeom([-n/4, (1-n)/4, (2-n)/4, (3-n)/4], [1, 1, 1], 1).

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A294037 a(n) = 4^n*hypergeom([-n/4, (1-n)/4, (2-n)/4, (3-n)/4], [1, 1, 1], -1).

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A098338 Expansion of 1/sqrt(1-6x+13x^2).

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A098331 Expansion of 1/sqrt(1 - 2x + 5x^2).