A098264 -id:A098264 - cp's OEIS frontend

A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1-4k)*x^2).

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 13, 19, 1, 1, 1, 9, 19, 49, 51, 1, 1, 1, 11, 25, 91, 161, 141, 1, 1, 1, 13, 31, 145, 331, 581, 393, 1, 1, 1, 15, 37, 211, 561, 1441, 2045, 1107, 1, 1, 1, 17, 43, 289, 851, 2841, 5797, 7393, 3139, 1

Offset: 0

Seiichi Manyama, May 01 2019

			Square array begins:
   1,   1,    1,    1,     1,     1,     1, ...
   1,   1,    1,    1,     1,     1,     1, ...
   1,   3,    5,    7,     9,    11,    13, ...
   1,   7,   13,   19,    25,    31,    37, ...
   1,  19,   49,   91,   145,   211,   289, ...
   1,  51,  161,  331,   561,   851,  1201, ...
   1, 141,  581, 1441,  2841,  4901,  7741, ...
   1, 393, 2045, 5797, 12489, 22961, 38053, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
T. D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Columns k=0..6 give A000012, A002426, A084601, A084603, A084605, A098264, A098265.
Main diagonal gives A187018.
Cf. A110180, A292627, A307847, A307860, A307910.

T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

A(n,k) is the coefficient of x^n in the expansion of (1 + x + k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * binomial(2*j,j).
D-finite with recurrence: n * A(n,k) = (2*n-1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).

A098443 Expansion of 1/sqrt(1-8x-4x^2).

Original entry on oeis.org

1, 4, 26, 184, 1366, 10424, 80996, 637424, 5064166, 40528984, 326251276, 2638751504, 21426682876, 174563719984, 1426219233416, 11681133293024, 95877105146246, 788433553532824, 6494463369141116, 53576199709855184

Offset: 0

Paul Barry, Sep 07 2004

Binomial transform of A098444. Second binomial transform of A084770. Third binomial transform of A098264.

			G.f. = 1 + 4*x + 26*x^2 + 184*x^3 + 1366*x^4 + 10424*x^5 + 80996*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Column k=2 of A386621.

Cf. A006139, A126869.

CoefficientList[Series[1/Sqrt[1 - 8*x - 4*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 15 2012, updated Mar 21 2024 *)

x='x+O('x^66); Vec(1/sqrt(1-8*x-4*x^2)) \\ Joerg Arndt, May 11 2013

E.g.f.: exp(4*x) * BesselI(0, 2*sqrt(5)*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k) * binomial(2(n-k), n) * 2^(n-2k).
D-finite with recurrence: n*a(n) = 4*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) ~ sqrt(50+20*sqrt(5))*(4+2*sqrt(5))^n/(10*sqrt(Pi*n)). Equivalently, a(n) ~ 2^(n-1/2) * phi^(3*n + 3/2) / (5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Oct 15 2012, updated Mar 21 2024
G.f.: 1/(1 - 2*x*(2+x)*Q(0)), where Q(k)= 1 + (4*k+1)*x*(2+x)/(k+1 - x*(2+x)*(2*k+2)*(4*k+3)/(2*x*(2+x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: Q(0), where Q(k) = 1 + 2*x*(x+2)*(4*k+1)/( 2*k+1 - x*(x+2)*(2*k+1)*(4*k+3)/(x*(x+2)*(4*k+3) + (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 16 2013
From Peter Bala, Mar 16 2024: (Start)
a(n) = (-2*i)^n * P(n, 2*i), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial.
Sum_{n >= 1} (-1)^(n+1)*4^n/(n*a(n-1)*a(n)) = 2*arctan(1/2) = 2*A073000. (End)
From Seiichi Manyama, Aug 29 2025: (Start)
a(n) = Sum_{k=0..n} (2-i)^k * (2+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.
a(n) = Sum_{k=0..floor(n/2)} 5^k * 4^(n-2*k) * binomial(n,2*k) * binomial(2*k,k).
a(n) = [x^n] (1+4*x+5*x^2)^n. (End)

A110180 Triangle of generalized central trinomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 5, 1, 1, 1, 19, 13, 7, 1, 1, 1, 51, 49, 19, 9, 1, 1, 1, 141, 161, 91, 25, 11, 1, 1, 1, 393, 581, 331, 145, 31, 13, 1, 1, 1, 1107, 2045, 1441, 561, 211, 37, 15, 1, 1, 1, 3139, 7393, 5797, 2841, 851, 289, 43, 17, 1, 1

Offset: 0

Paul Barry, Jul 14 2005

Rows sums are A110181. Diagonal sums are A110182. Columns include central trinomial coefficients A002426, A084601, A084603, A084605, A098264. T(n,k) = central coefficient (1 + x + kx^2)^n.

			Rows begin
  1;
  1,  1;
  1,  1,  1;
  1,  3,  1,  1;
  1,  7,  5,  1,  1;
  1, 19, 13,  7,  1,  1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[n_, 0] := 1; T[n_, k_] := Sum[Binomial[n - k, j]*Binomial[n - k - j, j]*k^j, {j, 0, Floor[(n - k)/2]}]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Mar 05 2017 *)

Number triangle T(n, k) = Sum_{j=0..floor((n-k)/2)} C(n-k, j)*C(n-k-j, j)*k^j.

A098265 G.f. : 1/(1-2x-23x^2)^(1/2).

Original entry on oeis.org

1, 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357, 232260610177, 1327920945601, 7627285093069, 43787832627493, 252042452907169, 1451244932278129, 8370001674641917, 48303478743113893, 279083099450496961

Offset: 0

Paul Barry, Aug 31 2004

Central coefficient of (1+x+6x^2)^n.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Cf. A084601, A084603, A084605, A098264.

Table[SeriesCoefficient[1/Sqrt[1-2*x-23*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

x='x+O('x^66); Vec(1/(1-2*x-23*x^2)^(1/2)) \\ Joerg Arndt, May 11 2013

E.g.f.: exp(x)*BesselI(0, 2*sqrt(6)x).
a(n) = sum{k=0..floor(n/2), binomial(n, k)*binomial(n-k, k)*6^k}.
a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*binomial(2k, k)*6^k}.
n*a(n) +(1-2n)*a(n-1) +23(1-n)*a(n-2)=0. (Recurrence (4) in the Noe paper).- Veka Gesell, Jun 26 2012
a(n) ~ sqrt(72+6*sqrt(6))*(1+2*sqrt(6))^n/(12*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

A098444 Expansion of 1/sqrt(1-6x-11x^2).

Original entry on oeis.org

1, 3, 19, 117, 771, 5193, 35629, 247467, 1734931, 12250953, 87006249, 620818047, 4447016781, 31959556983, 230331965379, 1664043517557, 12047551338771, 87387014213433, 634918255153369, 4619923954541247, 33661450900419001

Offset: 0

Paul Barry, Sep 07 2004

Binomial transform of A084770. Second binomial transform of A098264. Binomial transform is A098443.

Coefficient of x^n in (1 + 3 x + 5 x^2)^n = number of paths from the origin to (n,0) with steps U=(1,1), H=(1,0) and D=(1,-1); U can have 5 colors and H can have 3 colors. - N-E. Fahssi, Jan 28 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Table[SeriesCoefficient[1/Sqrt[1-6*x-11*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)

x='x+O('x^66); Vec(1/sqrt(1-6*x-11*x^2)) \\ Joerg Arndt, May 11 2013

E.g.f.: exp(3x)*BesselI(0, 2*sqrt(5)*x)
D-finite with recurrence: n*a(n) = 3*(2*n-1)*a(n-1) + 11*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) ~ sqrt(50+15*sqrt(5))*(3+2*sqrt(5))^n/(10*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012

A127947 Hankel transform of central coefficients of (1+k*x+5x^2)^n, k arbitrary integer.

Original entry on oeis.org

1, 10, 500, 125000, 156250000, 976562500000, 30517578125000000, 4768371582031250000000, 3725290298461914062500000000, 14551915228366851806640625000000000

Offset: 0

Paul Barry, Feb 08 2007

Hankel transform of A098264. The Hankel transform of e.g.f. Bessel_I(0,2*sqrt(5)x) and its k-th binomial transforms, are given by a(n). In general, the Hankel transform of e.g.f. Bessel_I(0,2*sqrt(m)x) and its binomial transforms is 2^n*m^C(n+1,2).

G. C. Greubel, Table of n, a(n) for n = 0..52

[2^n*5^Binomial(n+1,2): n in [0..30]]; // G. C. Greubel, May 03 2018

Table[2^n*5^Binomial[n+1,2], {n,0,30}] (* G. C. Greubel, May 03 2018 *)

for(n=0, 30, print1(2^n*5^binomial(n+1,2), ", ")) \\ G. C. Greubel, May 03 2018

a(n) = 2^n*5^C(n+1,2).

cp's OEIS Frontend

A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1-4*k)*x^2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A098443 Expansion of 1/sqrt(1-8*x-4*x^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A110180 Triangle of generalized central trinomial coefficients.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

Formula

A098265 G.f. : 1/(1-2x-23x^2)^(1/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A098444 Expansion of 1/sqrt(1-6x-11x^2).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula

A127947 Hankel transform of central coefficients of (1+k*x+5x^2)^n, k arbitrary integer.

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

PARI

Formula

A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1-4k)*x^2).

A098443 Expansion of 1/sqrt(1-8x-4x^2).