A025237 -id:A025237 - cp's OEIS frontend

A084603 Coefficients of 1/sqrt(1 - 2x - 11x^2); also, a(n) is the central coefficient of (1 + x + 3*x^2)^n.

1, 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139, 2778852979, 12043194163, 52286516821, 227323871929, 989675651041, 4313712072241, 18822940658947, 82215245701519

Offset: 0

Paul D. Hanna, Jun 01 2003

5th binomial transform of 2^n*LegendreP(n,-2) (signed version of A069835). - Paul Barry, Sep 03 2004
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U (or D) steps come in three colors. - N-E. Fahssi, Feb 05 2008
Diagonal of the rational function 1 / (1 - 3*x^2 - y^2 - x*y). - Ilya Gutkovskiy, Apr 22 2025

Seiichi Manyama, Table of n, a(n) for n = 0..1542 (terms 0..200 from Vincenzo Librandi)
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Cf. A002426, A084600-A084602, A084604-A084615, A025237.

Table[Sum[Binomial[n-k,k]*Binomial[n,k]*3^k,{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

for(n=0,30,t=polcoeff((1+x+3*x^2)^n,n,x); print1(t","))

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*binomial(n, k)3^k. - Paul Barry, Aug 26 2004
Binomial transform is A084609. Hankel transform is 6^n*3^C(n,2). - Paul Barry, Sep 16 2006
a(n) = (1/Pi)*Integral_{x=1-2*sqrt(3)..1+2*sqrt(3)} x^n/sqrt(-x^2 + 2*x + 11). - Paul Barry, Sep 16 2006
From Paul Barry, Sep 16 2006: (Start)
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(2k,k)*3^k;
a(n) = Sum_{k=0..floor(n/2)} C(n,k)*C(n-k,k)*3^k. (End)
From N-E. Fahssi, Feb 05 2008: (Start)
a(n) is also the central coefficient of (3+x+x^2)^n;
a(n) = Sum_{k=0..n} 2^(n-k)*C(n,k)*T(k,n), where T(k,n) is the triangle of trinomial coefficients = coefficient of x^n of (1+x+x^2)^k: A027907. (End)
D-finite with recurrence: a(n+2) = ( (2*n+3)*a(n+1) + 11*(n+1)*a(n) )/(n+2); a(0)=a(1)=1. - Sergei N. Gladkovskii, Aug 01 2012
a(n) ~ sqrt(18+3*sqrt(3))*(1+2*sqrt(3))^n/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012
E.g.f.: exp(x)*BesselI(0, 2*sqrt(3)*x). - Paul D. Hanna, Nov 09 2014, after Vladeta Jovovic in A084601
From Peter Bala, Jan 07 2022: (Start)
O.g.f. A(x) = 1 + x*d/dx(log(B(x))), where B(x) = (1 - x - sqrt(1 - 2*x - 11*x^2))/(6*x^2) is the o.g.f. of A025237.
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)

A217275 Expansion of 2/(1-x+sqrt(1-2x-27x^2)).

Original entry on oeis.org

1, 1, 8, 22, 141, 561, 3291, 15583, 88691, 459187, 2599570, 14136200, 80391235, 450046143, 2579291352, 14710321998, 85002979083, 491050703739, 2859262171872, 16674374605722, 97747766045679, 574231140306699, 3385974360904227, 20009363692187115, 118582649963026677

Offset: 0

Vaclav Kotesovec, Sep 29 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001006 (z=1), A025235 (z=2), A025237 (z=3), A091147 (z=4), A091148 (z=5), A091149 (z=6).

Table[SeriesCoefficient[2/(1-x+Sqrt[1-2*x-27*x^2]),{x,0,n}],{n,0,25}]
Table[Sum[Binomial[n,2k]*Binomial[2k,k]*7^k/(k+1),{k,0,n}],{n,0,25}]

Generally for G.f. = 2/(1-x+sqrt(1-2x-(4*z-1)*x^2)) is asymptotic
a(n) ~ (1+2*sqrt(z))^(n+3/2)/(2*sqrt(Pi)*z^(3/4)*n^(3/2)); here we have the case z=7.
D-finite with recurrence: (n+2)*a(n)=(2*n+1)*a(n-1)+(4*z-1)*(n-1)*a(n-2);; here with z=7.
G.f.: 1/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - ....))))), a continued fraction. - Ilya Gutkovskiy, May 26 2017

A014432 a(n) = Sum_{i=1..n-1} a(i)*a(n-1-i), with a(0) = 1, a(1) = 3.

Original entry on oeis.org

1, 3, 3, 12, 30, 111, 363, 1353, 4917, 18777, 71769, 280506, 1103556, 4395009, 17622309, 71220828, 289510662, 1183627137, 4862148753, 20061888924, 83100910530, 345457823493, 1440734205513, 6026408186457, 25275954499905, 106277040064191

Offset: 0

Wouter Meeussen

nonn

Robert Israel, Table of n, a(n) for n = 0..1545

Cf. A025237.

seq(coeff(convert(series((1+x-sqrt(1-2*x-11*x^2))/(2*x),x,50),polynom),x,i),i=0..30);
A014431:=proc(n) options remember: local i: if n<2 then RETURN([1,3][n+1]) else RETURN(add(A014431(i)*A014431(n-1-i),i=1..n-1)) fi:end;seq(A014431(n),n=0..30); # C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004

Rest[CoefficientList[Series[(1+x-Sqrt[1-2x-11x^2])/2,{x,0,30}],x]] (* Harvey P. Dale, Apr 17 2019 *)

a(n)=polcoeff((1+x-sqrt(1-2*x-11*x^2+x*O(x^n)))/2,n)

G.f.: ((1+x-sqrt(1-2*x-11*x^2)))/(2*x). - Michael Somos, Jun 08 2000; corrected by Robert Israel, Sep 10 2020
a(n) = (3/(11*n)) * ((3+n)*A025237(n+1) - (2*n+3)*A025237(n)) for n>0. [Mark van Hoeij, Jul 02 2010]
(n+1)*a(n) = (2*n-1)*a(n-1)+11*(n-2)*a(n-2). - Robert Israel, Sep 10 2020
G.f.: 1 + 3*x/(1 - x - 3*x^2/(1 - x - 3*x^2/(1 - x - 3*x^2/(1 - x - 3*x^2/(1 - ...))))) (continued fraction). - Nikolaos Pantelidis, Nov 24 2022

Corrected by C. Ronaldo (aga_new_ac(AT)hotmail.com) and Ralf Stephan, Dec 19 2004

A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2x + (1-4k)*x^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 7, 9, 1, 1, 1, 5, 10, 21, 21, 1, 1, 1, 6, 13, 37, 61, 51, 1, 1, 1, 7, 16, 57, 121, 191, 127, 1, 1, 1, 8, 19, 81, 201, 451, 603, 323, 1, 1, 1, 9, 22, 109, 301, 861, 1639, 1961, 835, 1

Offset: 0

Seiichi Manyama, May 06 2019

			Square array begins:
   1,   1,   1,    1,    1,    1,     1,     1, ...
   1,   1,   1,    1,    1,    1,     1,     1, ...
   1,   2,   3,    4,    5,    6,     7,     8, ...
   1,   4,   7,   10,   13,   16,    19,    22, ...
   1,   9,  21,   37,   57,   81,   109,   141, ...
   1,  21,  61,  121,  201,  301,   421,   561, ...
   1,  51, 191,  451,  861, 1451,  2251,  3291, ...
   1, 127, 603, 1639, 3445, 6231, 10207, 15583, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..7 give A000012, A001006, A025235, A025237, A091147, A091148, A091149, A217275.
Main diagonal gives A307906.
Cf. A000108, A107267, A247495, A307855.

T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, 2*j] * CatalanNumber[j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

A(n,k) is the coefficient of x^n in the expansion of 1/(n+1) * (1 + x + k*x^2)^(n+1).
A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * A000108(j).
(n+2) * A(n,k) = (2*n+1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).

A374488 Expansion of 1/(1 - 2x - 11x^2)^(3/2).

Original entry on oeis.org

1, 3, 24, 100, 555, 2541, 12628, 59004, 281655, 1315765, 6171132, 28692456, 133315273, 616780815, 2848833120, 13124483344, 60364983987, 277142478921, 1270586298520, 5817063737100, 26600252408961, 121501917998263, 554429553154044, 2527595449990500

Offset: 0

Seiichi Manyama, Jul 09 2024

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A102839, A102840, A374487.

Cf. A014432, A025237, A084603.

Module[{x}, CoefficientList[Series[1/(1 - (11*x + 2)*x)^(3/2), {x, 0, 25}], x]] (* Paolo Xausa, Aug 25 2025 *)

a(n) = binomial(n+2, 2)*sum(k=0, n\2, 3^k*binomial(n, 2*k)*binomial(2*k, k)/(k+1));

a(0) = 1, a(1) = 3; a(n) = ((2*n+1)*a(n-1) + 11*(n+1)*a(n-2))/n.
a(n) = binomial(n+2,2) * A025237(n).
From Seiichi Manyama, Aug 20 2025: (Start)
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 3^k * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (1/2)^k * (11/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)
a(n) ~ sqrt(n) * (1 + 2*sqrt(3))^(n + 3/2) / (4 * 3^(3/4) * sqrt(Pi)). - Vaclav Kotesovec, Aug 21 2025

cp's OEIS Frontend

A084603 Coefficients of 1/sqrt(1 - 2*x - 11*x^2); also, a(n) is the central coefficient of (1 + x + 3*x^2)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A217275 Expansion of 2/(1-x+sqrt(1-2*x-27*x^2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A014432 a(n) = Sum_{i=1..n-1} a(i)*a(n-1-i), with a(0) = 1, a(1) = 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1-4*k)*x^2)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A374488 Expansion of 1/(1 - 2*x - 11*x^2)^(3/2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A084603 Coefficients of 1/sqrt(1 - 2x - 11x^2); also, a(n) is the central coefficient of (1 + x + 3*x^2)^n.

A217275 Expansion of 2/(1-x+sqrt(1-2x-27x^2)).

A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2x + (1-4k)*x^2)).

A374488 Expansion of 1/(1 - 2x - 11x^2)^(3/2).