A084603 -id:A084603 - cp's OEIS frontend

A090042 a(n) = 2a(n-1) + 11a(n-2) for n > 1, a(0) = a(1) = 1.

1, 1, 13, 37, 217, 841, 4069, 17389, 79537, 350353, 1575613, 7005109, 31341961, 139740121, 624241813, 2785624957, 12437909857, 55517694241, 247852396909, 1106399430469, 4939175226937, 22048744189033, 98428415874373, 439393017828109, 1961498610274321, 8756320416657841

Offset: 0

Paul Barry, Nov 20 2003

Binomial transform of A001021 (powers of 12), with interpolated zeros.
For n > 0, a(n) = term (1,1) in the n-th power of the 2 X 2 matrix [1,3; 4,1]. - Gary W. Adamson, Aug 06 2010
a(n) is the number of compositions of n when there are 1 type of 1 and 12 types of other natural numbers. - Milan Janjic, Aug 13 2010

Muniru A Asiru, Table of n, a(n) for n = 0..319
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (2,11).

Cf. A049310, A015520, A084603.

a := [1, 1];; for n in [3..30] do a[n] := 2*a[n-1]+ 11*a[n-2]; od; a; # Muniru A Asiru, Feb 18 2018

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +11*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019

a := proc(n) option remember: if n=0 then 1 elif n=1 then 1 elif n>=2 then 2*procname(n-1) + 11*procname(n-2) fi; end:
seq(a(n), n=0..25); # Muniru A Asiru, Feb 18 2018

a[n_]:= Simplify[((1+Sqrt[12])^n +(1-Sqrt[12])^n)/2]; Array[a, 30, 0] (* or *)
CoefficientList[Series[(x-1)/(11x^2+2x-1), {x,0,30}], x] (* or *)
Table[ MatrixPower[{{1, 2}, {6, 1}}, n][[1, 1]], {n, 0, 30}] (* Robert G. Wilson v, Sep 18 2013 and modified per Wolfdieter Lang Feb 17 2018 *)
LinearRecurrence[{2, 11}, {1, 1}, 30] (* Ray Chandler, Aug 01 2015 *)

x='x+O('x^30); Vec((1-x)/(1-2*x-11*x^2)) \\ Altug Alkan, Feb 17 2018

((1-x)/(1-2*x-11*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019

E.g.f.: exp(x)*cosh(2*sqrt(3)*x).
a(n) = ((1 + 2*sqrt(3))^n + (1 - 2*sqrt(3))^n)/2.
a(n) = Sum_{k=0..n} A098158(n,k)*12^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=12, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, Apr 29 2010
From Wolfdieter Lang, Feb 17 2018: (Start)
G.f.: (1-x)/(1 - 2*x - 11*x^2). (See the Mathematica program.)
a(n) = b(n+1) - b(n), with b(n) = A015520(n). This leads to the Binet-de Moivre type formula given in the Mathematica program.
a(n) = (i*sqrt(11))^n*(S(n,-2*i/sqrt(11)) + (i/sqrt(11))*S(n-1,-2*i/sqrt(11))), n >= 0, with Chebyshev S polynomials (coefficients in A049310), with S(-2, x) = -1, S(-1, x) = 0 and i = sqrt(-1). Via Cayley-Hamilton. See the Gary W. Adamson comment above or the Mathematica program of Robert G. Wilson v with another matrix. (End)
From Peter Bala, Jan 07 2022: (Start)
a(n) = [x^n] (x + sqrt(1 + 12*x^2))^n.
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.
O.g.f.: 1 + x*d/dx(log(B(x))), where B(x) = 1/sqrt(1 - 2*x - 11*x^2) is the o.g.f. of A084603. (End)

A098264 G.f.: 1/(1-2x-19x^2)^(1/2).

Original entry on oeis.org

1, 1, 11, 31, 211, 851, 4901, 22961, 124531, 623011, 3313201, 17086301, 90453661, 473616781, 2509264811, 13250049551, 70368250451, 373539254611, 1989045489281, 10597110956861, 56566637447401, 302196871378601, 1616570627763311, 8654955238504531, 46384344189261661

Offset: 0

Paul Barry, Aug 31 2004

Central coefficient of (1+x+5x^2)^n.

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 can have five colors. - N-E. Fahssi, Mar 31 2008

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.

Table[SeriesCoefficient[1/Sqrt[1-2*x-19*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

x='x+O('x^66); Vec(1/(1-2*x-19*x^2)^(1/2)) \\ Joerg Arndt, May 11 2013

E.g.f. : exp(x)*BesselI(0, 2*sqrt(5)x).
a(n) = sum{k=0..floor(n/2), binomial(n, k)*binomial(n-k, k)*5^k}.
a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*binomial(2k, k)*5^k}.
n*a(n) +(1-2*n)*a(n-1) +19*(1-n)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
a(n) ~ sqrt(50+5*sqrt(5))*(1+2*sqrt(5))^n/(10*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

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).

Original entry on oeis.org

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).

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.

A084602 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x + 3x^2)^n.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 7, 6, 9, 1, 3, 12, 19, 36, 27, 27, 1, 4, 18, 40, 91, 120, 162, 108, 81, 1, 5, 25, 70, 185, 331, 555, 630, 675, 405, 243, 1, 6, 33, 110, 330, 726, 1441, 2178, 2970, 2970, 2673, 1458, 729, 1, 7, 42, 161, 539, 1386, 3157, 5797, 9471, 12474, 14553

Offset: 0

Paul D. Hanna, Jun 01 2003

			Rows:
{1},
{1,1, 3},
{1,2, 7,  6,  9},
{1,3,12, 19, 36, 27,  27},
{1,4,18, 40, 91,120, 162, 108,  81},
{1,5,25, 70,185,331, 555, 630, 675, 405, 243},
{1,6,33,110,330,726,1441,2178,2970,2970,2673,1458,729},

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

Cf. A002426, A084600-A084601, A084603-A084615.

With[{eq = (1 + x + 3*x^2)}, Flatten[Table[CoefficientList[Expand[eq^n], x], {n, 0, 10}]]] (* G. C. Greubel, Mar 02 2017 *)

for(n=0,15, for(k=0,2*n,t=polcoeff((1+x+3*x^2)^n,k,x); print1(t",")); print(" "))

A084604 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x + 4x^2)^n.

Original entry on oeis.org

1, 1, 1, 4, 1, 2, 9, 8, 16, 1, 3, 15, 25, 60, 48, 64, 1, 4, 22, 52, 145, 208, 352, 256, 256, 1, 5, 30, 90, 285, 561, 1140, 1440, 1920, 1280, 1024, 1, 6, 39, 140, 495, 1206, 2841, 4824, 7920, 8960, 9984, 6144, 4096, 1, 7, 49, 203, 791, 2261, 6027, 12489, 24108, 36176

Offset: 0

Paul D. Hanna, Jun 01 2003

			Rows:
{1},
{1,1, 4},
{1,2, 9,  8, 16},
{1,3,15, 25, 60,  48,  64},
{1,4,22, 52,145, 208, 352, 256, 256},
{1,5,30, 90,285, 561,1140,1440,1920,1280,1024},
{1,6,39,140,495,1206,2841,4824,7920,8960,9984,6144,4096},

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

Cf. A002426, A084600-A084603, A084605-A084615.

With[{eq=(1+x+4x^2)},Flatten[Table[CoefficientList[Expand[eq^n],x],{n,0,10}]]] (* Harvey P. Dale, May 19 2011 *)

for(n=0,10, for(k=0,2*n,t=polcoeff((1+x+4*x^2)^n,k,x); print1(t",")); print(" "))

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

A098329 Expansion of 1/(1-2x-31x^2)^(1/2).

Original entry on oeis.org

1, 1, 17, 49, 481, 2081, 16241, 85457, 600769, 3489601, 23391697, 143000177, 938797729, 5897385313, 38397492017, 244866166289, 1590355308929, 10231490804353, 66456634775441, 429898281869489, 2795449543782241, 18150017431150241, 118194927388259057, 769438418283309649

Offset: 0

Paul Barry, Sep 03 2004

Central coefficient of (1+x+8x^2)^n. 7th binomial transform of 2^n*LegendreP(n,-3) (signed version of A084773).

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 can have 8 colors. - N-E. Fahssi, Mar 31 2008

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

Cf. A084603.

Table[SeriesCoefficient[1/Sqrt[1-2*x-31*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
CoefficientList[Series[1/Sqrt[1-2x-31x^2],{x,0,30}],x] (* Harvey P. Dale, May 14 2017 *)
a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 1, 32];
Table[a[n], {n, 0, 23}] (* Peter Luschny, Mar 18 2018 *)

x='x+O('x^66); Vec(1/(1-2*x-31*x^2)^(1/2)) \\ Joerg Arndt, May 11 2013

a(n) = sum{k=0..floor(n/2), binomial(n-k, k)*binomial(n, k)*8^k}.
E.g.f.: exp(x)*BesselI(0, 4*sqrt(2)*x)
Recurrence: n*a(n) = (2*n-1)*a(n-1) + 31*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(8+sqrt(2))*(1+4*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012
a(n) = hypergeom([1/2 - n/2, -n/2], [1], 32). - Peter Luschny, Mar 18 2018

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

A090042 a(n) = 2*a(n-1) + 11*a(n-2) for n > 1, a(0) = a(1) = 1.

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A098264 G.f.: 1/(1-2x-19x^2)^(1/2).

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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).

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A110180 Triangle of generalized central trinomial coefficients.

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A084602 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x + 3x^2)^n.

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A084604 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x + 4x^2)^n.

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Mathematica

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

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A098329 Expansion of 1/(1-2x-31x^2)^(1/2).

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A090042 a(n) = 2a(n-1) + 11a(n-2) for n > 1, a(0) = a(1) = 1.

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).

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