A052907 -id:A052907 - cp's OEIS frontend

A387476 a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(k,n-2*k)^2.

1, 0, 2, 2, 4, 16, 12, 72, 88, 264, 608, 1056, 3280, 5504, 15328, 31904, 71104, 175488, 358080, 900736, 1925248, 4518016, 10404864, 23138304, 54970624, 122038272, 286077440, 651510272, 1492685824, 3465687040, 7876488192, 18322630656, 41904609280, 96788580352, 223335882752

Offset: 0

Seiichi Manyama, Aug 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A375276, A387477.

Cf. A052907.

[(&+[2^k * Binomial(k,n-2*k)^2: k in [0..Floor(n/2)]]): n in [0..40]]; // Vincenzo Librandi, Aug 31 2025

Table[Sum[2^k* Binomial[k,n-2*k]^2,{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Aug 31 2025 *)

a(n) = sum(k=0, n\2, 2^k*binomial(k, n-2*k)^2);

G.f.: 1/sqrt((1-2*x^2-2*x^3)^2 - 16*x^5).

A099040 Riordan array (1, 2+2x).

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 0, 8, 8, 0, 0, 4, 24, 16, 0, 0, 0, 24, 64, 32, 0, 0, 0, 8, 96, 160, 64, 0, 0, 0, 0, 64, 320, 384, 128, 0, 0, 0, 0, 16, 320, 960, 896, 256, 0, 0, 0, 0, 0, 160, 1280, 2688, 2048, 512, 0, 0, 0, 0, 0, 32, 960, 4480, 7168, 4608, 1024, 0, 0, 0, 0, 0, 0, 384, 4480, 14336, 18432, 10240, 2048

Offset: 0

Paul Barry, Sep 23 2004

Row sums give A002605. Diagonal sums give A052907.
The Riordan array (1,s+t*x) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).
T(n,k) is the number of compositions of n into two types of parts of size 1 and 2 that have exactly k parts. - Geoffrey Critzer, Aug 18 2012.
Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 22 2020

			Rows begin {1}, {0,2}, {0,2,4}, {0,0,8,8}, {0,0,4,24,16}, {0,0,0,24,64,32},...
T(3,2)=8 because we have: 1+2,1+2',1'+2,1'+2',2+1,2+1',2'+1,2'+1' where a part of the second type is designated by '. - _Geoffrey Critzer_, Aug 18 2012

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A002605, A026729, A038208, A052907.

nn = 8; CoefficientList[Series[1/(1 - 2 y x - 2 y x^2), {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 18 2012 *)

Number triangle T(n, k) = 2^k*binomial(k, n-k).
Columns have g.f. (2x+2x^2)^k.
T(n,k) = A026729(n,k)*2^k. - Philippe Deléham, Jul 28 2006
O.g.f.: 1/(1-2*y*x-2*y*x^2). - Geoffrey Critzer, Aug 18 2012.

A099493 Expansion of (1+x^2)^2/(1+x^2-2x^3+x^4+x^6).

Original entry on oeis.org

1, 0, 1, 2, -1, 0, 3, -4, -3, 8, -7, -10, 23, -8, -33, 56, 1, -104, 121, 58, -297, 232, 291, -780, 349, 1072, -1903, 174, 3407, -4272, -1505, 9840, -8543, -8752, 26321, -13902, -33777, 65456, -11805, -110356, 150173, 35192, -325303, 310054, 257319, -885496, 537919, 1054888, -2240927

Offset: 0

Paul Barry, Oct 19 2004

A Chebyshev transform of A052907, which has g.f. 1/(1-2x^2-2x^3). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).

Index entries for linear recurrences with constant coefficients, signature (0,-1,2,-1,0,-1).

a(n)=-a(n-2)+2a(n-3)-a(n-4)-a(n-6); a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k*sum{j=0.., floor(n-2k/2), C(j, n-2k-2j)2^j}}.

A191697 a(n) = r1^n + r2^n + r3^n where r1, r2, r3 are the three roots of x^3 - 2*x - 2 = 0.

Original entry on oeis.org

0, 4, 6, 8, 20, 28, 56, 96, 168, 304, 528, 944, 1664, 2944, 5216, 9216, 16320, 28864, 51072, 90368, 159872, 282880, 500480, 885504, 1566720, 2771968, 4904448, 8677376, 15352832, 27163648, 48060416, 85032960, 150448128, 266186752, 470962176, 833269760

Offset: 1

N. J. A. Sloane, Jun 19 2011

Definition 1.1 defines F_3^n as a Boolean function and definition 1.3 defines the Fourier transform of a Boolean function. - Michael Somos, Aug 04 2012

			G.f. = 4*x^2 + 6*x^3 + 8*x^4 + 20*x^5 + 28*x^6 + 56*x^7 + 96*x^8 + 168*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500
X. Zhang, H. Guo, R. Feng and Y. Li, Proof of a conjecture about rotation symmetric functions, Discrete Math., 311 (2011), 1281-1289.
Index entries for linear recurrences with constant coefficients, signature (0,2,2).

m:=50; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x^2*(2+3*x)/(1-2*x^2-2*x^3))); // G. C. Greubel, Aug 13 2018

I:=[0,4,6]; [n le 3 select I[n] else 2*Self(n-2)+2*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 14 2018

Rest[CoefficientList[Series[2*x^2*(2+3*x)/(1-2*x^2-2*x^3), {x, 0, 50}], x]] (* G. C. Greubel, Aug 13 2018 *)
LinearRecurrence[{0, 2, 2}, {0, 4, 6}, 40] (* Vincenzo Librandi, Aug 14 2018 *)

{a(n) = if( n<1, 0, polsym( x^3 - 2*x - 2, n)[n + 1])}; /* Michael Somos, Aug 04 2012 */

{a(n) = if( n<1, 0, sum( x=0, 2^n-1, (-1)^sum( i=0, n-1, bittest(x, i) * bittest(x, (i+1)%n) * bittest(x, (i+2)%n))))}; /* Michael Somos, Aug 04 2012 */

a(n) = hat{F_3^n}(0), the Fourier transform evaluated at 0 of the Boolean function F_3^n defined by F_3^n(x_0, ..., x_{n-1}) = Sum_{ 0Michael Somos, Aug 04 2012
From Michael Somos, Aug 04 2012: (Start)
G.f.: 2 * x^2 * (2 + 3*x) / (1 - 2*x^2 - 2*x^3).
a(n + 3) = 2*a(n + 1) + 2*a(n). (End)
a(n) = 4*A052907(n) +6*A052907(n-1). - R. J. Mathar, Aug 10 2012

A099092 Riordan array (1,2+4x).

Original entry on oeis.org

1, 0, 2, 0, 4, 4, 0, 0, 16, 8, 0, 0, 16, 48, 16, 0, 0, 0, 96, 128, 32, 0, 0, 0, 64, 384, 320, 64, 0, 0, 0, 0, 512, 1280, 768, 128, 0, 0, 0, 0, 256, 2560, 3840, 1792, 256, 0, 0, 0, 0, 0, 2560, 10240, 10752, 4096, 512, 0, 0, 0, 0, 0, 1024, 15360, 35840, 28672, 9216, 1024, 0, 0, 0

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A063727. Diagonal sums are A052907.

The Riordan array (1, s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).

			Rows begin
  {1},
  {0,  2},
  {0,  4,  4},
  {0,  0, 16,  8},
  {0,  0, 16, 48, 16}, ...

Cf. A026729, A038210, A052907, A063727, A113953.

Number triangle T(n,k) = binomial(k, n-k)*2^n; columns have g.f. (2x+4x^2)^k.

T(n,k) = A113953(n,k)*2^k = A026729(n,k)*2^n. - Philippe Deléham, Dec 11 2008

A104579 A Padovan-Jacobsthal convolution triangle.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 1, 4, 3, 0, 1, 4, 3, 6, 4, 0, 1, 5, 12, 6, 8, 5, 0, 1, 6, 16, 24, 10, 10, 6, 0, 1, 13, 24, 34, 40, 15, 12, 7, 0, 1, 16, 53, 60, 60, 60, 21, 14, 8, 0, 1, 25, 72, 135, 120, 95, 84, 28, 16, 9, 0, 1, 42, 126, 200, 275, 210, 140, 112, 36, 18, 10, 0, 1, 57, 220, 381

Offset: 0

Paul Barry, Mar 16 2005

First column is A052947. Row sums are A077947. Diagonal sums are A052907.

			Rows begin {1},{0,1},{1,0,1},{2,2,0,1},{1,4,3,0,1},{4,3,6,4,0,1},..

Riordan array (1/(1-x^2-2x^3), x/(1-x^2-2x^3))

T(n,k) = T(n-1,k-1)+T(n-2,k)+2*T(n-3,k), T(0,0)=1, T(n,k)=0 if k>n or if k<0. - Philippe Deléham, Jan 08 2014

cp's OEIS Frontend

A387476 a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(k,n-2*k)^2.

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A099040 Riordan array (1, 2+2x).

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A099493 Expansion of (1+x^2)^2/(1+x^2-2x^3+x^4+x^6).

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A191697 a(n) = r1^n + r2^n + r3^n where r1, r2, r3 are the three roots of x^3 - 2*x - 2 = 0.

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A099092 Riordan array (1,2+4x).

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A104579 A Padovan-Jacobsthal convolution triangle.

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