A276289 -id:A276289 - cp's OEIS frontend

A053220 a(n) = (3n-1) 2^(n-2).

1, 5, 16, 44, 112, 272, 640, 1472, 3328, 7424, 16384, 35840, 77824, 167936, 360448, 770048, 1638400, 3473408, 7340032, 15466496, 32505856, 68157440, 142606336, 297795584, 620756992, 1291845632, 2684354560, 5570035712, 11542724608, 23890755584, 49392123904

Offset: 1

Asher Auel, Jan 01 2000

Coefficients in the hypergeometric series identity 1 - 5*x/(x + 4) + 16*x*(x - 1)/((x + 4)*(x + 6)) - 44*x*(x - 1)*(x - 2)/((x + 4)*(x + 6)*(x + 8)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A276289. - Peter Bala, May 30 2019

For n>=2, a(n) is the total number of ones in runs of ones of length >=5 over all binary strings of length n+3. - Félix Balado, Aug 06 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci, Hypercubes and Isometric Words based on Swap and Mismatch Distance, arXiv:2303.09898 [math.CO], 2023.
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A053219, A053221, A132776, A276289.

Center elements from triangle A053218. Also a diagonal of triangle A056242.

a053220 n = a056242 (n + 1) n  -- Reinhard Zumkeller, May 08 2014

[(3*n-1)*2^(n-2): n in [1..50]]; // Vincenzo Librandi, May 09 2011

ListCorrelate[{1, 1}, Table[n 2^(n - 1), {n, 0, 28}]] (* or *) ListConvolve[{1, 1}, Table[n 2^(n - 1), {n, 0, 28}]] (* Ross La Haye, Feb 24 2007 *)
LinearRecurrence[{4, -4}, {1, 5}, 35] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
Array[(3# - 1) 2^(# - 2) &, 35] (* Alonso del Arte, Sep 04 2018 *)
CoefficientList[Series[(1 + x)/(1 - 2 * x)^2, {x, 0, 50}], x] (* Stefano Spezia, Sep 04 2018 *)

PARI
```
a(n)=if(n<1,0,(3*n-1)*2^(n-2))
```

a(n)=(3*n-1)<<(n-2) \\ Charles R Greathouse IV, Apr 17 2012

G.f.: x*(1+x)/(1-2*x)^2.
a(n) = (3*n-1) * 2^(n-2).
E.g.f.: exp(2*x)*(1+3*x). The sequence 0, 1, 5, 16, ... has a(n) = ((3n-1)*2^n + 0^n)/4 (offset 0). It is the binomial transform of A032766. The sequence 1, 5, 16, ... has a(n) = (2+3n)*2^(n-1) (offset 0). It is the binomial transform of A016777. - Paul Barry, Jul 23 2003
Row sums of A132776(n-1). - Gary W. Adamson, Aug 29 2007
a(n+1) = det(f(i-j+1)){1 <= i, j <= n}, where f(0) = 1, f(1) = 5 and for k > 0, we have f(k+1) = 9 and f(-k) = 0. - _Mircea Merca, Jun 23 2012

A077616 Binomial transform of n^2*2^n/2.

Original entry on oeis.org

1, 10, 63, 324, 1485, 6318, 25515, 99144, 373977, 1377810, 4979799, 17714700, 62178597, 215765046, 741360195, 2525407632, 8537599665, 28669116186, 95692860783, 317684800980, 1049522104701, 3451916556990, 11307641812443

Offset: 1

Karol A. Penson, Nov 12 2002

With a leading zero, this is second binomial transform of the hexagonal numbers A000384 (with leading zero). - Paul Barry, Jun 09 2003

Coefficients in the hypergeometric series identity 1 - 10*x/(x + 9) + 63*x*(x - 1)/((x + 9)*(x + 12)) - 324*x*(x - 1)*(x - 2)/((x + 9)*(x + 12)*(x + 15)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A276289 and A084901. - Peter Bala, May 30 2019

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A084901, A276289.

List([1..30], n-> 3^(n-2)*n*(1+2*n)) # G. C. Greubel, Jun 03 2019

[3^(n-2)*n*(1+2*n): n in [1..30]]; // G. C. Greubel, Jun 03 2019

LinearRecurrence[{9, -27, 27}, {1, 10, 63}, 30] (* Jean-François Alcover, May 23 2016 *)

a(n)=n*(2*n+1)*3^(n-2) \\ Charles R Greathouse IV, Mar 19 2017

[3^(n-2)*n*(1+2*n) for n in (1..30)] # G. C. Greubel, Jun 03 2019

E.g.f: x*(1+2*x)*exp(3*x).
O.g.f: ((1/3)*x^(3/4)*3^(3/4)/(-(3*x+1)/(3*x-1)+1)^(1/4))*(-(3*x+1)/(3*x-1)-1)^(1/4)*hypergeom([ -1, 2], [3/2], 3*x/(3*x-1))/(3*x-1)^2, which can also be represented as associated Legendre function: 1/6*x^(3/4)*Pi^(1/2)*3^(3/4)*LegendreP(1, -1/2, (3*x+1)/(1-3*x))/(3*x-1)^2.
G.f.: x*(1+x)/(1-3*x)^3. - Paul Barry, Jun 09 2003
a(n) = n*(2*n+1)*3^(n-2). - Paul Barry, Jul 24 2003

A084901 a(n) = 4^(n-2)n(5*n+3)/2.

Original entry on oeis.org

0, 1, 13, 108, 736, 4480, 25344, 136192, 704512, 3538944, 17367040, 83623936, 396361728, 1853882368, 8573157376, 39258685440, 178241142784, 803158884352, 3594887626752, 15994458210304, 70781061038080, 311711546474496

Offset: 0

Paul Barry, Jun 10 2003

Binomial transform of A084900. Third binomial transform of heptagonal numbers A000566. Fourth binomial transform of (0,1,5,0,0,0,...).

Coefficients in the hypergeometric series identity 1 - 13*x/(x + 12) + 108*x*(x - 1)/((x + 12)*(x + 16)) - 736*x*(x - 1)*(x - 2)/((x + 12)*(x + 16)*(x + 20)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A276289 and A077616. - Peter Bala, May 30 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A084902, A077616, A276289.

List([0..30], n-> 2^(2*n-5)*n*(5*n+3)); # G. C. Greubel, Jun 06 2019

[2^(2*n-5)*n*(5*n+3): n in [0..30]]; // G. C. Greubel, Jun 06 2019

Table[2^(2*n-5)*n*(5*n+3), {n,0,30}] (* G. C. Greubel, Jun 06 2019 *)
LinearRecurrence[{12,-48,64},{0,1,13},30] (* or *) CoefficientList[ Series[-((x (1+x))/(-1+4 x)^3),{x,0,30}],x] (* Harvey P. Dale, Jul 14 2021 *)

vector(30, n, n--; 2^(2*n-5)*n*(5*n+3)) \\ G. C. Greubel, Jun 06 2019

[2^(2*n-5)*n*(5*n+3) for n in (0..30)] # G. C. Greubel, Jun 06 2019

G.f.: x*(1+x)/(1-4*x)^3.
E.g.f.: x*(2 + 5*x)*exp(4*x)/2. - G. C. Greubel, Jun 06 2019
a(n) = 12*a(n-1)-48*a(n-2)+64*a(n-3). - Wesley Ivan Hurt, May 28 2021

A091320 Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges and k leaves.

Original entry on oeis.org

1, 2, 1, 4, 7, 1, 8, 30, 16, 1, 16, 104, 122, 30, 1, 32, 320, 660, 365, 50, 1, 64, 912, 2920, 2875, 903, 77, 1, 128, 2464, 11312, 17430, 9856, 1960, 112, 1, 256, 6400, 39872, 88592, 78974, 28560, 3864, 156, 1, 512, 16128, 130944, 396480, 512316, 294042, 73008, 7074, 210, 1

Offset: 1

Emeric Deutsch, Feb 24 2004

T(n,k) is the number of even trees with 2n edges and k-1 jumps. An even tree is an ordered tree in which each vertex has an even outdegree. In the preorder traversal of an ordered tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump. - Emeric Deutsch, Jan 19 2007

T(n,k) is the number of non-crossing set partitions of {1..3n} into n sets of 3 with k of the sets being a contiguous set of elements; also the number of non-crossing configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 3n, see [Young]. - Donovan Young, May 29 2020

			Triangle starts:
   1;
   2,   1;
   4,   7,   1;
   8,  30,  16,   1;
  16, 104, 122,  30,  1;
  32, 320, 660, 365, 50, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 28-29.
Alexander Burstein and Megan Martinez, Pattern classes equinumerous to the class of ternary forests, Permutation Patterns Virtual Workshop, Howard University (2020).
Philippe Flajolet and Marc Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
Marc Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.
Donovan Young, Polyomino matchings in generalised games of memory and linear k-chord diagrams, arXiv:2004.06921 [math.CO], 2020. See also J. Int. Seq., Vol. 23 (2020), Article 20.9.1.

Row sums give A001764.
Column 2 is A276289.
Cf. A072247.

T := proc(n,k) if k=n then 1 else (1/n)*binomial(n,k)*sum(2^(n+1-2*k+j)*binomial(n,j)*binomial(n-k,k-1-j),j=0..n) fi end: seq(seq(T(n,k),k=1..n),n=1..12);

T[n_, k_] := 1/n Binomial[n, k] Sum[2^(n+1-2k+j) Binomial[n, j] Binomial[n-k, k-1-j], {j, 0, n}];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 29 2018 *)

T(n,k) = binomial(n, k)*sum(j=0, n, 2^(n+1-2*k+j)*binomial(n, j)*binomial(n-k, k-1-j))/n; \\ Andrew Howroyd, Nov 06 2017

T(n, k) = (1/n)*binomial(n, k)*Sum_{j=0..n} 2^(n+1-2*k+j)*binomial(n, j)*binomial(n-k, k-1-j).

G.f.: G(t, z) satisfies z*G^3 - (1 + z - t*z)*G + 1 = 0.

cp's OEIS Frontend

A053220 a(n) = (3*n-1) * 2^(n-2).

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A077616 Binomial transform of n^2*2^n/2.

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A084901 a(n) = 4^(n-2)*n*(5*n+3)/2.

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A091320 Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges and k leaves.

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A053220 a(n) = (3n-1) 2^(n-2).

A084901 a(n) = 4^(n-2)n(5*n+3)/2.