A364553 -id:A364553 - cp's OEIS frontend

A026937 a(n) = Sum_{k=0..n} (k+1)*T(n, n-k), where T is given by A008288.

1, 3, 10, 30, 87, 245, 676, 1836, 4925, 13079, 34446, 90090, 234227, 605865, 1560200, 4002072, 10230201, 26069995, 66251090, 167941494, 424753615, 1072057117, 2700704172, 6791746500, 17052595573, 42752015487, 107035180630, 267634562754, 668407232235, 1667467065425

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Cf. A000129, A001333, A006645, A008288, A364553.

I:=[1, 3, 10, 30]; [n le 4 select I[n] else 4*Self(n-1)-2*Self(n-2)-4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 20 2012

with (combinat):seq(add(fibonacci(n,2),k=0..n)/2,n=1..27); # Zerinvary Lajos, May 25 2008

CoefficientList[Series[(1-x)/(1-2x-x^2)^2,{x,0,40}],x]  (* Harvey P. Dale, Mar 22 2011 *)
LinearRecurrence[{4,-2,-4,-1},{1,3,10,30},40] (* Vincenzo Librandi, Jun 20 2012 *)
Table[(1/2)*(n+2)*Fibonacci[n+1, 2], {n, 0, 40}] (* G. C. Greubel, May 25 2021 *)

my(x='x+O('x^40)); Vec((1-x)/(1-2*x-x^2)^2) \\ Altug Alkan, Sep 20 2018

a(n) = my(w=quadgen(8)); (n/8)*((2+w)*(1+w)^n - (w-2)*(1-w)^n); \\ Michel Marcus, Jul 31 2023

[(1/2)*(n+2)*lucas_number1(n+1,2,-1) for n in (0..40)] # G. C. Greubel, May 25 2021

G.f.: (1-x)/(1 - 2*x - x^2)^2.
a(n) = Sum_{k=0..n+1} A000129(k)*A001333(n+1-k). - Graeme McRae, Aug 03 2006 and Michel Marcus, Aug 01 2023
a(n) = A006645(n+2) - A006645(n+1). - R. J. Mathar, Jan 27 2011
a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 20 2012
a(n) = ((n+2)/2)*A000129(n+1). - G. C. Greubel, May 25 2021
a(n) = ((n+2)/8)*((sqrt(2) + 2)*(1 + sqrt(2))^n - (sqrt(2) - 2)*(1 - sqrt(2))^n). - Peter Luschny, Jul 31 2023
a(n) = A361732(n+2)/2. - R. J. Mathar, Jun 30 2025

A364361 Table read by rows. T(n, k) = Sum_{j=0..n-k} kbinomial(k, j)binomial(n - j, k).

Original entry on oeis.org

0, 0, 1, 0, 3, 2, 0, 5, 10, 3, 0, 7, 26, 21, 4, 0, 9, 50, 75, 36, 5, 0, 11, 82, 189, 164, 55, 6, 0, 13, 122, 387, 516, 305, 78, 7, 0, 15, 170, 693, 1284, 1155, 510, 105, 8, 0, 17, 226, 1131, 2724, 3405, 2262, 791, 136, 9, 0, 19, 290, 1725, 5156, 8415, 7734, 4025, 1160, 171, 10

Offset: 0

Peter Luschny, Jul 30 2023

			The triangle begins:
  [0] 0;
  [1] 0,  1;
  [2] 0,  3,   2;
  [3] 0,  5,  10,   3;
  [4] 0,  7,  26,   21,    4;
  [5] 0,  9,  50,   75,   36,    5;
  [6] 0, 11,  82,  189,  164,   55,    6;
  [7] 0, 13, 122,  387,  516,  305,   78,   7;
  [8] 0, 15, 170,  693, 1284, 1155,  510, 105,   8;
  [9] 0, 17, 226, 1131, 2724, 3405, 2262, 791, 136, 9;
Seen as an array:
  [0] 0,  1,   2,   3,     4,     5,      6,      7, ...  A001477
  [1] 0,  3,  10,   21,   36,    55,     78,    105, ...  A014105
  [2] 0,  5,  26,   75,  164,   305,    510,    791, ...  A048395
  [3] 0,  7,  50,  189,  516,  1155,   2262,   4025, ...
  [4] 0,  9,  82,  387, 1284,  3405,   7734,  15687, ...
  [5] 0, 11, 122,  693, 2724,  8415,  21918,  50281, ...
  [6] 0, 13, 170, 1131, 5156, 18265,  53934, 138775, ...
  [7] 0, 15, 226, 1725, 8964, 35915, 118950, 340473, ...
    A005408|A069894

Cf. A364553 (row sums),  A364634 (main diagonal).
Rows: A001477, A014105, A048395.
Columns: A005408, A069894.

T := (n, k) -> local j; add(k*binomial(k, j)*binomial(n-j, k), j = 0..n-k):
seq(seq(T(n, k), k = 0..n), n = 0..10);

T(2*n, n) = n * LegendreP(n, 3).

A364636 a(n) = ((1 - sqrt(2))^n + (1 + sqrt(2))^n)*n/2.

Original entry on oeis.org

0, 1, 6, 21, 68, 205, 594, 1673, 4616, 12537, 33630, 89309, 235212, 615173, 1599402, 4137105, 10653712, 27327857, 69856182, 178017061, 452390740, 1146776253, 2900399106, 7320463897, 18441561624, 46376946025, 116442406158, 291929022189, 730881930716, 1827523107829

Offset: 0

Peter Luschny, Jul 30 2023

Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Cf. A093967, A364553.

A364636 := n -> ((1 - sqrt(2))^n + (1 + sqrt(2))^n)*n / 2:
seq(simplify(A364636(n)), n = 0..29);

a(n) = ((1 - quadgen(8))^n + (1 + quadgen(8))^n)*n/2; \\ Michel Marcus, Jul 31 2023

The sequence can be continued to all ZZ, and a(-n) = -(-1)^n*a(n).
a(n) = [x^n] (x + 2*x^2 - x^3)/(-1 + x*(2 + x))^2.
a(n) = 2*A364553(n) - A093967(n).

cp's OEIS Frontend

A026937 a(n) = Sum_{k=0..n} (k+1)*T(n, n-k), where T is given by A008288.

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A364361 Table read by rows. T(n, k) = Sum_{j=0..n-k} kbinomial(k, j)binomial(n - j, k).

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Maple

Formula

A364636 a(n) = ((1 - sqrt(2))^n + (1 + sqrt(2))^n)*n/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

cp's OEIS Frontend

A026937 a(n) = Sum_{k=0..n} (k+1)*T(n, n-k), where T is given by A008288.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A364361 Table read by rows. T(n, k) = Sum_{j=0..n-k} k*binomial(k, j)*binomial(n - j, k).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A364636 a(n) = ((1 - sqrt(2))^n + (1 + sqrt(2))^n)*n/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

A364361 Table read by rows. T(n, k) = Sum_{j=0..n-k} kbinomial(k, j)binomial(n - j, k).