A103516 -id:A103516 - cp's OEIS frontend

A135852 A007318 * A103516 as a lower triangular matrix.

1, 3, 2, 8, 4, 3, 20, 6, 9, 4, 48, 8, 18, 16, 5, 112, 10, 30, 40, 25, 6, 256, 12, 45, 80, 75, 36, 7, 576, 14, 63, 140, 175, 126, 49, 8, 1280, 16, 84, 224, 350, 336, 196, 64, 9, 2816, 18, 108, 336, 630, 756, 588, 288, 81, 10

Offset: 0

Gary W. Adamson, Dec 01 2007

Binomial transform of triangle A103516.

			First few rows of the triangle are:
    1;
    3,  2;
    8,  4,  3;
   20,  6,  9,  4;
   48,  8, 18, 16,  5;
  112, 10, 30, 40, 25,  6;
  256, 12, 45, 80, 75, 36,  7;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318, A103516.
Cf. A001792 (1st column), A099035 (row sums).
Cf. A135853 (= A103516 * A007318).

T[n_, k_]:= If[n==0, 1, If[k==0, (n+2)*2^(n-1), (k+1)*Binomial[n, k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 07 2016 *)

def A135852(n,k):
    if (n==0): return 1
    elif (k==0): return (n+2)*2^(n-1)
    else: return (k+1)*binomial(n, k)
flatten([[A135852(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 07 2022

T(n, k) = (A007318 * A103516)(n, k).
T(n, 0) = A001792(n).
Sum_{k=0..n} T(n, k) = A099035(n+1).
T(n, k) = (k+1)*binomial(n, k), with T(n, 0) = (n+2)*2^(n-1), T(n, n) = n+1. - G. C. Greubel, Dec 07 2016

Offset changed to 0 by G. C. Greubel, Feb 07 2022

A135853 A103516 * A007318 as an infinite lower triangular matrix.

Original entry on oeis.org

1, 4, 2, 6, 6, 3, 8, 12, 12, 4, 10, 20, 30, 20, 5, 12, 30, 60, 60, 30, 6, 14, 42, 105, 140, 105, 42, 7, 16, 56, 168, 280, 280, 168, 56, 8, 18, 72, 252, 504, 630, 504, 252, 72, 9, 20, 90, 360, 840, 1260, 1260, 840, 360, 90, 10

Offset: 0

Gary W. Adamson, Dec 01 2007

			First few rows of the triangle are:
   1;
   4,   2;
   6,   6,  3;
   8,  12,  12,   4;
  10,  20,  30,  20,   5;
  12,  30,  60,  60,  30,   6;
  14,  42, 105, 140, 105,  42,   7;
  ...

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

Cf. A007318, A103516.
Cf. A103517 (1st column), A135854 (row sums).
Cf. A135852 (= A007318 * A103516).

T[n_, k_]:= If[k==n, n+1, If[k==0, 2*(n+1), (k+1)*Binomial[n+1, k+1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//flatten (* G. C. Greubel, Dec 07 2016 *)

def A135853(n,k):
    if (n==0): return 1
    elif (k==0): return 2*(n+1)
    else: return (k+1)*binomial(n+1, k+1)
flatten([[A135853(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 06 2022

T(n, k) = (A103516 * A007318)(n, k).
Sum_{k=0..n} T(n, k) = A135854(n).
T(n, k) = (k+1)*binomial(n+1, k+1), with T(n, n) = n+1, T(n, 0) = 2*(n+1). - G. C. Greubel, Dec 07 2016
T(n, 0) = A103517(n). - G. C. Greubel, Feb 06 2022

A187012 Antidiagonal sums of A103516.

Original entry on oeis.org

1, 2, 5, 4, 8, 6, 11, 8, 14, 10, 17, 12, 20, 14, 23, 16, 26, 18, 29, 20, 32, 22, 35, 24, 38, 26, 41, 28, 44, 30, 47, 32, 50, 34, 53, 36, 56, 38, 59, 40, 62, 42, 65, 44, 68, 46, 71, 48, 74, 50, 77, 52, 80, 54, 83, 56, 86, 58, 89, 60, 92, 62, 95, 64

Offset: 2

Michel Marcus, Aug 30 2013

This sequence differs from A081556 at least for n=24 (see comment about n=24 in A081556).

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

CoefficientList[Series[(1 + 2 x + 3 x^2 - x^4)/((1 - x)^2 (1 + x)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 24 2014 *)

a(n) = sum(k=0, n\2, 0^(k*(n-2*k))*(n-k+1)); \\ Michel Marcus, Aug 30 2013

a(n) = sum{k=0..floor(n/2), 0^(k(n-2k))*(n-k+1)}. - Paul Barry, Aug 30 2013
G.f. : x^2*(1+2*x+3*x^2-x^4)/((1-x)^2*(1+x)^2).
a(n) = A080512(n) - 1 for n>2.

A103517 Expansion of (1+2*x-x^2)/(1-x)^2.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126

Offset: 0

Paul Barry, Feb 09 2005

Row sums of A103516.
Also the number of maximal and maximum cliques in the (n+1) X (n+1) rook graph. - Eric W. Weisstein, Sep 14 2017
Also the number of maximal and maximum independent vertex sets in the (n+1) X (n+1) rook complement graph. - Eric W. Weisstein, Sep 14 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set
Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set
Eric Weisstein's World of Mathematics, Rook Complement Graph
Eric Weisstein's World of Mathematics, Rook Graph
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A103516.
Essentially the same as A004277, A005843, A051755, and A076032. - R. J. Mathar, Jul 31 2010
Cf. A272651 (for which this sequence is a conjectured continuation for large n).

[1] cat [2*n+2 : n in [1..60]]; // Wesley Ivan Hurt, Dec 07 2016

a := n -> 2*(n + 1) - 0^n: seq(a(n), n = 0..62); # Peter Luschny, May 12 2023

CoefficientList[Series[(-z^2 + 2*z + 1)/(z - 1)^2, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 10 2011 *)
CoefficientList[Series[2/(x - 1)^2 - 1, {x, 0, 62}], x] (* Robert G. Wilson v, Jan 29 2015 *)
Join[{1}, 2 Range[2, 20]] (* Eric W. Weisstein, Sep 14 2017 *)
Join[{1}, LinearRecurrence[{2, -1}, {4, 6}, 20]] (* Eric W. Weisstein, Sep 14 2017 *)

a(n) = 2*n + 2 - 0^n.
a(n) = Sum_{k=0..n} 0^(k(n-k))*(n+1).
Equals binomial transform of [1, 3, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 23 2008
a(n) = 2*a(n-1) - a(n-2) for n > 2. - Eric W. Weisstein, Sep 14 2017
G.f.: (1 + 2*x - x^2)/(-1 + x)^2. - Eric W. Weisstein, Sep 14 2017

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A135852 A007318 * A103516 as a lower triangular matrix.

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A135853 A103516 * A007318 as an infinite lower triangular matrix.

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A187012 Antidiagonal sums of A103516.

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A103517 Expansion of (1+2*x-x^2)/(1-x)^2.

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