A130129 -id:A130129 - cp's OEIS frontend

A062111 Upper-right triangle resulting from binomial transform calculation for nonnegative integers.

0, 1, 1, 4, 3, 2, 12, 8, 5, 3, 32, 20, 12, 7, 4, 80, 48, 28, 16, 9, 5, 192, 112, 64, 36, 20, 11, 6, 448, 256, 144, 80, 44, 24, 13, 7, 1024, 576, 320, 176, 96, 52, 28, 15, 8, 2304, 1280, 704, 384, 208, 112, 60, 32, 17, 9, 5120, 2816, 1536, 832, 448, 240, 128, 68, 36, 19, 10

Offset: 0

Henry Bottomley, May 30 2001

From Philippe Deléham, Apr 15 2007: (Start)
This triangle can be found in the Laisant reference in the following form:
  .......................5...11..
  ...................4...9...20..
  ...............3...7..16...36..
  ...........2...5..12..28.......
  .......1...3...8..20..48.......
  ...0...1...4..12..32..80....... (End)
Triangle A152920 reversed. - Philippe Deléham, Apr 21 2009

			As a lower triangle (T(n, k)):
    0;
    1,   1;
    4,   3,   2;
   12,   8,   5,  3;
   32,  20,  12,  7,  4;
   80,  48,  28, 16,  9,  5;
  192, 112,  64, 36, 20, 11,  6;
  448, 256, 144, 80, 44, 24, 13, 7;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
F. Ellermann, Illustration of binomial transforms
C.-A. Laisant, Sur les tableaux de sommes - Nouvelles applications, Compt. Rendus de l'Association Francaise pour l'Avancement des Sciences, Aout 04 1893, pp. 206-216 (table given on p. 212).

Rows include (essentially) A001787, A001792, A034007, A045623, A045891.
Diagonals include (essentially) A001477, A005408, A008586, A008598, A017113.
Column sums are A058877.
Cf. A111297, A159694, A159695, A159696, A159697. - Philippe Deléham, Apr 21 2009
Cf. A058877, A073371, A130129, A152920.

[2^(n-k-1)*(n+k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 28 2022

Table[2^(n-k-1)*(n+k), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 28 2022 *)

def A062111(n,k): return 2^(n-k-1)*(n+k)
flatten([[A062111(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Sep 28 2022

A(n, k) = A(n, k-1) + A(n+1, k) if k > n with A(n, n) = n.
A(n, k) = (k+n)*2^(k-n-1) if k >= n.
T(2*n, n) = 3*n*2^(n-1) = 3*A001787(n). - Philippe Deléham, Apr 21 2009
From G. C. Greubel, Sep 28 2022: (Start)
T(n, k) = 2^(n-k-1)*(n+k) for 0 <= k <= n, n >= 0.
T(m*n, n) = 2^((m-1)*n-1)*(m+1)*A001477(n), m >= 1.
T(2*n-1, n-1) = A130129(n-1).
T(2*n+1, n-1) = 12*A001787(n).
Sum_{k=0..n} T(n, k) = A058877(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = 3*A073371(n-2), n >= 2.
T(n, k) = A152920(n, n-k). (End)

A066373 a(n) = (3n-2)2^(n-3).

Original entry on oeis.org

2, 7, 20, 52, 128, 304, 704, 1600, 3584, 7936, 17408, 37888, 81920, 176128, 376832, 802816, 1703936, 3604480, 7602176, 15990784, 33554432, 70254592, 146800640, 306184192, 637534208, 1325400064, 2751463424, 5704253440, 11811160064, 24427626496, 50465865728, 104152956928

Offset: 2

N. J. A. Sloane, Jan 04 2002

An elephant sequence, see A175654. For the corner squares 16 A[5] vectors, with decimal values between 59 and 440, lead to this sequence (with a leading 1 added). For the central square these vectors lead to the companion sequence A098156 (without a(1)). - Johannes W. Meijer, Aug 15 2010

a(n) is the total number of 1's in runs of 1's of length >= 2 over all binary words with n bits. - Félix Balado, Jan 15 2024

Harry J. Smith, Table of n, a(n) for n = 2..200
M. Azaola and F. Santos, The number of triangulations of the cyclic polytope C(n,n-4), Discrete Comput. Geom., 27 (2002), 29-48.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Column k=2 of A229079.

Cf. A016789, A098156, A130129, A288834.

seq((3*n-2)*2^(n-3),n=2..30); # Emeric Deutsch, Jul 23 2006

Array[(3 # - 2)*2^(# - 3) &, 28, 2] (* or *)
Drop[CoefficientList[Series[x^2*(2 - x)/(1 - 2 x)^2, {x, 0, 29}], x], 2] (* Michael De Vlieger, Jun 30 2018 *)

a(n) = { (3*n - 2)*2^(n - 3) } /* Harry J. Smith, Feb 11 2010 */

G.f.: x^2*(2-x)/(1-2x)^2. - Emeric Deutsch, Jul 23 2006
a(n) = 2*a(n-1) +3*2^(n-3). - Vincenzo Librandi, Mar 20 2011
a(n+1) - a(n) = A098156(n). - R. J. Mathar, Apr 25 2013
From Paul Curtz, Jun 29 2018: (Start)
a(n) = A130129(n-2) - A130129(n-3) for n >= 2.
Binomial transform of A016789.
Inverse binomial transform of A288834.
Also the main diagonal of the difference table of m -> (-1)^m*(m+2).
    2,  -3,   4,  -5, ...
   -5,   7,  -9,  11, ...
   12, -16,  20, -24, ...
  -28,  36, -44,  52, ... . (End)

A193730 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (2x+1)^n and q(n,x) = (2x+1)^n.

Original entry on oeis.org

1, 2, 1, 4, 8, 3, 8, 28, 30, 9, 16, 80, 144, 108, 27, 32, 208, 528, 648, 378, 81, 64, 512, 1680, 2880, 2700, 1296, 243, 128, 1216, 4896, 10800, 14040, 10692, 4374, 729, 256, 2816, 13440, 36288, 60480, 63504, 40824, 14580, 2187, 512, 6400, 35328, 112896, 229824, 308448, 272160, 151632, 48114, 6561

Offset: 0

Clark Kimberling, Aug 04 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Triangle T(n,k), read by rows, given by (2,0,0,0,0,0,0,0,...) DELTA (1,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

			First six rows:
   1;
   2,   1;
   4,   8,   3;
   8,  28,  30,   9;
  16,  80, 144, 108,  27;
  32, 208, 528, 648, 378, 81;

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

Cf. A000079, A005053, A084938, A130129, A133494, A165326.

Cf. A193722, A193731, A199923.

function T(n, k) // T = A193730
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 2*T(n-1, k) + 3*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2023

(* First program *)
z = 8; a = 2; b = 1; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]     (* A193730 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]     (* A193731 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 2*T[n-1, k] + 3*T[n-1, k-1]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2023 *)

def T(n, k): # T = A193730
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 2*T(n-1, k) + 3*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2023

T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x*y)/(1-2*x-3*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 20 2023: (Start)
T(n, 0) = A000079(n).
T(n, 1) = A130129(n-1).
T(n, n) = A133494(n).
T(n, n-1) = A199923(n).
Sum_{k=0..n} T(n, k) = A005053(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A165326(n). (End)

A193731 Mirror of the triangle A193730.

Original entry on oeis.org

1, 1, 2, 3, 8, 4, 9, 30, 28, 8, 27, 108, 144, 80, 16, 81, 378, 648, 528, 208, 32, 243, 1296, 2700, 2880, 1680, 512, 64, 729, 4374, 10692, 14040, 10800, 4896, 1216, 128, 2187, 14580, 40824, 63504, 60480, 36288, 13440, 2816, 256, 6561, 48114, 151632, 272160, 308448, 229824, 112896, 35328, 6400, 512

Offset: 0

Clark Kimberling, Aug 04 2011

A193731 is obtained by reversing the rows of the triangle A193730.

Triangle T(n,k), read by rows, given by (1,2,0,0,0,0,0,0,0,...) DELTA (2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

			First six rows:
   1;
   1,   2;
   3,   8,   4;
   9,  30,  28,   8;
  27, 108, 144,  80,  16;
  81, 378, 648, 528, 208, 32;

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

Cf. A000012, A000079, A005053, A006234, A007483, A080420, A080421.
Cf. A080422, A080423, A084938, A130129, A133494, A153881.
Cf. A193722, A193730.

function T(n, k) // T = A193731
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return k+1;
  else return 3*T(n-1, k) + 2*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2023

(* First program *)
z = 8; a = 2; b = 1; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193730 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]     (* A193731 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, 3*T[n-1, k] + 2*T[n -1, k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2023 *)

def T(n, k): # T = A193731
    if (k<0 or k>n): return 0
    elif (n<2): return k+1
    else: return 3*T(n-1, k) + 2*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2023

T(n,k) = A193730(n,n-k).
T(n,k) = 2*T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x)/(1-3*x-2*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 20 2023: (Start)
T(n, 0) = A133494(n).
T(n, 1) = 2*A006234(n+2).
T(n, 2) = 4*A080420(n-2).
T(n, 3) = 8*A080421(n-3).
T(n, 4) = 16*A080422(n-4).
T(n, 5) = 32*A080423(n-5).
T(n, n) = A000079(n).
T(n, n-1) = A130129(n-1).
Sum_{k=0..n} T(n, k) = A005053(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A153881(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A007483(n-1).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A000012(n). (End)

A236538 Triangle read by rows: T(n,k) = (n+1)2^(n-2)+(k-1)2^(n-1) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 5, 8, 12, 16, 20, 28, 36, 44, 48, 64, 80, 96, 112, 112, 144, 176, 208, 240, 272, 256, 320, 384, 448, 512, 576, 640, 576, 704, 832, 960, 1088, 1216, 1344, 1472, 1280, 1536, 1792, 2048, 2304, 2560, 2816, 3072, 3328, 2816, 3328, 3840, 4352, 4864, 5376

Offset: 1

Fedor Igumnov, Jan 28 2014

1, 9, 45, 161, 497, 1409, ... is the sequence of perimeters (sum of border elements) of the triangle.
1, 5, 80, 3520, 394240, 107233280, 68629299200, ... is the sequence of determinants of the triangle.
Only the first three terms are odd.

			Triangle begins:
================================================
\k |    1     2     3     4     5     6     7
n\ |
================================================
1  |    1;
2  |    3,    5;
3  |    8,   12,   16;
4  |   20,   28,   36,   44;
5  |   48,   64,   80,   96,  112;
6  |  112,  144,  176,  208,  240,  272;
7  |  256,  320,  384,  448,  512,  576,  640;
...

Fedor Igumnov, T(n,k) for n = 1..26

Cf. A001792 (column 1), A053220 (right border). Also:
A014477, row sums;
A036826, partial sums;
A058962, central elements in odd rows;
A045623, second column;
A045891, third column;
A034007, fourth column;
A167667, subdiagonal;
A130129, second subdiagonal.

int a(int n, int k) {return (n+1)*pow(2,n-2)+(k-1)*pow(2,n-1);}

/* As triangle: */ [[(n+1)*2^(n-2)+(k-1)*2^(n-1): k in [1..n]]: n in [1..10]]; // Bruno Berselli, Jan 28 2014

t[n_, k_] := (n + 1)*2^(n - 2) + (k - 1)*2^(n - 1); Table[t[n, k], {n, 10}, {k, n}] // Flatten (* Bruno Berselli, Jan 28 2014 *)

T(n,k) = T(n-1,k) + T(n-1,k+1).

Sum_{k=1..n} T(n,k) = n^2*2^(n-1) = A014477(n-1).

More terms from Bruno Berselli, Jan 28 2014

cp's OEIS Frontend

A062111 Upper-right triangle resulting from binomial transform calculation for nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A066373 a(n) = (3*n-2)*2^(n-3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A193730 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (2x+1)^n and q(n,x) = (2x+1)^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A193731 Mirror of the triangle A193730.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A236538 Triangle read by rows: T(n,k) = (n+1)*2^(n-2)+(k-1)*2^(n-1) for 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Magma

Mathematica

Formula

Extensions

A066373 a(n) = (3n-2)2^(n-3).

A236538 Triangle read by rows: T(n,k) = (n+1)2^(n-2)+(k-1)2^(n-1) for 1 <= k <= n.