A014480 -id:A014480 - cp's OEIS frontend

A105851 Binomial transform triangle, read by rows.

1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 20, 12, 5, 1, 32, 48, 32, 16, 6, 1, 64, 112, 80, 44, 20, 7, 1, 128, 256, 192, 112, 56, 24, 8, 1, 256, 576, 448, 272, 144, 68, 28, 9, 1, 512, 1280, 1024, 640, 352, 176, 80, 32, 10, 1, 1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1

Offset: 0

Gary W. Adamson, Apr 23 2005

Let P = Pascal's triangle as an infinite lower triangular matrix and A is the infinite array of arithmetic sequences as shown in A077028:
  1, 1, 1,  1,  1, ...
  1, 2, 3,  4,  5, ...
  1, 3, 5,  7,  9, ...
  1, 4, 7, 10, 13, ...
  1, 5, 9, 13, 17, ...
Perform the operation P * A, getting a new array with each column being the binomial transform of an arithmetic sequence. Take antidiagonals of the new array, then by rows = the triangle of A105851.

			Column 3: 1, 5, 16, 44, 112, ... (A053220) is the binomial transform of 3k+1 (A016777: 1, 4, 7, ...).
Triangle begins:
     1;
     2,    1;
     4,    3,    1;
     8,    8,    4,    1;
    16,   20,   12,    5,   1;
    32,   48,   32,   16,   6,   1;
    64,  112,   80,   44,  20,   7,   1;
   128,  256,  192,  112,  56,  24,   8,  1;
   256,  576,  448,  272, 144,  68,  28,  9,  1;
   512, 1280, 1024,  640, 352, 176,  80, 32, 10,  1;
  1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1;
  ...

Cf. A077028, A001792, A001787, A053220, A016777, A014480.

/* As triangle */ [[(2+k*(n-k))*2^(n-k-1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 26 2015

seq(seq((2 + k*(n - k))*2^(n-k-1),k=0..n),n=0..10); # Peter Bala, Jul 26 2015

t[n_, k_]:=(2 + k (n - k)) 2^(n - k - 1); Table[t[n - 1, k - 1], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Jul 26 2015 *)

n-th column of the triangle is the binomial transform of the arithmetic sequence (n*k + 1), (k = 0, 1, 2, ...).
From Peter Bala, Jul 26 2015: (Start)
T(n,k) = (2 + k*(n - k))*2^(n-k-1) for 0 <= k <= n.
O.g.f.: (1 - x*(2 + t) + 3*t*x^2)/((1 - 2*x)^2*(1 - t*x)^2) = 1 + (2 + t)*x + (4 + 3*t + t^2)*x^2 + ....
k-th column g.f.: (1 + (k - 2)*x)/(1 - 2*x)^2. Cf. A077028. (End)

More terms from Philippe Deléham, Mar 31 2007

A126063 Triangle read by rows: see A128196 for definition.

Original entry on oeis.org

1, 1, 2, 3, 6, 4, 15, 30, 20, 8, 105, 210, 140, 56, 16, 945, 1890, 1260, 504, 144, 32, 10395, 20790, 13860, 5544, 1584, 352, 64, 135135, 270270, 180180, 72072, 20592, 4576, 832, 128, 2027025, 4054050, 2702700, 1081080, 308880, 68640, 12480, 1920, 256

Offset: 0

N. J. A. Sloane, Feb 28 2007

			Triangle begins:
       1
       1,       2
       3,       6,       4
      15,      30,      20,       8
     105,     210,     140,      56,     16
     945,    1890,    1260,     504,    144,    32
   10395,   20790,   13860,    5544,   1584,   352,    64
  135135,  270270,  180180,   72072,  20592,  4576,   832,  128

Ivan Neretin, Rows n = 0..100, flattened
P. Luschny, Variants of Variations.

First column is A001147, second column is A097801.

The diagonal is A000079, the subdiagonal is A014480.

A126063 := (n,k) -> 2^k*doublefactorial(2*n-1)/ doublefactorial(2*k-1); seq(print(seq(A126063(n,k),k=0..n)),n=0..7); # Peter Luschny, Dec 20 2012

Flatten[Table[2^k (2n - 1)!!/(2k - 1)!!, {n, 0, 8}, {k, 0, n}]] (* Ivan Neretin, May 11 2015 *)

Let H be the diagonal matrix diag(1,2,4,8,...) and
let G be the matrix (n!! defined as A001147(n), -1!! = 1):
(-1)!!/(-1)!!
1!!/(-1)!! 1!!/1!!
3!!/(-1)!! 3!!/1!! 3!!/3!!
5!!/(-1)!! 5!!/1!! 5!!/3!! 5!!/5!!
...
Then T = G*H. [Gottfried Helms]
T(n,k) = 2^k*(2n - 1)!!/(2k - 1)!!. - Ivan Neretin, May 13 2015

A204203 Triangle based on (0,1/4,1) averaging array.

Original entry on oeis.org

1, 1, 5, 1, 6, 13, 1, 7, 19, 29, 1, 8, 26, 48, 61, 1, 9, 34, 74, 109, 125, 1, 10, 43, 108, 183, 234, 253, 1, 11, 53, 151, 291, 417, 487, 509, 1, 12, 64, 204, 442, 708, 904, 996, 1021, 1, 13, 76, 268, 646, 1150, 1612, 1900, 2017, 2045, 1, 14, 89, 344, 914

Offset: 1

Clark Kimberling, Jan 12 2012

See A204201 for a discussion and guide to other averaging arrays.

			First six rows:
1
1...5
1...6...13
1...7...19...29
1...8...26...48...61
1...9...34...74...109...125

Cf. A204201.

a = 0; r = 1/4; b = 1;  t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]    (* averaging array *)
u = Table[(1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]  (* A204203 triangle *)
Flatten[u]    (* A204203 sequence *)

From Philippe Deléham, Dec 24 2013: (Start)
T(n,n) = A036563(n+1).
Sum_{k=1..n} T(n,k) = A014480(n-1).
T(n,k) = T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=1, T(2,1)=1, T(2,2)=5, T(n,k)=0 if k<1 or if k>n. (End)

A257791 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 2^(n+1)(2k - 1), n,k >= 1.

Original entry on oeis.org

4, 8, 12, 16, 24, 20, 32, 48, 40, 28, 64, 96, 80, 56, 36, 128, 192, 160, 112, 72, 44, 256, 384, 320, 224, 144, 88, 52, 512, 768, 640, 448, 288, 176, 104, 60, 1024, 1536, 1280, 896, 576, 352, 208, 120, 68, 2048, 3072, 2560, 1792, 1152, 704, 416, 240, 136, 76

Offset: 1

L. Edson Jeffery, May 08 2015

Lemma: The sequence is a permutation of A008586\{0} = {4*m : m = 1,2,...}.

Proof: Write A(n,k)/4 = A054582(n-1,k-1). The sequence A054582 is known to be a permutation of the natural numbers, and the result follows. QED

			Array A begins:
.       4    12     20     28     36     44     52     60     68     76
.       8    24     40     56     72     88    104    120    136    152
.      16    48     80    112    144    176    208    240    272    304
.      32    96    160    224    288    352    416    480    544    608
.      64   192    320    448    576    704    832    960   1088   1216
.     128   384    640    896   1152   1408   1664   1920   2176   2432
.     256   768   1280   1792   2304   2816   3328   3840   4352   4864
.     512  1536   2560   3584   4608   5632   6656   7680   8704   9728
.    1024  3072   5120   7168   9216  11264  13312  15360  17408  19456
.    2048  6144  10240  14336  18432  22528  26624  30720  34816  38912

Cf. A000079 (powers of 2), A005408 (odd numbers), A008586 (multiples of 4), A014480, A054582.

Cf. A257499.

(* Array: *)
A257791[n_, k_] := 2^(n + 1)*(2*k - 1); Grid[Table[A257791[n, k], {n, 10}, {k, 10}]]
(* Array antidiagonals flattened: *)
Flatten[Table[2^(n - k + 2)*(2*k - 1), {n, 10}, {k, n}]]

A(n,n) = 4*A014480(n-1).

A339771 a(n) = Sum_{i=0..n} Sum_{j=0..n} 2^max(i,j).

Original entry on oeis.org

1, 7, 27, 83, 227, 579, 1411, 3331, 7683, 17411, 38915, 86019, 188419, 409603, 884739, 1900547, 4063235, 8650755, 18350083, 38797315, 81788931, 171966467, 360710147, 754974723, 1577058307, 3288334339, 6845104131, 14227079171, 29527900163, 61203283971

Offset: 0

Bernard Schott, Dec 16 2020

			a(3) = 5*2^4 + 3 = 83.

Eric Billault, Walter Damin, Robert Ferréol, Rodolphe Garin, MPSI Classes Prépas - Khôlles de Maths, Exercices corrigés, Ellipses, 2012, exercice 2.22 (2), pp. 26, 43-44.

Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A142964 (with min instead of max).
Cf. A027981, A066524.
Partial sums of A014480.

Maple
```
seq((2*n-1)*2^(n+1)+3,n=0..40);
```

Table[(2*n - 1)*2^(n + 1) + 3, {n, 0, 29}] (* Amiram Eldar, Dec 16 2020 *)

a(n) = sum(i=0, n, sum(j=0, n, 2^max(i,j))); \\ Michel Marcus, Dec 16 2020

def A339771():
    a, b, c = 1, 7, 27
    yield(a); yield(b)
    while True:
        yield c
        z = 4*a - 8*b + 5*c
        a, b, c = b, c, z
a = A339771()
print([next(a) for  in range(30)]) # _Peter Luschny, Dec 17 2020

a(n) = (2*n-1) * 2^(n+1) + 3.
G.f.: -(2*x+1)/((x-1)*(2*x-1)^2). - Alois P. Heinz, Dec 16 2020
E.g.f: 3*exp(x) + 2*exp(2*x)*(4*x - 1). - Stefano Spezia, Dec 16 2020
a(n) = 2*A066524(n+1) - A142964(n). - Kevin Ryde, Dec 17 2020
a(n) = (2*A027981(n)+1)/3 for n >= 1. - Hugo Pfoertner, Dec 17 2020

A132775 A007818 * A132774.

Original entry on oeis.org

1, 3, 3, 5, 10, 5, 7, 21, 21, 7, 9, 36, 54, 36, 9, 11, 55, 110, 110, 55, 11, 13, 78, 195, 260, 195, 78, 13, 15, 105, 315, 525, 525, 315, 105, 15, 17, 136, 476, 952, 1190, 952, 476, 136, 17, 19, 171, 684, 1596, 2394, 2394, 1596, 684, 171, 19

Offset: 0

Gary W. Adamson, Aug 29 2007

Row sums = A014480: (1, 6, 20, 56, 144, 352, 832, ...).

			First few rows of the triangle:
   1;
   3,   3;
   5,  10,   5;
   7,  21,  21,   7;
   9,  36,  54,  36,   9;
  11,  55, 110, 110,  55,  11;
  13,  78, 195, 260, 195,  78,  13;
  15, 105, 315, 525, 525, 315, 105, 15;
  ...
Row 3 = (7, 21, 21, 7) = 7 * (1, 3, 3, 1).

Cf. A132774, A014480.

Binomial transform of A132774. T(n,k) = (2n+1) * A007318(n,k).

A134233 (A007318 * A134082 + A134082 * A007318) - A007318 as infinite lower triangular matrices.

Original entry on oeis.org

1, 5, 1, 9, 10, 1, 13, 27, 15, 1, 17, 52, 54, 20, 1, 21, 85, 130, 90, 25, 25, 126, 255, 260, 135, 30, 1, 29, 175, 441, 595, 455, 189, 35, 33, 232, 700, 1176, 1190, 728, 252, 40, 1, 37, 297, 1044, 2100, 2646, 2142, 1092, 324, 45, 1

Offset: 0

Gary W. Adamson, Oct 14 2007

Row sums = A014480: (1, 6, 20, 56, 144, 352, ...).

Also 2*A134083-A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 15 2007

			First few rows of the triangle:
   1;
   5,   1;
   9,  10,   1;
  13,  27,  15,   1;
  17,  52,  54,  20,   1;
  21,  85, 130,  90,  25,   1;
  ...

Cf. A134082, A014480.

A185342 Triangle of successive recurrences in columns of A117317(n).

Original entry on oeis.org

2, 4, -4, 6, -12, 8, 8, -24, 32, -16, 10, -40, 80, -80, 32, 12, -60, 160, -240, 192, -64, 14, -84, 280, -560, 672, -448, 128, 16, -112, 448, -1120, 1792, -1792, 1024, -256, 18, -144, 672, -2016, 4032, -5376, 4608, -2304, 512, 20, -180, 960, -3360, 8064

Offset: 0

Paul Curtz, Jan 26 2012

A117317 (A):
1
2    1
4    5   1
8   16   9   1
16  44  41  14   1
32 112 146  85  20  1
64 272 456 377 155 27 1
have for their columns successive signatures
(2) (4,-4) (6,-12,8) (8,-24, 32, -16) (10,-40,80,-80,32) i.e. a(n).
Take based on abs(A133156) (B):
1
2   0
4   1  0
8   4  0 0
16 12  1 0 0
32 32  6 0 0 0
64 80 24 1 0 0 0.
The recurrences of successive columns are also a(n). a(n) columns: A005843(n+1), A046092(n+1), A130809, A130810, A130811, A130812, A130813.
A053220 + A001787 = A014480.

			Triangle T(n,k),for 1<=k<=n, begins :
2                                         (1)
4    -4                                   (2)
6   -12   8                               (3)
8   -24  32   -16                         (4)
10  -40  80   -80   32                    (5)
12  -60 160  -240  192   -64              (6)
14  -84 280  -560  672  -448  128         (7)
16 -112 448 -1120 1792 -1792 1024 -256    (8)
Successive rows can be divided by A171977.

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

Cf. For (A): A053220, A056243. For (B): A000079, A001787, A001788, A001789. For A193862: A115068 (a Coxeter group). For (2): A014480 (also (3),(4),(5),..); also A053220 and A001787.

Cf. A007318.

Table[(-1)*Binomial[n, k]*(-2)^k, {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Jun 27 2017 *)

for(n=1,20, for(k=1,n, print1((-2)^(k+1)*binomial(n,k)/2, ", "))) \\ G. C. Greubel, Jun 27 2017

Take A133156(n) without 1's or -1's double triangle (C)=
2
4
8      -4
16    -12
32    -32   6
64    -80  24
128  -192  80   -8
256  -448 240  -40
512 -1024 672 -160 10;
a(n) is increasing odd diagonals and increasing (sign changed) even diagonals. Rows sum of (C) = A201629 (?) Another link between Chebyshev polynomials and cos( ).
Absolute values: A013609(n) without 1's. Also 2*A193862 = (2*A002260)*A135278.
T(n,k) = T(n-1,k) - 2*T(n-1,k-1) for k>1, T(n,1) = 2*n = 2*T(n-1,1) - T(n-2,1). - Philippe Deléham, Feb 11 2012
T(n,k) = (-1)* Binomial(n,k)*(-2)^k, 1<=k<=n. - Philippe Deléham, Feb 11 2012

A261349 T(n,k) is the decimal equivalent of a code for k that maximizes the sum of the Hamming distances between (cyclical) adjacent code words; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 3, 1, 2, 0, 7, 1, 6, 3, 4, 2, 5, 0, 15, 1, 14, 3, 12, 2, 13, 6, 9, 7, 8, 5, 10, 4, 11, 0, 31, 1, 30, 3, 28, 2, 29, 6, 25, 7, 24, 5, 26, 4, 27, 12, 19, 13, 18, 15, 16, 14, 17, 10, 21, 11, 20, 9, 22, 8, 23, 0, 63, 1, 62, 3, 60, 2, 61, 6, 57, 7, 56, 5

Offset: 0

Alois P. Heinz, Nov 18 2015

This code might be called "Anti-Gray code".

The sum of the Hamming distances between (cyclical) adjacent code words of row n gives 0, 2, 6, 20, 56, 144, 352, ... = A014480(n-1) for n>1.

			Triangle T(n,k) begins:
  0;
  0,  1;
  0,  3, 1,  2;
  0,  7, 1,  6, 3,  4, 2,  5;
  0, 15, 1, 14, 3, 12, 2, 13, 6,  9, 7,  8, 5, 10, 4, 11;
  0, 31, 1, 30, 3, 28, 2, 29, 6, 25, 7, 24, 5, 26, 4, 27, 12, 19, ... ;
  0, 63, 1, 62, 3, 60, 2, 61, 6, 57, 7, 56, 5, 58, 4, 59, 12, 51, ... ;

Alois P. Heinz, Rows n = 0..13, flattened
Wikipedia, Gray code
Wikipedia, Hamming distance

Columns k=0-3 give: A000004, A000225, A000012 (for n>1), A000918 (for n>1).
Row lengths give A000079.
Row sums give A006516.
Cf. A003188, A014480, A083329, A083420, A101080.

g:= n-> Bits[Xor](n, iquo(n, 2)):
T:= (n, k)-> (t-> `if`(m=0, t, 2^n-1-t))(g(iquo(k, 2, 'm'))):
seq(seq(T(n, k), k=0..2^n-1), n=0..6);

T(n,k) = A003188(k/2) if k even, T(n,k) = 2^n-1-A003188((k-1)/2) else.
A101080(T(n,2k),T(n,2k+1)) = n, A101080(T(n,2k),T(n,2k-1)) = n-1.
T(n,2^n-1) = A083329(n-1) for n>0.
T(n,2^n-2) = A000079(n-2) for n>1.
T(2n,2n) = A003188(n).
T(2n+1,2n+1) = 2*4^n - 1 - A003188(n) = A083420(n) - A003188(n).

A264735 a(n) = prime(2^(n-1)(2n-1)), n >= 1.

Original entry on oeis.org

2, 13, 71, 263, 827, 2377, 6379, 16603, 41611, 101573, 243473, 573637, 1333613, 3065983, 6975971, 15746347, 35277211, 78510373, 173717329, 382404863, 837870343, 1828047503, 3973066781, 8604768101, 18576139177, 39983413319

Offset: 1

L. Edson Jeffery, Nov 22 2015

nonn

Cf. A000040, A014480, A264731 (main diagonal).

[NthPrime(2^(n-1)*(2*n-1)): n in [1..20]]; // Vincenzo Librandi, Nov 23 2015

Table[Prime[2^(n - 1)*(2*n - 1)], {n, 26}]

a(n)=if(n<1,print1("Error n < 1"),prime(2^(n-1)*(2*n-1))) \\ Anders Hellström, Nov 23 2015

use ntheory ":all"; say "$ ",nth_prime(2**($-1) * (2*$-1))  for 1..50; # _Dana Jacobsen, Apr 28 2017

a(n) = A000040(2^(n-1)*(2*n-1)).

a(n) = A000040(A014480(n-1)). - Michel Marcus, Nov 23 2015

cp's OEIS Frontend

A105851 Binomial transform triangle, read by rows.

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A126063 Triangle read by rows: see A128196 for definition.

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A204203 Triangle based on (0,1/4,1) averaging array.

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A257791 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 2^(n+1)*(2*k - 1), n,k >= 1.

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A339771 a(n) = Sum_{i=0..n} Sum_{j=0..n} 2^max(i,j).

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A132775 A007818 * A132774.

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A134233 (A007318 * A134082 + A134082 * A007318) - A007318 as infinite lower triangular matrices.

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A185342 Triangle of successive recurrences in columns of A117317(n).

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A257791 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 2^(n+1)(2k - 1), n,k >= 1.

A264735 a(n) = prime(2^(n-1)(2n-1)), n >= 1.