A045883 -id:A045883 - cp's OEIS frontend

A122438 Riordan array (1/(1-2x), x(1+2x)).

1, 2, 1, 4, 4, 1, 8, 8, 6, 1, 16, 16, 16, 8, 1, 32, 32, 32, 28, 10, 1, 64, 64, 64, 64, 44, 12, 1, 128, 128, 128, 128, 120, 64, 14, 1, 256, 256, 256, 256, 256, 208, 88, 16, 1, 512, 512, 512, 512, 512, 496, 336, 116, 18, 1, 1024, 1024, 1024, 1024, 1024, 1024, 912, 512, 148

Offset: 0

Paul Barry, Sep 05 2006

Generalized Whitney triangle. Row sums are A045883(n+1). Diagonal sums are A122439.

			Number triangle begins
1,
2, 1,
4, 4, 1,
8, 8, 6, 1,
16, 16, 16, 8, 1,
32, 32, 32, 28, 10, 1,
64, 64, 64, 64, 44, 12, 1

Cf. A004070.

T[n_, k_] := Sum[ Binomial[k, n - k - j]*2^(n - k), {j, 0, n - k}]; Table[ T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Sep 14 2006 *)

Number triangle T(n,k)=sum{j=0..n-k, C(k,n-k-j)}*2^(n-k).

More terms from Robert G. Wilson v, Sep 14 2006

A102841 a(n) = ((9n^2 + 33n + 26)*2^n + (-1)^n)/27.

Original entry on oeis.org

1, 5, 19, 61, 179, 493, 1299, 3309, 8211, 19949, 47635, 112109, 260627, 599533, 1366547, 3089901, 6937107, 15476205, 34331155, 75769325, 166451731, 364127725, 793500179, 1723082221, 3729512979, 8048092653, 17319057939

Offset: 0

Creighton Dement, Feb 27 2005

nonn

A floretion-generated sequence relating the number of edges and faces in n-dimensional hypercube.
Equals A001787, (1, 4, 12, 32, 80, ...) convolved with A001045, the Jacobsthal sequence. - Gary W. Adamson, May 23 2009
The sum of the sizes of all inversions in compositions of n. - Arnold Knopfmacher, Jan 22 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
Index entries for linear recurrences with constant coefficients, signature (5,-6,-4,8).

Cf. A045883, A001788, A001793, A102301.

Cf. A001787, A001045. - Gary W. Adamson, May 23 2009

[((9*n^2 + 33*n + 26)*2^n + (-1)^n)/27 : n in [0..40]]; // Wesley Ivan Hurt, Jul 03 2020

Table[(1/27)*((9 n^2 + 33 n + 26) 2^n + (-1)^n), {n, 0, 50}] (* or *) LinearRecurrence[{5,-6,-4,8}, {1,5,19,61}, 50] (* G. C. Greubel, Sep 27 2017 *)

G.f.: 1/((1+x)*(1-2*x)^3).
a(n+1) - 2*a(n) = A045883(n+2).
a(n) + a(n+1) = A001788(n+2).
a(n) = 5*a(n-1) - 6*a(n-2) - 4*a(n-3) + 8*a(n-4). - Wesley Ivan Hurt, Jul 03 2020

Corrected by T. D. Noe, Nov 08 2006

A225114 Number of skew partitions of n whose diagrams have no empty rows and columns.

Original entry on oeis.org

1, 1, 3, 9, 28, 87, 272, 850, 2659, 8318, 26025, 81427, 254777, 797175, 2494307, 7804529, 24419909, 76408475, 239077739, 748060606, 2340639096, 7323726778, 22915525377, 71701378526, 224349545236, 701976998795, 2196446204672, 6872555567553, 21503836486190, 67284284442622, 210528708959146

Offset: 0

Joerg Arndt, Apr 29 2013

nonn

A skew partition S of size n is a pair of partitions [p1,p2] where p1 is a partition of the integer n1, p2 is a partition of the integer n2, p2 is an inner partition of p1, and n=n1-n2. We say that p1 and p2 are respectively the inner and outer partitions of S. A skew partition can be depicted by a diagram made of rows of cells, in the same way as a partition. Only the cells of the outer partition p1 which are not in the inner partition p2 appear in the picture. [from the Sage manual, see links]

			The a(4)=28 skew partitions of 4 are
01:  [[4], []]
02:  [[3, 1], []]
03:  [[4, 1], [1]]
04:  [[2, 2], []]
05:  [[3, 2], [1]]
06:  [[4, 2], [2]]
07:  [[2, 1, 1], []]
08:  [[3, 2, 1], [1, 1]]
09:  [[3, 1, 1], [1]]
10:  [[4, 2, 1], [2, 1]]
11:  [[3, 3], [2]]
12:  [[4, 3], [3]]
13:  [[2, 2, 1], [1]]
14:  [[3, 3, 1], [2, 1]]
15:  [[3, 2, 1], [2]]
16:  [[4, 3, 1], [3, 1]]
17:  [[2, 2, 2], [1, 1]]
18:  [[3, 3, 2], [2, 2]]
19:  [[3, 2, 2], [2, 1]]
20:  [[4, 3, 2], [3, 2]]
21:  [[1, 1, 1, 1], []]
22:  [[2, 2, 2, 1], [1, 1, 1]]
23:  [[2, 2, 1, 1], [1, 1]]
24:  [[3, 3, 2, 1], [2, 2, 1]]
25:  [[2, 1, 1, 1], [1]]
26:  [[3, 2, 2, 1], [2, 1, 1]]
27:  [[3, 2, 1, 1], [2, 1]]
28:  [[4, 3, 2, 1], [3, 2, 1]]

Sage Development Team, Skew Partitions, Sage Reference Manual.

\\ The following program is significantly faster.
A225114(n)=
{
    my( C=vector(n, j, 1) );
    my(m=n, z, t, ret);
    while ( 1,  /* for all compositions C[1..m] of n */
\\        print( vector(m, n, C[n] ) ); /* print composition */
        t = prod(j=2,m, min(C[j-1], C[j]) + 1 );  /* A225114 */
\\        t = prod(j=2,m, min(C[j-1], C[j]) + 0 );  /* A006958 */
\\        t = prod(j=2,m, C[j-1] + C[j] + 0 );  /* A059716 */
\\        t = prod(j=2,m, C[j-1] + C[j] + 1 );  /* A187077 */
\\        t = sum(j=2,m, C[j-1] > C[j] );  /* A045883 */
        ret += t;
        if ( m<=1, break() ); /* last composition? */
        /* create next composition: */
        C[m-1] += 1;
        z = C[m];
        C[m] = 1;
        m += z - 2;
    );
    return(ret);
}
for (n=0, 30, print1(A225114(n),", "));
\\ Joerg Arndt, Jul 09 2013

[SkewPartitions(n).cardinality() for n in range(16)]

Conjectured g.f.: 1/(2 - 1/(1 - x/(1 - x/(1 - x^2/(1 - x^2/(1 - x^3/(1 - x^3/(1 - ...)))))))). - Mikhail Kurkov, Sep 03 2024

Edited by Max Alekseyev, Dec 22 2015

A248810 Signed version of A164984.

Original entry on oeis.org

1, -1, 1, 3, -3, 1, -5, 9, -5, 1, 11, -23, 19, -7, 1, -21, 57, -61, 33, -9, 1, 43, -135, 179, -127, 51, -11, 1, -85, 313, -493, 433, -229, 73, -13, 1, 171, -711, 1299, -1359, 891, -375, 99, -15, 1, -341, 1593, -3309, 4017, -3141, 1641, -573, 129, -17, 1, 683, -3527, 8211, -11343, 10299, -6423, 2787, -831, 163, -19, 1

Offset: 0

Derek Orr, Oct 14 2014

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+2)^0 + A_1*(x+2)^1 + A_2*(x+2)^2 + ... + A_n*(x+2)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

			1;
-1,      1;
3,      -3,     1;
-5,      9,    -5,      1;
11,    -23,    19,     -7,     1;
-21,    57,   -61,     33,    -9,     1;
43,   -135,   179,   -127,    51,   -11,    1;
-85,   313,  -493,    433,  -229,    73,  -13,    1;
171,  -711,  1299,  -1359,   891,  -375,   99,  -15,   1;
-341, 1593, -3309,   4017, -3141,  1641, -573,  129, -17,   1;
683, -3527,  8211, -11343, 10299, -6423, 2787, -831, 163, -19, 1;

Cf. A164984, A193845, A077925, A045883.

for(n=0,20,for(k=0,n,print1(1/k!*sum(i=0,n,((-2)^(i-k)*prod(j=0,k-1,i-j))),", ")))

T(n,n-1) = -2*n+1 for n > 0.
T(n,n-2) = 2*(n-1)^2+1 for n > 1.
T(n,0) = A077925(n).
T(n,1) = (-1)^(n+1)*A045883(n).
Rows with odd n sum to 0.
Rows with even n sum to 1.

A348407 a(n) = ((n+1)32^(n+1) + 29*2^n + (-1)^n)/9.

Original entry on oeis.org

4, 9, 21, 47, 105, 231, 505, 1095, 2361, 5063, 10809, 22983, 48697, 102855, 216633, 455111, 953913, 1995207, 4165177, 8679879, 18058809, 37515719, 77827641, 161247687, 333680185, 689729991, 1424199225, 2937876935, 6054710841, 12467335623, 25650499129, 52732654023, 108328619577

Offset: 0

Paul Curtz and Thomas Scheuerle, Oct 17 2021

The ratio (count of ones)/(count of zeros) in the binary expansion of a(n) is > 1/2 and <= 5 for all n > 0, this is because the division by 9 adds a repeating pattern 111000... after some binary digits.

This sequence has in its "partial binomial transform" (see formula section) no other constants than 2 and 1 despite of its more complicated looking closed form expression. This transform has a deep connection to the Grünwald-Letnikov fractional derivative if we replace the order of the derivative with the variable x: D^x*f(x).

Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Cf. A001045, A001787, A034007, A045883, A052951, A113861, A348405.

Array[((# + 1)*3*2^(# + 1) + 29*2^# + (-1)^#)/9 &, 33, 0] (* Michael De Vlieger, Oct 19 2021 *)
LinearRecurrence[{3,0,-4},{4,9,21},40] (* Harvey P. Dale, Aug 12 2023 *)

a(n) = round(((n+1)*3*2^(n+1) + 29*2^n)/9).
a(n) = 2^(n+2) + A113861(n).
a(n) = 2^(n+2) + n*2^n - A045883(n) = 2^(n+2) + n*2^n - round(((3*n+1)*2^n)/9).
a(n+1) - 2*a(n) = A001045(n+2).
a(n) = A034007(n+3) + A045883(n-1) for n > 0.
A partial binomial transform in two parts:
(Partial means a diagonal in a difference table a(0), a(2)-a(1), ... . This is partial because one diagonal alone is no invertible transform.)
A001787(n+2) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(2*n-k)
             = (n+2)*2^(n+1).
A052951(n+1) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*(a(1+2*n-k) - a(2*n-k))
             = (n+2)*2^(n+1) + 2^n.
The inverse transform:
a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(k+2)*2^(k+1)
       + Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k)*((k+2)*2^(k+1) + 2^k).
From Stefano Spezia, Oct 20 2021: (Start)
G.f.: (4 - 3*x - 6*x^2)/((1 + x)*(1 - 2*x)^2).
a(n) = 3*a(n-1) - 4*a(n-3) for n > 2. (End)

A191007 a(n) = n*2^(n+1) + (2^(n+3)+(-1)^n)/3.

Original entry on oeis.org

3, 9, 27, 69, 171, 405, 939, 2133, 4779, 10581, 23211, 50517, 109227, 234837, 502443, 1070421, 2271915, 4805973, 10136235, 21321045, 44739243, 93672789, 195734187, 408245589, 850045611, 1767200085, 3668617899, 7605671253, 15748213419, 32570168661, 67287820971

Offset: 0

Edward Omey, Jun 16 2011

Another renewal type of sequence: Let X, X(1),X(2),... denote independent random variables with pdf P(X=1) = P(X=2) = P(X=4) = 1/3. Let N(x) denote the first value of k such that X(1)*X(2)...*X(k) > x, and let H(x) = E(N(x)). The sequence a(n) is given by a(n) = 2^(n+1)*H(2^n).

Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

3 times A045883.

[n*2^(n+1)+(2^(n+3)+(-1)^n)/3: n in [0..30]]; // Vincenzo Librandi, Oct 16 2014

Table[n 2^(n + 1) + (2^(n + 3) + (-1)^n)/3, {n, 0, 70}] (* Vincenzo Librandi, Oct 16 2014 *)
LinearRecurrence[{3,0,-4},{3,9,27},40] (* Harvey P. Dale, Feb 11 2024 *)

a(n) = n*2^(n+1) + (2^(n+3)+(-1)^n)/3; \\ Michel Marcus, Oct 16 2014

a(n) = n*2^(n+1) + (2^(n+3)+(-1)^n)/3.
a(n) = 3 * A045883(n+1).
G.f.: 3/((1 + x)*(1 - 2*x)^2). [Bruno Berselli, Oct 16 2014]

Formula corrected and more terms from Michel Marcus, Oct 16 2014

A246798 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x+3)^k.

Original entry on oeis.org

1, -5, 2, 22, -16, 3, -86, 92, -33, 4, 319, -448, 237, -56, 5, -1139, 1982, -1383, 484, -85, 6, 3964, -8224, 7122, -3296, 860, -120, 7, -13532, 32600, -33702, 19384, -6700, 1392, -161, 8, 45517, -124864, 150006, -103088, 44330, -12216, 2107, -208, 9, -151313, 465626, -637314, 509272, -261850, 89844, -20573, 3032, -261, 10

Offset: 0

Derek Orr, Nov 15 2014

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x+3)^0 + A_1*(x+3)^1 + A_2*(x+3)^2 + ... + A_n*(x+3)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

			Triangle starts:
1;
-5,           2;
22,         -16,       3;
-86,         92,     -33,       4;
319,       -448,     237,     -56,       5;
-1139,     1982,   -1383,     484,     -85,      6;
3964,     -8224,    7122,   -3296,     860,   -120,      7;
-13532,   32600,  -33702,   19384,   -6700,   1392,   -161,    8;
45517,  -124864,  150006, -103088,   44330, -12216,   2107, -208,    9;
-151313, 465626, -637314,  509272, -261850,  89844, -20573, 3032, -261, 10;
...

Cf. A246788, A045944, A191008.

T(n, k) = (k+1)*sum(i=0, n-k, (-3)^i*binomial(i+k+1, k+1))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))

T(n,0) = (1-(4*n+5)*(-3)^(n+1))/16, for n >= 0.
T(n,n-1) = -n*(3*n+2), for n >= 1.
Row n sums to (-1)^n*A045883(n+1) = T(n,0) of A246788.

A352692 a(n) + a(n+1) = 2^n for n >= 0 with a(0) = 4.

Original entry on oeis.org

4, -3, 5, -1, 9, 7, 25, 39, 89, 167, 345, 679, 1369, 2727, 5465, 10919, 21849, 43687, 87385, 174759, 349529, 699047, 1398105, 2796199, 5592409, 11184807, 22369625, 44739239, 89478489, 178956967, 357913945, 715827879, 1431655769, 2863311527, 5726623065, 11453246119, 22906492249

Offset: 0

Paul Curtz, Mar 29 2022

Difference table D(n,k) = D(n-1,k+1) - D(n-1,k), D(0,k) = a(k):
    4,  -3,   5,  -1,   9,   7, 25, ...
   -7,   8,  -6,  10,  -2,  18, 14,  50, ...
   15, -14,  16, -12,  20,  -4, 36,  28, 100, ...
  -29,  30, -28,  32, -24,  40, -8,  72,  56, 200, ...
   59, -58,  60, -56,  64, -48, 80, -16, 144, 112, 400, ...
  ...
The diagonals are given by D(n,n+k) = a(k)*2^n.
D(n,1) = -(-1)^n* A340627(n).
a(n)   - a(n) =  0,  0,  0,  0,   0, ...       (trivially)
a(n+1) + a(n) =  1,  2,  4,  8,  16, ... = 2^n (by definition)
a(n+2) - a(n) =  1,  2,  4,  8,  16, ... = 2^n
a(n+3) + a(n) =  3,  6, 12, 24,  48, ... = 2^n*3
a(n+4) - a(n) =  5, 10, 20, 40,  80, ... = 2^n*5
a(n+5) + a(n) = 11, 22, 44, 88, 176, ... = 2^n*11
     (...)
This table is given by T(r,n) = A001045(r)*2^n with r, n >= 0.
Sums of antidiagonals are A045883(n).
Main diagonal: A192382(n).
First upper diagonal: A054881(n+1).
First subdiagonal: A003683(n+1).
Second subdiagonal: A246036(n).
Now consider the array from c(n) = (-1)^n*a(n) with its difference table:
   4,   3,   5,    1,    9,   -7,   25,   -39, ...  = c(n)
  -1,   2,  -4,    8,  -16,   32,  -64,   128, ...  = -A122803(n)
   3,  -6,  12,  -24,   48,  -96,  192,  -384, ...  =
  -9,  18, -36,   72, -144,  288, -576,  1152, ...
  27, -54, 108, -216,  432, -864, 1728, -3456, ...
...
The first subdiagonal is -A000400(n). The second is A169604(n).

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

If a(0) = k then A001045 (k=0), A078008 (k=1), A140966 (k=2), A154879 (k=3), this sequence (k=4).
Essentially the same as A115335.
Cf. A000079, A002450, A340627.
Cf. A020714, A175805.
Cf. A045883, A192382, A003683, A246036.
Cf. A054881, A000400, A169604, A122803.
Cf. A132805, A082311, A082365.
Cf. A024495, A132804, A163868.

a := proc(n) option remember; ifelse(n = 0, 4, 2^(n-1) - a(n-1)) end: # Peter Luschny, Mar 29 2022
A352691 := proc(n)
    (11*(-1)^n + 2^n)/3
end proc: # R. J. Mathar, Apr 26 2022

LinearRecurrence[{1, 2}, {4, -3}, 40] (* Amiram Eldar, Mar 29 2022 *)

a(n) = (11*(-1)^n + 2^n)/3; \\ Thomas Scheuerle, Mar 29 2022

abs(a(n)) = A115335(n-1) for n >= 1.
a(3*n) - (-1)^n*4 = A132805(n).
a(3*n+1) + (-1)^n*4 = A082311(n).
a(3*n+2) - (-1)^n*4 = A082365(n).
From Thomas Scheuerle, Mar 29 2022: (Start)
G.f.: (-4 + 7*x)/(-1 + x + 2*x^2).
Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(m + 2*n-k) = a(m)*2^n.
Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(1 + n-k) = -(-1)^n*A340627(n).
a(n) = (11*(-1)^n + 2^n)/3.
a(n + 2*m) = a(n) + A002450(m)*2^n.
a(2*n) = A192382(n+1) + (-1)^n*a(n).
a(n) = ( A045883(n) - Sum_{k=0..n-1}(-1)^k*a(k) )/n, for n > 0. (End)
a(n) = A001045(n) + 4*(-1)^n.
a(n+1) = 2*a(n) -11*(-1)^n.
a(n+2) = a(n) + 2^n.
a(n+4) = a(n) + A020714(n).
a(n+6) = a(n) + A175805(n).
a(2*n) = A163868(n).
a(2*n+1) = (2^(2*n+1) - 11)/3.

Warning: The DATA is correct, but there may be errors in the COMMENTS, which should be rechecked. - Editors of OEIS, Apr 26 2022

Edited by M. F. Hasler, Apr 26 2022.

cp's OEIS Frontend

A122438 Riordan array (1/(1-2x), x(1+2x)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A102841 a(n) = ((9*n^2 + 33*n + 26)*2^n + (-1)^n)/27.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A225114 Number of skew partitions of n whose diagrams have no empty rows and columns.

Views

Author

Keywords

Comments

Examples

Links

Programs

PARI

Sage

Formula

Extensions

A248810 Signed version of A164984.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A348407 a(n) = ((n+1)*3*2^(n+1) + 29*2^n + (-1)^n)/9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A191007 a(n) = n*2^(n+1) + (2^(n+3)+(-1)^n)/3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A246798 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x+3)^k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A352692 a(n) + a(n+1) = 2^n for n >= 0 with a(0) = 4.

A102841 a(n) = ((9n^2 + 33n + 26)*2^n + (-1)^n)/27.

A348407 a(n) = ((n+1)32^(n+1) + 29*2^n + (-1)^n)/9.

A246798 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x+3)^k.