A060801 -id:A060801 - cp's OEIS frontend

A291000 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3.

1, 3, 9, 26, 74, 210, 596, 1692, 4804, 13640, 38728, 109960, 312208, 886448, 2516880, 7146144, 20289952, 57608992, 163568448, 464417728, 1318615104, 3743926400, 10630080640, 30181847168, 85694918912, 243312448256, 690833811712, 1961475291648, 5569190816256

Offset: 0

Clark Kimberling, Aug 22 2017

Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).  Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
In the following guide to p-INVERT sequences using s = (1,1,1,1,1,...) = A000012, in some cases t(1,1,1,1,1,...) is a shifted version of the cited sequence:
p(S)                             t(1,1,1,1,1,...)
1 - S                                A000079
1 - S^2                              A000079
1 - S^3                              A024495
1 - S^4                              A000749
1 - S^5                              A139761
1 - S^6                              A290993
1 - S^7                              A290994
1 - S^8                              A290995
1 - S - S^2                          A001906
1 - S - S^3                          A116703
1 - S - S^4                          A290996
1 - S^3 - S^6                        A290997
1 - S^2 - S^3                        A095263
1 - S^3 - S^4                        A290998
1 - 2 S^2                            A052542
1 - 3 S^2                            A002605
1 - 4 S^2                            A015518
1 - 5 S^2                            A163305
1 - 6 S^2                            A290999
1 - 7 S^2                            A291008
1 - 8 S^2                            A291001
(1 - S)^2                            A045623
(1 - S)^3                            A058396
(1 - S)^4                            A062109
(1 - S)^5                            A169792
(1 - S)^6                            A169793
(1 - S^2)^2                          A024007
1 - 2 S - 2 S^2                      A052530
1 - 3 S - 2 S^2                      A060801
(1 - S)(1 - 2 S)                     A053581
(1 - 2 S)(1 - 3 S)                   A291002
(1 - S)(1 - 2 S)(1 - 3 S)(1 - 4 S)   A291003
(1 - 2 S)^2                          A120926
(1 - 3 S)^2                          A291004
1 + S - S^2                          A000045  (Fibonacci numbers starting with -1)
1 - S - S^2 - S^3                    A291000
1 - S - S^2 - S^3 - S^4              A291006
1 - S - S^2 - S^3 - S^4 - S^5        A291007
1 - S^2 - S^4                        A290990
(1 - S)(1 - 3 S)                     A291009
(1 - S)(1 - 2 S)(1 - 3 S)            A291010
(1 - S)^2 (1 - 2 S)                  A291011
(1 - S^2)(1 - 2 S)                   A291012
(1 - S^2)^3                          A291013
(1 - S^3)^2                          A291014
1 - S - S^2 + S^3                    A045891
1 - 2 S - S^2 + S^3                  A291015
1 - 3 S + S^2                        A136775
1 - 4 S + S^2                        A291016
1 - 5 S + S^2                        A291017
1 - 6 S + S^2                        A291018
1 - S - S^2 - S^3 + S^4              A291019
1 - S - S^2 - S^3 - S^4 + S^5        A291020
1 - S - S^2 - S^3 + S^4 + S^5        A291021
1 - S - 2 S^2 + 2 S^3                A175658
1 - 3 S^2 + 2 S^3                    A291023
(1 - 2 S^2)^2                        A291024
(1 - S^3)^3                          A291143
(1 - S - S^2)^2                      A209917

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, -4, 2)

Cf. A000012, A289780.

z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291000 *)

G.f.: (-1 + x - x^2)/(-1 + 4 x - 4 x^2 + 2 x^3).

a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-3) for n >= 4.

A291219 p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - S - S^3.

Original entry on oeis.org

1, 1, 3, 5, 11, 21, 42, 83, 163, 323, 635, 1255, 2473, 4880, 9625, 18985, 37451, 73869, 145715, 287421, 566954, 1118331, 2205947, 4351307, 8583091, 16930447, 33395857, 65874464, 129939569, 256310161, 505580371, 997274197, 1967156763, 3880282533, 7653987242

Offset: 0

Clark Kimberling, Aug 24 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
In the following guide to p-INVERT sequences using s = (1,0,1,0,1,...) = A000035, in some cases t(1,0,1,0,1,...) is a shifted version of the indicated sequence.
p(S)                             t(1,0,1,0,1,...)
1 - S                                A000045 (Fibonacci numbers)
1 - S^2                              A147600
1 - S^3                              A291217
1 - S^5                              A291218
1 - S - S^2                          A289846
1 - S - S^3                          A291219
1 - S - S^4                          A291220
1 - S^3- S^6                         A291221
1 - S^2- S^3                         A291222
1 - S^3- S^4                         A291223
1 - 2S                               A052542
1 - 3S                               A006190
(1 - S)^2                            A239342
(1 - S)^3                            A276129
(1 - S)^4                            A291224
(1 - S)^5                            A291225
(1 - S)^6                            A291226
1 - S - 2 S^2                        A291227
1 - 2 S - 2 S^2                      A291228
1 - 3 S - 2 S^2                      A060801
(1 - S)(1 - 2 S)                     A291229
(1 - S)(1 - 2 S)(1 - 3 S)            A291230
(1 - S)(1 - 2 S)(1 - 3 S)( 1 - 4 S)  A291231
(1 - 2 S)^2                          A291264
(1 - 3 S)^2                          A291232
1 - S - S^2 - S^3                    A291233
1 - S - S^2 - S^3 - S^4              A291234
1 - S - S^2 - S^3 - S^4 - S^5        A291235
(1 - S)(1 - 3 S)                     A291236
(1 - S)(1 - 2S)( 1 - 4S)             A291237
(1 - S)^2 (1 - 2S)                   A291238
(1 - S^2) (1 - 2S)                   A291239
(1 - S^3)^2                          A291240
1 - S - S^2 + S^3                    A291241
1 - 2 S - S^2 + S^3                  A291242
1 - 3 S + S^2                        A291243
1 - 4 S + S^2                        A291244
1 - 5 S + S^2                        A291245
1 - 6 S + S^2                        A291246
1 - S - S^2 - S^3 + S^4              A291247
1 - S - S^2 - S^3 - S^4 + S^5        A291248
1 - S - S^2 - S^3 + S^4 + S^5        A291249
1 - S - 2 S^2 + 2 S^3                A291250
1 - 3 S^2 + 2 S^3                    A291251 (includes negative terms)
(1 - S^3)^3                          A291252
(1 - S - S^2)^2                      A291253
(1 - 2 S - S^2)^2                    A291254
(1 - S - 2 S^2)^2                    A291255

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-3,1,1)

Cf. A000035, A290890, A291000.

I:=[1,1,3,5,11,21]; [n le 6 select I[n] else Self(n-1)+3*Self(n-2)-Self(n-3)-3*Self(n-4)+Self(n-5)+Self(n-6): n in [1..45]]; // Vincenzo Librandi, Aug 25 2017

z = 60; s = x/(1 - x^2); p = 1 - s - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000035 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291219 *)
LinearRecurrence[{1, 3, -1, -3, 1, 1}, {1, 1, 3, 5, 11, 21}, 50] (* Vincenzo Librandi, Aug 25 2017 *)

G.f.: -(1 - x^2 + x^4)/(-1 + x + 3*x^2 - x^3 - 3*x^4 + x^5 + x^6).

a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 3*a(n-4) + a(n-5) + a(n-6) for n >= 7.

A220562 T(n,k)=Number of ways to reciprocally link elements of an nXk array either to themselves or to exactly one horizontal or antidiagonal neighbor.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 5, 14, 13, 1, 8, 47, 64, 34, 1, 13, 149, 430, 292, 89, 1, 21, 481, 2604, 3966, 1332, 233, 1, 34, 1544, 16310, 45966, 36640, 6076, 610, 1, 55, 4965, 101052, 561636, 810778, 338581, 27716, 1597, 1, 89, 15957, 628269, 6743873, 19333688, 14298089

Offset: 1

R. H. Hardin Dec 16 2012

Table starts
.1.....2........3............5...............8..................13
.1.....5.......14...........47.............149.................481
.1....13.......64..........430............2604...............16310
.1....34......292.........3966...........45966..............561636
.1....89.....1332........36640..........810778............19333688
.1...233.....6076.......338581........14298089...........665748170
.1...610....27716......3128843.......252139015.........22926570957
.1..1597...126428.....28913910......4446314533........789539775889
.1..4181...576708....267196106.....78407942556......27190037181789
.1.10946..2630684...2469184016...1382674335890.....936366170726784
.1.28657.12000004..22817958901..24382584703194...32246430390993213
.1.75025.54738652.210862878471.429971411669361.1110497486239608510

			Some solutions for n=3 k=4 0=self 3=ne 4=w 6=e 7=sw (reciprocal directions total 10)
..0..0..0..0....0..6..4..0....0..7..6..4....0..7..6..4....6..4..0..0
..0..7..6..4....6..4..7..0....3..0..0..0....3..0..0..7....6..4..7..7
..3..6..4..0....0..3..0..0....0..0..0..0....0..0..3..0....0..3..3..0

R. H. Hardin, Table of n, a(n) for n = 1..337

Column 2 is A001519(n+1)
Column 3 is A060801
Row 1 is A000045(n+1)

A221828 T(n,k)=Number of nXk arrays of occupancy after each element moves to some horizontal, vertical or antidiagonal neighbor, without 2-loops or left turns.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 0, 16, 14, 0, 0, 73, 202, 64, 0, 0, 333, 2879, 2930, 292, 0, 0, 1519, 40983, 115417, 41618, 1332, 0, 0, 6929, 583419, 4715043, 4755445, 592601, 6076, 0, 0, 31607, 8304016, 192788993, 570066685, 194407111, 8434490, 27716, 0, 0, 144177

Offset: 1

R. H. Hardin Jan 26 2013

Table starts
.0......0..........0............0.............0..............0.............0
.0......3.........16...........73...........333...........1519..........6929
.0.....14........202.........2879.........40983.........583419.......8304016
.0.....64.......2930.......115417.......4715043......192788993....7888216533
.0....292......41618......4755445.....570066685....68573607669.8233561918895
.0...1332.....592601....194407111...68489860450.24121109763712
.0...6076....8434490...7954432829.8224749361083
.0..27716..120053263.325446165881
.0.126428.1708776751
.0.576708
.0

			Some solutions for n=3 k=4
..0..1..1..0....1..2..1..1....1..1..0..2....0..2..0..1....1..1..1..1
..1..2..0..1....1..1..2..1....2..1..1..1....2..1..0..1....1..0..1..2
..2..1..2..1....0..0..1..1....1..1..1..0....1..2..1..1....1..1..2..0

R. H. Hardin, Table of n, a(n) for n = 1..71

Column 2 is A060801(n-1)

Empirical for col 2: a(n) = 5*a(n-1) -2*a(n-2) for n>3

Empirical for row 2: a(k)=5*a(k-1)-2*a(k-2) for k>4

A181308 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with an odd sum (0<=k<=n). A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

Original entry on oeis.org

1, 0, 2, 3, 0, 4, 0, 16, 0, 8, 14, 0, 52, 0, 16, 0, 104, 0, 144, 0, 32, 64, 0, 460, 0, 368, 0, 64, 0, 616, 0, 1624, 0, 896, 0, 128, 292, 0, 3428, 0, 5056, 0, 2112, 0, 256, 0, 3456, 0, 14688, 0, 14528, 0, 4864, 0, 512, 1332, 0, 23132, 0, 53920, 0, 39488, 0, 11008, 0, 1024, 0

Offset: 0

Emeric Deutsch, Oct 13 2010

The sum of entries in row n is A003480(n).
T(n,k) = 0 if n and k have opposite parities.
T(2n,0) = A060801(n).
Sum(k*T(n,k), k=0..n) = A181326(n).
For the statistic "number of column with an even sum" see A181327.

			T(2,2) = 4 because we have (1,0/0,1), (0,1/1,0), (1,1/0,0), and (0,0/1,1) (the 2-compositions are written as (top row/bottom row)).
Triangle starts:
1;
0,  2;
3,  0,  4;
0, 16,  0, 8;
14, 0, 52, 0, 16;

Alois P. Heinz, Table of n, a(n) for n = 0..140, flattened
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

Cf. A003480, A060801, A181326, A181327.

G := (1-z^2)^2/(1-5*z^2+2*z^4-2*t*z): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 11 do P[n] := sort(coeff(Gser, z, n)) end do; for n from 0 to 11 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
       expand(add(add(`if`(i=0 and j=0, 0, b(n-i-j)*
       `if`(irem(i+j,2)=1, x, 1)), i=0..n-j), j=0..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15); # Alois P. Heinz, Mar 16 2014

b[n_] := b[n] = If[n == 0, 1, Expand[Sum[Sum[If[i == 0 && j == 0, 0, b[n-i-j]* If[Mod[i+j, 2] == 1, x, 1]], {i, 0, n-j}], {j, 0, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

G.f.: G(t,z) = (1-z)^2*(1+z)^2/(1-5z^2+2z^4-2tz).
The g.f. of column k is (2z)^k*(1-z^2)^2/(1-5z^2+2z^4)^{k+1} (we have a Riordan array).
The g.f. H(t,s,z), where z marks size and t (s) marks number of columns with an odd (even) sum, is H=(1-z^2)^2/(1-2z^2+z^4-2tz-3sz^2+sz^4).

A292744 a(0) = 1; a(n) = Sum_{k=1..n} prime(k+1)*a(n-k).

Original entry on oeis.org

1, 3, 14, 64, 294, 1346, 6166, 28242, 129362, 592538, 2714096, 12431808, 56943398, 260826950, 1194707382, 5472309246, 25065693008, 114812401444, 525893599720, 2408834540066, 11033569993066, 50538824799712, 231491059896394, 1060335514811206, 4856824295820082, 22246488881086116

Offset: 0

Ilya Gutkovskiy, Sep 22 2017

nonn

Invert transform of the odd primes.

Number of compositions (ordered partitions) of n where there are prime(k+1) sorts of part k.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index entries for sequences related to compositions

Cf. A030017, A060801, A065091.

a[0] = 1; a[n_] := a[n] = Sum[Prime[k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]
nmax = 25; CoefficientList[Series[1/(1 - Sum[Prime[k + 1] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

t=26; Vec(1/(1-sum(k=1, t, prime(k+1)*x^k)) + O(x^t)) \\ Felix Fröhlich, Sep 22 2017

G.f.: 1/(1 - Sum_{k>=1} prime(k+1)*x^k).

cp's OEIS Frontend

A291000 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3.

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A291219 p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - S - S^3.

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A220562 T(n,k)=Number of ways to reciprocally link elements of an nXk array either to themselves or to exactly one horizontal or antidiagonal neighbor.

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A221828 T(n,k)=Number of nXk arrays of occupancy after each element moves to some horizontal, vertical or antidiagonal neighbor, without 2-loops or left turns.

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A181308 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with an odd sum (0<=k<=n). A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

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A292744 a(0) = 1; a(n) = Sum_{k=1..n} prime(k+1)*a(n-k).

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