A073388 -id:A073388 - cp's OEIS frontend

A291385 a(n) = (1/4)*A073388(n+1).

1, 4, 14, 47, 152, 480, 1488, 4548, 13744, 41152, 122272, 360944, 1059584, 3095552, 9005568, 26101824, 75404544, 217191424, 623928832, 1788071680, 5113137152, 14592352256, 41569120256, 118219097088, 335685021696, 951817715712, 2695241605120, 7622609858560

Offset: 0

Clark Kimberling, Sep 04 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,-8,-4)

Cf. A019590, A291382, A073388, A155112.

z = 60; s = x + x^2; p = (1 - 2 s)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A019590 *)
u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A073388 *)
u / 2  (* A291385 *)
LinearRecurrence[{4,0,-8,-4},{1,4,14,47},30] (* Harvey P. Dale, Aug 24 2022 *)

G.f.: -(((1 + x) (-1 + x + x^2))/(-1 + 2 x + 2 x^2)^2).
a(n) = 4*a(n-1) - 8*a(n-3) + 4*a(n-4) for n >= 5.
a(n) = Sum_{k=0..n+1} k * A155112(n+1,k). - Alois P. Heinz, Sep 29 2022

A291382 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2.

Original entry on oeis.org

2, 7, 22, 70, 222, 705, 2238, 7105, 22556, 71608, 227332, 721705, 2291178, 7273743, 23091762, 73308814, 232731578, 738846865, 2345597854, 7446508273, 23640235416, 75050038224, 238259397096, 756395887969, 2401310279090, 7623377054503, 24201736119310

Offset: 0

Clark Kimberling, Sep 04 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 = (1,1,0,0,0,...) = A019590, in some cases t(1,1,0,0,0,...) is a shifted version of the cited sequence:
p(S)                     t(1,1,0,0,0,...)
1 - S                       A000045 (Fibonacci numbers)
1 - S^2                     A094686
1 - S^3                     A115055
1 - S^4                     A291379
1 - S^5                     A281380
1 - S^6                     A281381
1 - 2 S                     A002605
1 - 3 S                     A125145
(1 - S)^2                   A001629
(1 - S)^3                   A001628
(1 - S)^4                   A001629
(1 - S)^5                   A001873
(1 - S)^6                   A001874
1 - S - S^2                 A123392
1 - 2 S - S^2               A291382
1 - S - 2 S^2               A124861
1 - 2 S - S^2               A291383
(1 - 2 S)^2                 A073388
(1 - 3 S)^2                 A291387
(1 - 5 S)^2                 A291389
(1 - 6 S)^2                 A291391
(1 - S)(1 - 2 S)            A291393
(1 - S)(1 - 3 S)            A291394
(1 - 2 S)(1 - 3 S)          A291395
(1 - S)(1 - 2 S)            A291393
(1 - S)(1 - 2 S)(1 - 3 S)   A291396
1 - S - S^3                 A291397
1 - S^2 - S^3               A291398
1 - S - S^2 - S^3           A186812
1 - S - S^2 - S^3 - S^4     A291399
1 - S^2 - S^4               A291400
1 - S - S^4                 A291401
1 - S^3 - S^4               A291402
1 - 2 S^2 - S^4             A291403
1 - S^2 - 2 S^4             A291404
1 - 2 S^2 - 2 S^4           A291405
1 - S^3 - S^6               A291407
(1 - S)(1 - S^2)            A291408
(1 - S^2)(1 - S)^2          A291409
1 - S - S^2 - 2 S^3         A291410
1 - 2 S - S^2 + S^3         A291411
1 - S - 2 S^2 + S^3         A291412
1 - 3 S + S^2 + S^3         A291413
1 - 2 S + S^3               A291414
1 - 3 S + S^2               A291415
1 - 4 S + S^2               A291416
1 - 4 S + 2 S^2             A291417

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 3, 2, 1)

Cf. A019590, A290890, A291000, A291219, A291728, A292479, A292480.

z = 60; s = x + x^2; p = 1 - 2 s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A019590 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291382 *)

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

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

A073387 Convolution triangle of A002605(n) (generalized (2,2)-Fibonacci), n>=0.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 16, 16, 6, 1, 44, 56, 30, 8, 1, 120, 188, 128, 48, 10, 1, 328, 608, 504, 240, 70, 12, 1, 896, 1920, 1872, 1080, 400, 96, 14, 1, 2448, 5952, 6672, 4512, 2020, 616, 126, 16, 1, 6688, 18192, 23040, 17856, 9352, 3444, 896, 160, 18, 1

Offset: 0

Wolfdieter Lang, Aug 02 2002

The g.f. for the row polynomials P(n,x) = Sum_{m=0..n} T(n,m)*x^m is 1/(1-(2+x+2*z)*z). See Shapiro et al. reference and comment under A053121 for such convolution triangles.

T(n, k) is the number of words of length n over {0,1,2,3} having k letters 3 and avoiding runs of odd length for the letters 0,1. - Milan Janjic, Jan 14 2017

			Lower triangular matrix, T(n,k), n >= k >= 0, else 0:
    1;
    2,    1;
    6,    4,    1;
   16,   16,    6,    1;
   44,   56,   30,    8,   1;
  120,  188,  128,   48,  10,   1;
  328,  608,  504,  240,  70,  12,   1;
  896, 1920, 1872, 1080, 400,  96,  14,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Wolfdieter Lang, First 10 rows.

Cf. A002605, A007482 (row sums), A053121, A073403, A073404.
Cf. A001045, A005563, A015518, A159920.
Columns: A002605 (k=0), A073388 (k=1), A073389 (k=2), A073390 (k=3), A073391 (k=4), A073392 (k=5), A073393 (k=6), A073394 (k=7), A073397 (k=8), A073398 (k=9).

A073387:= func< n,k | (&+[2^(n-k-j)*Binomial(n-j,k)*Binomial(n-k-j,j): j in [0..Floor((n-k)/2)]]) >;
[A073387(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 03 2022

T := (n,k) -> `if`(n=0,1,2^(n-k)*binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -2)): seq(seq(simplify(T(n,k)),k=0..n),n=0..10); # Peter Luschny, Apr 25 2016

T[n_, k_]:=T[n,k]=Sum[2^(n-k-j)*Binomial[n-j,k]*Binomial[n-k-j,j], {j,0,(n-k)/2}];
Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, Jun 04 2019 *)

def A073387(n,k): return sum(2^(n-k-j)*binomial(n-j,k)*binomial(n-k-j,j) for j in range(((n-k+2)//2)))
flatten([[A073387(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Oct 03 2022

T(n, k) = 2*(p(k-1, n-k)*(n-k+1)*T(n-k+1) + q(k-1, n-k)*(n-k+2)*T(n-k))/(k!*12^k), n >= k >= 1, with T(n) = T(n, k=0) = A002605(n), else 0; p(m, n) = Sum_{j=0..m} A(m, j)*n^(m-j) and q(m, n) = Sum_{j=0..m} B(m, j)*n^(m-j) with the number triangles A(k, m) = A073403(k, m) and B(k, m) = A073404(k, m).
T(n, k) = Sum_{j=0..floor((n-k)/2)} 2^(n-k-j)*binomial(n-j, k)*binomial(n-k-j, j) if n > k, else 0.
T(n, k) = ((n-k+1)*T(n, k-1) + 2*(n+k)*T(n-1, k-1))/(6*k), n >= k >= 1, T(n, 0) = A002605(n+1), else 0.
Sum_{k=0..n} T(n, k) = A007482(n).
G.f. for column m (without leading zeros): 1/(1-2*x*(1+x))^(m+1), m>=0.
T(n,k) = 2^(n-k)*binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -2) for n>=1. - Peter Luschny, Apr 25 2016
From G. C. Greubel, Oct 03 2022: (Start)
T(n, n-1) = A005843(n), n >= 1.
T(n, n-2) = 2*A005563(n-1), n >= 2.
T(n, n-3) = 4*A159920(n-1), n >= 2.
Sum_{k=0..n} (-1)^k*T(n, k) = A001045(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A015518(n+1). (End)

A073389 Second convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

Original entry on oeis.org

1, 6, 30, 128, 504, 1872, 6672, 23040, 77616, 256288, 832416, 2666496, 8441600, 26454528, 82174464, 253280256, 775316736, 2358812160, 7137023488, 21487386624, 64401106944, 192229535744, 571630694400, 1693996941312, 5004131659776, 14738997288960, 43293528760320

Offset: 0

Wolfdieter Lang, Aug 02 2002

Muniru A Asiru, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-6,-16,12,24,8).

Third (m=2) column of triangle A073387, A073388.

Cf. A002605.

List([0..25], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+2,2)*Binomial(n-k,k)*(1/2)^k)); # Muniru A Asiru, Jun 12 2018

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^3 )); // G. C. Greubel, Oct 03 2022

CoefficientList[Series[1/(1-2x(1+x))^3,{x,0,25}],x]  (* Harvey P. Dale, Mar 14 2011 *)

taylor( 1/(1-2*x-2*x^2)^3, x, 0, 25).list() # Zerinvary Lajos, Jun 03 2009; modified by G. C. Greubel, Oct 03 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k) with b(k) = A002605(k) and c(k) = A073388(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, 2)*binomial(n-k, k)*2^(n-k).
a(n) = (n+3)*((n+1)*U(n+1) + (n+2)*U(n))/12, with U(n) = A002605(n), n >= 0.
G.f.: 1/(1-2*x*(1+x))^3.

A099432 Convolution of A030195(n) (generalized (3,3)-Fibonacci) with itself.

Original entry on oeis.org

1, 6, 33, 162, 756, 3402, 14931, 64314, 273051, 1145988, 4764744, 19656756, 80561061, 328316814, 1331513397, 5377120038, 21633427836, 86747114430, 346810621815, 1382826606210, 5500378861551, 21830478128136, 86469557676048

Offset: 0

Paul Barry, Oct 15 2004

Index entries for linear recurrences with constant coefficients, signature (6,-3,-18,-9).

Cf. A073388.

LinearRecurrence[{6,-3,-18,-9},{1,6,33,162},30] (* Harvey P. Dale, May 20 2011 *)

G.f.: 1/(1 - 3*x - 3*x^2)^2.
a(n) = 6*a(n-1) - 3*a(n-2) - 18*a(n-3) - 9*a(n-4). [corrected by Harvey P. Dale, May 20 2011]
a(n) = Sum_{k=0..floor((n+2)/2)} k*binomial(n-k+2, k)*3^(n-k+1).
a(n) = (sqrt(7)*n + 2*sqrt(7) - sqrt(3))*(5*sqrt(7)/98 + sqrt(3)/14)*(3*sqrt(21)/2 + 15/2)^(n/2) + (15/2 - 3*sqrt(21)/2)^(n/2)*(sqrt(7)*n + 2*sqrt(7) + sqrt(3))*(5*sqrt(7)/98 - sqrt(3)/14)*(-1)^n.

A118357 Triangle read by rows: T(n,k) is the number of ternary sequences of length n containing k subsequences 00 (n>=0, 0<=k<=max(0,n-1)).

Original entry on oeis.org

1, 3, 8, 1, 22, 4, 1, 60, 16, 4, 1, 164, 56, 18, 4, 1, 448, 188, 68, 20, 4, 1, 1224, 608, 248, 80, 22, 4, 1, 3344, 1920, 864, 312, 92, 24, 4, 1, 9136, 5952, 2928, 1152, 380, 104, 26, 4, 1, 24960, 18192, 9696, 4128, 1472, 452, 116, 28, 4, 1, 68192, 54976, 31536, 14400

Offset: 0

Emeric Deutsch, May 24 2006

Sum of entries in row n is 3^n (A000244). T(n,0) = A028859(n). T(n,1) = A073388(n-2). Sum(k*T(n,k),k=0..n-1) = (n-1)*3^(n-2) (A027471).

			T(4,2) = 4 because we have 0001, 0002, 1000 and 2000.
Triangle starts:
1;
3;
8,1;
22,4,1;
60,16,4,1;

Alois P. Heinz, n = 0..141, flattened

Cf. A000244, A028859, A073388, A027471.

G:=(1+(1-t)*z)/(1-(2+t)*z-2*(1-t)*z^2): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: 1; for n from 1 to 12 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form

nn=15;a=1/(1-2x);b=x/(1-y x)+1;f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[a b/(1-2x^2/((1-y x)(1-2x))),{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Nov 19 2012 *)

G.f.: G-1, where G = G(t,z) = [1+(1-t)z]/[1-(2+t)z-2(1-t)z^2]. G.f. of column k is z^(k+1)*(1-2z)^(k-1)/(1-2z-2z^2)^(k+1) (k>=1).

A109634 Number of 1's that appear among all ternary strings of length n that contain no consecutive 1's.

Original entry on oeis.org

0, 1, 4, 16, 56, 188, 608, 1920, 5952, 18192, 54976, 164608, 489088, 1443776, 4238336, 12382208, 36022272, 104407296, 301618176, 868765696, 2495715328, 7152286720, 20452548608, 58369409024, 166276481024, 472876388352

Offset: 0

Ralph P. Grimaldi (ralph.grimaldi(AT)rose-hulman.edu), Aug 03 2005

nonn

Ralph P. Grimaldi, Ternary Strings with No Consecutive 1's, Ars Combin. 89 (2008), 321-343.

f[n_] := Simplify[(2Sqrt[3] (-(1 - Sqrt[3])^n + (1 + Sqrt[3])^n) + 3n*((1 - Sqrt[3])^(1 + n) + (1 + Sqrt[3])^(1 + n)))/36]; Table[ f[n], {n, 0, 25}] (* Robert G. Wilson v *)

(sqrt(3)/18)*((1+sqrt(3))^n - (1-sqrt(3))^n)+(n/12)*((1+sqrt(3))^(n+1)+(1-sqrt(3))^(n+1))

Conjecture: a(n)=A073388(n-1). [From R. J. Mathar, Aug 18 2008]

More terms from Robert G. Wilson v, Aug 05 2005

cp's OEIS Frontend

A291385 a(n) = (1/4)*A073388(n+1).

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A291382 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2.

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A073387 Convolution triangle of A002605(n) (generalized (2,2)-Fibonacci), n>=0.

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A073389 Second convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

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A099432 Convolution of A030195(n) (generalized (3,3)-Fibonacci) with itself.

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A118357 Triangle read by rows: T(n,k) is the number of ternary sequences of length n containing k subsequences 00 (n>=0, 0<=k<=max(0,n-1)).

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A109634 Number of 1's that appear among all ternary strings of length n that contain no consecutive 1's.

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