A099087 -id:A099087 - cp's OEIS frontend

A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx + kx^2).

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, -1, 0, 1, 4, 6, 0, -1, 0, 1, 5, 12, 9, -4, 0, 0, 1, 6, 20, 32, 9, -8, 1, 0, 1, 7, 30, 75, 80, 0, -8, 1, 0, 1, 8, 42, 144, 275, 192, -27, 0, 0, 0, 1, 9, 56, 245, 684, 1000, 448, -81, 16, -1, 0, 1, 10, 72, 384, 1421, 3240, 3625, 1024, -162, 32, -1, 0

Offset: 0

Seiichi Manyama, Feb 28 2021

			Square array begins:
  1,  1,  1, 1,   1,    1, ...
  0,  1,  2, 3,   4,    5, ...
  0,  0,  2, 6,  12,   20, ...
  0, -1,  0, 9,  32,   75, ...
  0, -1, -4, 9,  80,  275, ...
  0,  0, -8, 0, 192, 1000, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns 0..10 give A000007, A010892, A099087, A057083, A001787(n+1), A030191, A030192, A030240, A057084, A057085(n+1), A057086.
Rows 0..1 give A000012, A001477.
Main diagonal gives (-1) * A109519(n+1).
Cf. A342120, A342133, A342134.

T:= (n, k)-> (<<0|1>, <-k|k>>^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Mar 01 2021

T[n_, k_] := (-1)^n * Sum[If[k == j == 0, 1, (-k)^j] * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

T(n, k) = (-1)^n*sum(j=0, n\2, (-k)^(n-j)*binomial(n-j, j));

T(n, k) = (-1)^n*sum(j=0, n, (-k)^j*binomial(j, n-j));

T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)/2));

T(0,k) = 1, T(1,k) = k and T(n,k) = k*(T(n-1,k) - T(n-2,k)) for n > 1.
T(n,k) = (-1)^n * Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n-j,j) = (-1)^n * Sum_{j=0..n} (-k)^j * binomial(j,n-j).
T(n,k) = sqrt(k)^n * S(n, sqrt(k)) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind.

A100215 Expansion of (4 - 7x + 2x^2)/((1-2x)(1 - 2x + 2x^2)).

Original entry on oeis.org

4, 9, 14, 18, 24, 44, 104, 248, 544, 1104, 2144, 4128, 8064, 16064, 32384, 65408, 131584, 263424, 525824, 1049088, 2095104, 4189184, 8382464, 16775168, 33562624, 67129344, 134242304, 268443648, 536838144

Offset: 0

Creighton Dement, Nov 11 2004

a(n) = (-1)^n*A009116(n+3) + A100216(n) + A038503(n+1), where A009116, A100216 and A038503 can be generated by the operators jes, les and tes of the Floretion algebra, which is a product factor space Q x Q /{(1,1), (-1,-1)}.

Binomial transform of the sequence 4,5,0,-1 (repeated with period length 4). - R. J. Mathar, Apr 18 2009

			a(2) = 14 because (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e)^3 = 1'j + 1'k + 1j' + 1k' + 3'ii' + 2'jj' + 2'kk' + 1'jk' + 1'kj' + 1e and the sum of these coefficients is 1 + 1 + 1 + 1 + 3 + 2 + 2 + 1 + 1 + 1 = 14 (see comment).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Creighton Dement, Floretion Online Multiplier.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4).

Cf. A009116, A038503, A099087, A100213, A100214, A100216.

I:=[4, 9, 14]; [n le 3 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Jun 25 2012

LinearRecurrence[{4,-6,4},{4,9,14},40] (* Vincenzo Librandi, Jun 25 2012 *)

A099087=BinaryRecurrenceSequence(2,-2,1,2)
def A100215(n): return 2^(n+1) + 2*A099087(n) + A099087(n-1)
[A100215(n) for n in range(41)] # G. C. Greubel, Mar 29 2024

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3).
a(n) = (-1)^n*A009116(n+3) + A100216(n) + A038503(n+1).
a(n) = vesseq(.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e), where ves sums over all floretion basis vector coefficients.
a(n) = 2^(n+1) + 2*A099087(n) + A099087(n-1). - R. J. Mathar, Apr 18 2009

Definition replaced with the more precise g.f. by R. J. Mathar, Nov 17 2010

A139687 Basis of degenerate cases of sequences identical to its p-th differences. Complement to A140344 which is based on natural Catalan's triangle. Triangle without first term (probable 1) on line.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 3, 6, 9, 9, 1, 4, 10, 19, 28, 28, 1, 4, 10, 20, 34, 48, 48, 1, 5, 15, 35, 69, 117, 165, 165, 1, 5, 15, 35, 70, 125, 200, 275, 275, 1, 6, 21, 56, 126, 251, 451, 726, 1001, 1001, 1, 6, 21, 56, 126, 252, 461, 780, 1209, 1638, 1638

Offset: 0

Paul Curtz, Jun 13 2008

Triangle from A140344:
(1;)
0, 1, 1;
0, 0, 1, 2, 2;
0, 0, 0, 1, 3, 5, 5; see A138112,
0, 0, 0, 0, 1, 4, 9, 14, 14; see A140343,
begins (without 0's) like a(n).

Cf. A135356, A140344.

First four rows of triangle from second row: 1, 1; 1, 2, 2; see A099087, 1, 3, 5, 5; 1, 3, 6, 9, 9; see A057083 which can be preceded with 3 leading 0's, are, as said, from natural Catalan's triangle A009766. Origin of a(n) explained later.

A133212 a(n) = 4a(n-1) - 6a(n-2) + 4*a(n-3), n > 3; a(0) = 1, a(1) = 4, a(2) = 12, a(3) = 32.

Original entry on oeis.org

1, 4, 12, 32, 72, 144, 272, 512, 992, 1984, 4032, 8192, 16512, 33024, 65792, 131072, 261632, 523264, 1047552, 2097152, 4196352, 8392704, 16781312, 33554432, 67100672, 134201344, 268419072, 536870912, 1073774592, 2147549184

Offset: 0

Paul Curtz, Oct 11 2007

Conjecture: a(n) = 2*A038503(n+3) if n > 0. - R. J. Mathar, Oct 23 2007

Index entries for linear recurrences with constant coefficients, signature (4,-6,4).

Cf. A009116, A038503, A099087.

A133212 := proc(n) option remember ; if n <= 3 then op(n+1,[1,4,12,32]) ; else 4*A133212(n-1)-6*A133212(n-2)+4*A133212(n-3) ; fi ; end: seq(A133212(n),n=0..50) ; # R. J. Mathar, Oct 23 2007

Join[{1},LinearRecurrence[{4, -6, 4},{4, 12, 32},29]] (* Ray Chandler, Sep 23 2015 *)

Sequence is identical to its fourth differences.
From R. J. Mathar, Nov 18 2007: (Start)
G.f.: -(1 + 2*x^2 + 4*x^3)/((2*x - 1)*(2*x^2 - 2*x + 1)). - [Corrected by Georg Fischer, May 12 2019]
a(n) = -2*(-1)^n*A009116(n)+3*2^n. (End)
E.g.f.: exp(x)*(3*cosh(x) - 2*(cos(x) + sin(x)) + 5*sinh(x)). - Stefano Spezia, Jan 03 2023

More terms from R. J. Mathar, Oct 23 2007

A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 0, 2, 6, 12, -1, 0, 9, 32, 75, -1, -4, 9, 80, 275, 684, 0, -8, 0, 192, 1000, 3240, 8232, 1, -8, -27, 448, 3625, 15336, 47677, 122368, 1, 0, -81, 1024, 13125, 72576, 276115, 835584, 2158569, 0, 16, -162, 2304, 47500, 343440, 1599066, 5705728, 16953624, 44010000

Offset: 0

Roger L. Bagula, Nov 15 2009

			Triangle begins as:
   0;
   1,  1;
   1,  2,   3;
   0,  2,   6,   12;
  -1,  0,   9,   32,    75;
  -1, -4,   9,   80,   275,   684;
   0, -8,   0,  192,  1000,  3240,   8232;
   1, -8, -27,  448,  3625, 15336,  47677, 122368;
   1,  0, -81, 1024, 13125, 72576, 276115, 835584, 2158569;

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

Cf. A009545, A030191, A030192, A030240, A057083, A057084, A057085, A057086, A099087, A128834, A190871, A190873.

A167925:= func< n,k | Round((Sqrt(k+1))^(n-1)*Evaluate(ChebyshevSecond(n), Sqrt(k+1)/2)) >;
[A167925(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 11 2023

(* First program *)
m[k_]= {{k,1}, {-1,1}};
v[0, k_]:= {0,1};
v[n_, k_]:= v[n, k]= m[k].v[n-1,k];
T[n_, k_]:= v[n, k][[1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
A167925[n_, k_]:= (Sqrt[k+1])^(n-1)*ChebyshevU[n-1, Sqrt[k+1]/2];
Table[A167925[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 11 2023 *)

def A167925(n,k): return (sqrt(k+1))^(n-1)*chebyshev_U(n-1, sqrt(k+1)/2)
flatten([[A167925(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 11 2023

T(n, k) = (-1)^(n+1) * [x^(n-1)]( 1/(1 + (k+1)*x + (k+1)*x^2) ). - Francesco Daddi, Aug 04 2011 (modified by G. C. Greubel, Sep 11 2023)
From G. C. Greubel, Sep 11 2023: (Start)
T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2).
T(n, 0) = A128834(n).
T(n, 1) = A009545(n) = A099087(n-1).
T(n, 2) = A057083(n-1).
T(n, 3) = A001787(n).
T(n, 4) = A030191(n-1).
T(n, 5) = A030192(n-1).
T(n, 6) = A030240(n-1).
T(n, 7) = A057084(n-1).
T(n, 8) = A057085(n).
T(n, 9) = A057086(n-1).
T(n, 10) = A190871(n).
T(n, 11) = A190873(n). (End)

Edited by G. C. Greubel, Sep 11 2023

A172250 Triangle, read by rows, given by [0,1,-1,0,0,0,0,0,0,0,...] DELTA [1,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 0, 2, -1, 0, 0, 1, 1, -1, 0, 0, 0, 3, -2, 0, 0, 0, 0, 1, 3, -4, 1, 0, 0, 0, 0, 4, -2, -2, 1, 0, 0, 0, 0, 1, 6, -9, 3, 0, 0, 0, 0, 0, 0, 5, 0, -9, 6, -1, 0, 0, 0, 0, 0, 1, 10, -15, 3, 3, -1, 0, 0, 0, 0, 0, 0, 6, 5, -24, 18, -4, 0, 0, 0, 0, 0, 0, 0, 1, 15, -20, -6, 18, -8, 1

Offset: 0

Philippe Deléham, Jan 29 2010

			Triangle begins:
  1;
  0,  1;
  0,  1,  0;
  0,  0,  2, -1;
  0,  0,  1,  1, -1;
  0,  0,  0,  3, -2,  0;
  0,  0,  0,  1,  3, -4,  1;
  0,  0,  0,  0,  4, -2, -2,  1; ...

Cf. A101950.

T(n,k) = T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0)=1, T(n,k) = 0 if k > n or if k < 0.
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A088139(n+1), A001607(n+1), A000007(n), A000012(n), A099087(n), A190960(n+1) for x = -2, -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Feb 15 2012
G.f.: 1/(1-y*x+y*(y-1)*x^2). - Philippe Deléham, Feb 15 2012

A106664 Expansion of g.f.: (1-3x+x^2)/((1-x)(1+x)(1-2x+2*x^2)).

Original entry on oeis.org

-1, 1, 2, 5, 4, 1, -8, -15, -16, 1, 32, 65, 64, 1, -128, -255, -256, 1, 512, 1025, 1024, 1, -2048, -4095, -4096, 1, 8192, 16385, 16384, 1, -32768, -65535, -65536, 1, 131072, 262145, 262144, 1, -524288, -1048575, -1048576, 1, 2097152, 4194305, 4194304, 1, -8388608, -16777215, -16777216, 1, 33554432

Offset: 0

Creighton Dement, May 13 2005

Superseeker finds that a(n+2) - a(n) = A090131(n+1) (or with different signs, see A078069).

Floretion Algebra Multiplication Program, FAMP Code: 2ibaseiseq[ + .5'i + .5i' - .5'ii' + .5'jj' + .5'kk' + .5e]

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,-2,2).

Cf. A000035, A010673, A078069, A090131, A099087, A146559.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!(  (1-3*x+x^2)/((1-x^2)*(1-2*x+2*x^2)) )); // G. C. Greubel, Sep 08 2021

CoefficientList[Series[(1-3x+x^2)/((1-x)(1+x)(1-2x+2x^2)),{x,0,60}],x] (* Harvey P. Dale, Mar 20 2013 *)

def A106664_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( sinh(x) -exp(x)*(cos(x)-sin(x)) ).egf_to_ogf().list()
A106664_list(50) # G. C. Greubel, Sep 08 2021

a(n) = (1/2)*(A010673(n) - A099087(n+2)).
a(n) = (1/2)*(1 - (-1)^n - (1-i)^(n+1) - (1+i)^(n+1)), with i=sqrt(-1). - Ralf Stephan, Nov 16 2010
From G. C. Greubel, Sep 08 2021: (Start)
a(n) = (1-(-1)^n)/2 - 2^((n+1)/2)*cos((n+1)*Pi/4).
a(n) = A000035(n) - A146559(n).
E.g.f.: sinh(x) - exp(x)*(cos(x) - sin(x)). (End)

A133209 a(n) = 4a(n-1) - 6a(n-2) + 4a(n-3), n > 3; a(0) = 3, a(1) = 2, a(2) = a(3) = 0.

Original entry on oeis.org

3, 2, 0, 0, 8, 32, 80, 160, 288, 512, 960, 1920, 3968, 8192, 16640, 33280, 66048, 131072, 261120, 522240, 1046528, 2097152, 4198400, 8396800, 16785408, 33554432, 67092480, 134184960, 268402688, 536870912, 1073807360, 2147614720

Offset: 0

Paul Curtz, Oct 11 2007

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4).

Cf. A131470, A009116, A099087.

a[0]:=3: a[1]:=2: a[2]:=0: a[3]:=0; for n from 4 to 27 do a[n]:=4*a[n-1]-6*a[n-2]+4*a[n-3] end do: seq(a[n],n=0..27); # Emeric Deutsch, Oct 14 2007

a = {3, 2, 0, 0}; Do[AppendTo[a, 4*a[[ -1]] - 6*a[[ -2]] + 4*a[[ -3]]], {30}]; a (* Stefan Steinerberger, Oct 14 2007 *)
LinearRecurrence[{4, -6, 4},{3, 2, 0},32] (* Ray Chandler, Sep 23 2015 *)

Sequence is identical to its fourth differences.
a(n) = 2^n + 2^[(n+3)/2]*cos((n+1)Pi/4); a(n)=2^n + (1+i)^(n+1) + (1-i)^(n+1), where i=sqrt(-1). - Emeric Deutsch, Oct 14 2007
G.f.: -(3-10*x+10*x^2)/(2*x-1)/(2*x^2-2*x+1). - R. J. Mathar, Nov 14 2007

More terms from Stefan Steinerberger and Emeric Deutsch, Oct 14 2007

A202551 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, -1, 0, -1, 1, -1, 1, 1, -1, -1, 3, -2, -1, 1, 0, 2, -5, 3, 1, -1, 1, -2, -2, 7, -4, -1, 1, 1, -5, 7, 1, -9, 5, 1, -1, 0, -3, 12, -15, 1, 11, -6, -1, 1, -1, 3, 3, -21, 26, -4, -13, 7, 1, -1

Offset: 0

Philippe Deléham, Dec 21 2011

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

			Triangle begins :
1
1, -1
0, -1, 1
-1, 1, 1, -1
-1, 3, -2, -1, 1
0, 2, -5, 3, 1, -1

Cf. A129267, A199324

T(n,k) = T(n-1,k) + T(n-2,k-1) - T(n-1,k-1) - T(n-2,k).
G.f.: 1/(1+(y-1)*x+(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A190873(n+1), A190871(n+1), A057086(n), A057085(n+1), A057084(n), A030240(n), A030192(n), A030191(n), A001787(n+1), A057083(n), A099087(n), A010892(n), A000007(n), (-1)^n*A000045(n+1) for x = -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2 respectively.

A202603 Triangle T(n,k), read by rows, given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -3, -1, 1, 1, -3, -5, -2, 0, 1, 1, -4, -7, -2, 2, 1, 1, 1, -5, -9, -1, 7, 5, 1, 1, 1, -6, -11, 1, 15, 12, 3, 0, 1, 1, -7, -13, 4, 26, 21, 3, -3, -1, 1, 1, -8, -15, 8, 40, 31, -3, -15, -7, -1

Offset: 0

Philippe Deléham, Dec 21 2011

Mirror image of triangle in A129267.

			Triangle begins :
1
1, 1
1, 1, 0
1, 1, -1, -1
1, 1, -2, -3, -1
1, 1, -3, -5, -2, 0
1, 1, -4, -7, -2, 2, 1
1, 1, -5, -9, -1, 7, 5, 1

Cf. A129267, A010892, A005449

T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) - T(n-2,l-2) with T(0,0)= T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if n

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000012(n), A099087(n), A190960(n+1) for x = 0, 1, 2 respectively.

G.f.: 1/(1-(1+y)*x+(y+y^2)*x^2).

Previous Showing 11-20 of 20 results.

cp's OEIS Frontend

A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - k*x + k*x^2).

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A100215 Expansion of (4 - 7*x + 2*x^2)/((1-2*x)*(1 - 2*x + 2*x^2)).

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A139687 Basis of degenerate cases of sequences identical to its p-th differences. Complement to A140344 which is based on natural Catalan's triangle. Triangle without first term (probable 1) on line.

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A133212 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3), n > 3; a(0) = 1, a(1) = 4, a(2) = 12, a(3) = 32.

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A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.

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A172250 Triangle, read by rows, given by [0,1,-1,0,0,0,0,0,0,0,...] DELTA [1,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A106664 Expansion of g.f.: (1-3*x+x^2)/((1-x)*(1+x)*(1-2*x+2*x^2)).

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A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx + kx^2).

A100215 Expansion of (4 - 7x + 2x^2)/((1-2x)(1 - 2x + 2x^2)).

A133212 a(n) = 4a(n-1) - 6a(n-2) + 4*a(n-3), n > 3; a(0) = 1, a(1) = 4, a(2) = 12, a(3) = 32.

A106664 Expansion of g.f.: (1-3x+x^2)/((1-x)(1+x)(1-2x+2*x^2)).