A112555 -id:A112555 - cp's OEIS frontend

A166035 a(n) = (3^n+6*(-4)^n)/7.

1, -3, 15, -51, 231, -843, 3615, -13731, 57111, -221883, 907215, -3569811, 14456391, -57294123, 230770815, -918300291, 3687550071, -14707153563, 58957754415, -235443597171, 942936650151, -3768259816203, 15083499618015

Offset: 0

Philippe Deléham, Oct 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,12).

Cf. A077925, A091004.

LinearRecurrence[{-1, 12}, {1, -3}, 100] (* G. C. Greubel, Apr 24 2016 *)

a(n)= (3^n+6*(-4)^n)/7;
for(n=0,33,print1(a(n),", "));

a(n) = -a(n-1) + 12*a(n-2), a(0) = 1, a(1) = -3, for n>1.
a(n) = Sum_{0<=k<=n} A112555(n,k)*(-4)^k.
G.f.: (1-2x)/(1+x-12x^2).
E.g.f.: (1/7)*(exp(3*x) + 6*exp(-4*x)). - G. C. Greubel, Apr 24 2016

a(5) corrected by Tilman Neumann, Dec 31 2010

A091004 Expansion of x(1-x)/((1-2x)(1+3x)).

Original entry on oeis.org

0, 1, -2, 8, -20, 68, -188, 596, -1724, 5300, -15644, 47444, -141308, 425972, -1273820, 3829652, -11472572, 34450484, -103285916, 309988820, -929704316, 2789637236, -8367863132, 25105686548, -75312865340, 225946984628, -677824176668, 2033506084436

Offset: 0

Paul Barry, Dec 13 2003

Inverse binomial transform of A091001.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,6).

Cf. A091001, A091003, A091005, A112555.

Concatenation([0], List([1..30], n -> (3*2^n - 8*(-3)^n)/30)); # G. C. Greubel, Feb 01 2019

[0] cat [(3*2^n - 8*(-3)^n)/30: n in [1..30]]; // G. C. Greubel, Feb 01 2019

CoefficientList[Series[x(1-x)/((1-2x)(1+3x)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 17 2017 *)
Join[{0}, LinearRecurrence[{-1, 6}, {1, -2}, 30]] (* G. C. Greubel, Feb 01 2019 *)

vector(30, n, n--; (3*2^n - 8*(-3)^n + 5*0^n)/30) \\ G. C. Greubel, Feb 01 2019

[0] + [(3*2^n - 8*(-3)^n)/30 for n in (1..30)] # G. C. Greubel, Feb 01 2019

G.f.: x*(1-x)/((1-2*x)*(1+3*x)).
a(n) = (3*2^n - 8*(-3)^n + 5*0^n)/30.
2^n = A091003(n) + 3*a(n) + 6*A091005(n).
a(n+1) = Sum_{k=0..n} A112555(n,k)*(-3)^k. - Philippe Deléham, Sep 11 2009
E.g.f.: (3*exp(2*x) - 8*exp(-3*x) + 5)/30. - G. C. Greubel, Feb 01 2019

A166036 a(n) = (4^n+8*(-5)^n)/9.

Original entry on oeis.org

1, -4, 24, -104, 584, -2664, 14344, -67624, 354504, -1706984, 8797064, -42936744, 218878024, -1077612904, 5455173384, -27007431464, 136110899144, -676259528424, 3398477511304, -16923668079784

Offset: 0

Philippe Deléham, Oct 05 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,20).

Cf. A077925, A091004, A166035.

Table[(4^n+8(-5)^n)/9,{n,0,30}] (* or *) LinearRecurrence[{-1,20},{1,-4},30] (* Harvey P. Dale, Mar 05 2016 *)

vector(100, n, n--; (4^n+8*(-5)^n)/9) \\ Altug Alkan, Apr 24 2016

a(n) = -a(n-1) + 20*a(n-2), a(0) = 1, a(1) = -4, for n>1.
a(n) = Sum_{k=0..n} A112555(n,k)*(-5)^k.
G.f.: (1-3x)/(1+x-20x^2).
E.g.f.: (1/9)*(exp(4*x) + 8*exp(-5*x)). - G. C. Greubel, Apr 24 2016

A166149 a(n) = (5^n + 10*(-6)^n)/11.

Original entry on oeis.org

1, -5, 35, -185, 1235, -6785, 43835, -247385, 1562435, -8983985, 55857035, -325376585, 2001087635, -11762385185, 71795014235, -424666569785, 2578516996835, -15318514090385, 92674023995435, -552229446706985

Offset: 0

Philippe Deléham, Oct 08 2009

From Klaus Brockhaus, Oct 14 2009: (Start)
Fourth binomial transform of A014992.
Sixth binomial transform is A001020 preceded by 1.
Lim_{n -> infinity} a(n)/a(n-1) = -6. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (-1,30).

Cf. A166035, A166036.

Cf. A014992 (q-integers for q=-10), A001020 (powers of 11).

[(5^n+10*(-6)^n)/11: n in [0..30]]; // Vincenzo Librandi, May 02 2011

CoefficientList[Series[(1-4x)/(1+x-30x^2), {x,0,40}], x]  (* Harvey P. Dale, Mar 11 2011 *)
LinearRecurrence[{-1,30},{1,-5},20] (* Harvey P. Dale, Jan 20 2022 *)

a(n)=(5^n+10*(-6)^n)/11 \\ Charles R Greathouse IV, May 02 2016

a(n) = 30*a(n-2)-a(n-1), a(0)= 1, a(1)= -5.
G.f.: (1-4x)/(1+x-30*x^2).
a(n) = Sum_{k=0..n} A112555(n,k)*(-6)^k.
E.g.f.: (1/11)*(exp(5*x) + 10*exp(-6*x)). - G. C. Greubel, May 01 2016

A080242 Table of coefficients of polynomials P(n,x) defined by the relation P(n,x) = (1+x)*P(n-1,x) + (-x)^(n+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 1, 1, 4, 7, 6, 3, 1, 5, 11, 13, 9, 3, 1, 1, 6, 16, 24, 22, 12, 4, 1, 7, 22, 40, 46, 34, 16, 4, 1, 1, 8, 29, 62, 86, 80, 50, 20, 5, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 1, 10, 46, 128, 239, 314, 296, 200, 95

Offset: 0

Paul Barry, Feb 12 2003

Values generate solutions to the recurrence a(n) = a(n-1) + k(k+1)* a(n-2), a(0)=1, a(1) = k(k+1)+1. Values and sequences associated with this table are included in A072024.

			Rows are {1}, {1,1,1}, {1,2,2}, {1,3,4,2,1}, {1,4,7,6,3}, ... This is the same as table A035317 with an extra 1 at the end of every second row.
Triangle begins
  1;
  1,  1,  1;
  1,  2,  2;
  1,  3,  4,  2,  1;
  1,  4,  7,  6,  3;
  1,  5, 11, 13,  9,  3,  1;
  1,  6, 16, 24, 22, 12,  4;
  1,  7, 22, 40, 46, 34, 16,  4,  1;
  1,  8, 29, 62, 86, 80, 50, 20,  5;

G. C. Greubel, Rows n = 0..100 of coefficients, flattened

Similar to the triangles A059259, A035317, A108561, A112555. Cf. A059260.

Cf. A001045 (row sums).

Table[CoefficientList[Series[((1+x)^(n+2) -(-1)^n*x^(n+2))/(1+2*x), {x, 0, n+2}], x], {n, 0, 10}]//Flatten (* G. C. Greubel, Feb 18 2019 *)

Rows are generated by P(n,x) = ((x+1)^(n+2) - (-x)^(n+2))/(2*x+1).
The polynomials P(n,-x), n > 0, satisfy a Riemann hypothesis: their zeros lie on the vertical line Re x = 1/2 in the complex plane.
O.g.f.: (1+x*t+x^2*t)/((1+x*t)(1-t-x*t)) = 1 + (1+x+x^2)*t + (1+2x+2x^2)*t^2 + ... . - Peter Bala, Oct 24 2007
T(n,k) = if(k<=2*floor((n+1)/2), Sum_{j=0..floor((n+1)/2)} binomial(n-2j,k-2j), 0). - Paul Barry, Apr 08 2011 (This formula produces the odd numbered rows correctly, but not the even. - G. C. Greubel, Feb 22 2019)

A166152 a(n) = (6^n+12*(-7)^n)/13.

Original entry on oeis.org

1, -6, 48, -300, 2316, -14916, 112188, -738660, 5450556, -36474276, 265397628, -1797317220, 12944017596, -88431340836, 632080079868, -4346196394980, 30893559749436, -213433808338596, 1510963317814908

Offset: 0

Philippe Deléham, Oct 08 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1, 42).

Cf. A166035, A166036, A166149.

LinearRecurrence[{-1,42}, {1,-6}, 50] (* G. C. Greubel, May 01 2016 *)
Table[(6^n+12*(-7)^n)/13,{n,0,30}] (* Harvey P. Dale, Jun 01 2019 *)

a(n)=(6^n+12*(-7)^n)/13 \\ Charles R Greathouse IV, May 02 2016

a(n) = 42*a(n-2)-a(n-1), a(0)= 1, a(1)= -6, for n>1.
G.f.: (1-5x)/(1+x-42*x^2).
a(n)= Sum_{k, 0<=k<=n} A112555(n,k)*(-7)^k.
E.g.f.: (1/13)*(exp(6*x) + 12*exp(-7*x)). - G. C. Greubel, May 01 2016

A279006 Alternating Jacobsthal triangle read by rows (second version).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, 1, 1, 1, -2, 2, 0, 1, 1, -3, 4, -2, 1, 1, 1, -4, 7, -6, 3, 0, 1, 1, -5, 11, -13, 9, -3, 1, 1, 1, -6, 16, -24, 22, -12, 4, 0, 1, 1, -7, 22, -40, 46, -34, 16, -4, 1, 1, 1, -8, 29, -62, 86, -80, 50, -20, 5, 0, 1, 1, -9, 37, -91, 148, -166, 130, -70, 25, -5, 1, 1

Offset: 0

N. J. A. Sloane, Dec 07 2016

			Triangle begins:
  1,
  1,   1,
  1,   0,   1,
  1,  -1,   1,   1,
  1,  -2,   2,   0,   1,
  1,  -3,   4,  -2,   1,   1,
  1,  -4,   7,  -6,   3,   0,   1,
  1,  -5,  11, -13,   9,  -3,   1,   1,
  1,  -6,  16, -24,  22, -12,   4,   0,   1,
  ...

Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

See A112468, A112555 and A108561 for other versions.

Columns give A000124, A003600, A223718, A257890, A223659.

T := (n, k) -> local j; add((-1)^j*binomial(n-k-1+j, j), j = 0..k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..9);  # Peter Luschny, Aug 30 2024

T[i_, i_] = T[, 0] = 1; T[i, j_] := T[i, j] = T[i-1, j] - T[i-1, j-1];
Table[T[i, j], {i, 0, 11}, {j, 0, i}] // Flatten (* Jean-François Alcover, Sep 06 2018 *)
T[n_, k_] := 2^k*Hypergeometric2F1[-n, -k, -k, 1/2]; Table[T[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* Detlef Meya, Aug 30 2024 *)

\\ using arxiv (3.1) and (3.7) formulas where A is A220074 and B is this sequence
A(i, j) = if ((i < 0), 0, if (j==0, 1, A(i - 1, j - 1) - A(i - 1, j))); \\ A220074
B(i, j) = A(i, i-j);
tabl(nn) = for (i=0, nn, for (j=0, i, print1(B(i,j), ", ")); print()); \\ Michel Marcus, Jun 17 2017

T(i, j) = A220074(i, i-j). See (3.7) in arxiv link. - Michel Marcus, Jun 17 2017

T(n, k) = 2^k*hypergeom([-n, -k], [-k], 1/2). - Detlef Meya, Aug 30 2024

More terms from Michel Marcus, Jun 17 2017

A165405 a(0)=1, a(1)=3,a(n)=6*a(n-2)-a(n-1).

Original entry on oeis.org

1, 3, 3, 15, 3, 87, -69, 591, -1005, 4551, -10581, 37887, -101373, 328695, -936933, 2909103, -8530701, 25985319, -77169525, 233081439, -696098589, 2094587223, -6271178757, 18838702095, -56465774637, 169497987207, -508292635029

Offset: 0

Philippe Deléham, Sep 17 2009

sign

a(n)/a(n-1) tends to -3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,6).

nxt[{a_,b_}]:={b,6a-b}; NestList[nxt,{1,3},30][[;;,1]] (* or *) LinearRecurrence[{-1,6},{1,3},30] (* Harvey P. Dale, May 24 2024 *)

G.f.: (1+4x)/(1+x-6x^2). a(n)= Sum_{k, 0<=k<=n}A112555(n,k)*2^k.

a(n) = (6*2^n-(-3)^n)/5. [From Klaus Brockhaus, Sep 26 2009]

A165751 a(n) = 4 - 3*2^n.

Original entry on oeis.org

1, -2, -8, -20, -44, -92, -188, -380, -764, -1532, -3068, -6140, -12284, -24572, -49148, -98300, -196604, -393212, -786428, -1572860, -3145724, -6291452, -12582908, -25165820, -50331644, -100663292, -201326588, -402653180, -805306364, -1610612732, -3221225468

Offset: 0

Philippe Deléham, Sep 26 2009

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A131128.

Table[4 - 3*2^n, {n, 0, 50}] (* or *) LinearRecurrence[{3,-2}, {1,-2}, 50] (* G. C. Greubel, Apr 07 2016 *)

my(x='x+O('x^99)); Vec((1-5*x)/(1-3*x+2*x^2)) \\ Altug Alkan, Apr 07 2016

a(n) = 2*a(n-1) - 4, a(0)=1.
a(n) = Sum_{0<=k<=n} A112555(n,k)*(-3)^(n-k).
G.f.: (1-5x)/(1-3x+2x^2).
From G. C. Greubel, Apr 07 2016: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
E.g.f.: 4*exp(x) - 3*exp(2*x). (End)
a(n) = -A131128(n) for n>=1. - R. J. Mathar, Feb 27 2019

A165759 a(n) = (7-4*7^n)/3.

Original entry on oeis.org

1, -7, -63, -455, -3199, -22407, -156863, -1098055, -7686399, -53804807, -376633663, -2636435655, -18455049599, -129185347207, -904297430463, -6330082013255, -44310574092799, -310174018649607, -2171218130547263, -15198526913830855, -106389688396815999

Offset: 0

Philippe Deléham, Sep 26 2009

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8, -7).

(7-4*7^Range[0,20])/3 (* or *) LinearRecurrence[{8,-7},{1,-7},30] (* Harvey P. Dale, Dec 21 2014 *)

a(n) = 7*a(n-1)-14, a(0)=1.
a(n) = 8*a(n-1) - 7*a(n-2), a(0)= 1, a(1)= -7, for n>1.
G.f.: (1-15x)/(1-8x+7x^2).
a(n) = Sum_{0<=k<=n} A112555(n,k)*(-8)^(n-k).
E.g.f.: (1/3)*(7*exp(x) - 4*exp(7*x)). - G. C. Greubel, Apr 07 2016

Corrected and extended by Harvey P. Dale, Dec 21 2014

cp's OEIS Frontend

A166035 a(n) = (3^n+6*(-4)^n)/7.

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A091004 Expansion of x*(1-x)/((1-2*x)*(1+3*x)).

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A166036 a(n) = (4^n+8*(-5)^n)/9.

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A166149 a(n) = (5^n + 10*(-6)^n)/11.

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A080242 Table of coefficients of polynomials P(n,x) defined by the relation P(n,x) = (1+x)*P(n-1,x) + (-x)^(n+1).

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A166152 a(n) = (6^n+12*(-7)^n)/13.

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A279006 Alternating Jacobsthal triangle read by rows (second version).

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A091004 Expansion of x(1-x)/((1-2x)(1+3x)).