A095121 -id:A095121 - cp's OEIS frontend

A193815 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = x^n + x^(n-1) + ... + x+1 and q(n,x)=(x+1)^n.

1, 1, 1, 1, 3, 2, 1, 4, 6, 3, 1, 5, 10, 10, 4, 1, 6, 15, 20, 15, 5, 1, 7, 21, 35, 35, 21, 6, 1, 8, 28, 56, 70, 56, 28, 7, 1, 9, 36, 84, 126, 126, 84, 36, 8, 1, 10, 45, 120, 210, 252, 210, 120, 45, 9, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 10, 1, 12, 66, 220, 495

Offset: 0

Clark Kimberling, Aug 06 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
Triangle T(n,k), read by rows, given by (1,0,-1,1,0,0,0,0,0,0,0,...) DELTA (1,1,-1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 08 2011
Row sums are A095121. - Philippe Deléham, Nov 24 2011

			First six rows:
  1;
  1,  1;
  1,  3,  2;
  1,  4,  6,  3;
  1,  5, 10, 10,  4;
  1,  6, 15, 20, 15,  5;

Branko Malesevic, Yue Hu, Cristinel Mortici, Accurate Estimates of (1+x)^{1/x} Involved in Carleman Inequality and Keller Limit, arXiv:1801.04963 [math.CA], 2018.

Cf. A193722, A153861, A193818.

z = 10; c = 1; d = 1;
p[0, x_] := 1
p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;
q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193815 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A153861 *)
t[0, 0] = t[1, 0] = t[1, 1] = t[2, 0] = 1; t[2, 1] = 3; t[2, 2] = 2; t[n_, k_] /; k<0 || k>n = 0; t[n_, k_] := t[n, k] = t[n-1, k]+2*t[n-1, k-1]-t[n-2, k-1]-t[n-2, k-2]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 16 2013, after Philippe Deléham *)

T(n,k) = A153861(n,n-k). - Philippe Deléham, Oct 08 2011
G.f.: (1-y*x+y*(y+1)*x^2)/((1-y*x)*(1-(y+1)*x)). - Philippe Deléham, Nov 24 2011
Sum_{k=0..n} T(n,k)*x^k = (x+1)*((x+1)^n - x^n) + 0^n. - Philippe Deléham, Nov 24 2011
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(0,0)=T(1,0)=T(1,1)=T(2,0)=1, T(2,1)=3, T(2,2)=2, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Dec 15 2013

A002783 a(n) = 2*(3^n - 2^n) + 1.

Original entry on oeis.org

1, 3, 11, 39, 131, 423, 1331, 4119, 12611, 38343, 116051, 350199, 1054691, 3172263, 9533171, 28632279, 85962371, 258018183, 774316691, 2323474359, 6971471651, 20916512103, 62753730611, 188269580439, 564825518531, 1694510110023, 5083597438931, 15250926534519

Offset: 0

N. J. A. Sloane

Binomial transform of A095121. - R. J. Mathar, Oct 05 2012
Create a triangle having its left and right border both equal to the n-th row of Pascal's triangle, and internal terms m(i,j) = m(i-1,j-1) + m(i-1,j). Then the sum of all elements equals a(n). - J. M. Bergot, Oct 07 2012, edited by M. F. Hasler, Oct 10 2012
First differences of A090326 (with offset 1). - Wesley Ivan Hurt, Jul 08 2014

			From _J. M. Bergot_ and _M. F. Hasler_, Oct 10 2012: (Start)
For n=3, the triangle with left and right border (1,3,3,1) and internal terms m(i,j) = m(i-1,j-1) + m(i-1,j) is
     1
    3 3
   3 6 3
  1 9 9 1
and the sum of all the elements is 39 = a(3). (End)

H. Gupta, On a problem in parity, Indian J. Math., 11 (1969), 157-163.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
H. Gupta, On a problem in parity, Indian J. Math., 11 (1969), 157-163. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A027649, A090326, A095121.

[2*(3^n - 2^n)+1 : n in [0..30]]; // Wesley Ivan Hurt, Jul 08 2014

A002783:=n->2*(3^n - 2^n)+1: seq(A002783(n), n=0..30); # Wesley Ivan Hurt, Jul 08 2014

CoefficientList[Series[(-1 + 3*x - 4*x^2)/((x - 1)*(3*x - 1)*(2*x - 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jul 08 2014 *)

a(n)=2*(3^n-2^n)+1 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: ( -1+3*x-4*x^2 ) / ( (x-1)*(3*x-1)*(2*x-1) ). - Simon Plouffe in his 1992 dissertation
a(n+1) - a(n) = 2*A027649(n). - R. J. Mathar, Oct 05 2012
E.g.f.: exp(x)*(1 - 2*exp(x) + 2*exp(2*x)). - Stefano Spezia, May 18 2024

More terms from Wesley Ivan Hurt, Jul 08 2014

A134067 Row sums of triangle A134066.

Original entry on oeis.org

1, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590, 17179869182

Offset: 0

Gary W. Adamson, Oct 06 2007

Essentially the same as A095121. - R. J. Mathar, Mar 28 2012

			a(3) = 30 = sum of row 3 terms of triangle A134066: (2 + 12 + 12 + 4).
a(3) = 30 = (1, 3, 3, 1) dot (1, 5, 3, 5) = (1 + 15 + 9 + 5).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A095121, A134066.

Join[{1}, LinearRecurrence[{3, -2}, {6, 14}, 50]] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)

Vec(-(2*x^2-3*x-1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Mar 13 2014

Binomial transform of (1, 5, 3, 5, 3, 5, ...).
From Colin Barker, Mar 13 2014: (Start)
a(n) = 2^(2+n) - 2 for n > 0.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 0.
G.f.: -(2*x^2-3*x-1) / ((x-1)*(2*x-1)). (End)
E.g.f.: 2*exp(x)*(2*exp(x) - 1) - 1. - Stefano Spezia, May 07 2023

A153861 Triangle read by rows, binomial transform of triangle A153860.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 4, 10, 10, 5, 1, 5, 15, 20, 15, 6, 1, 6, 21, 35, 35, 21, 7, 1, 7, 28, 56, 70, 56, 28, 8, 1, 8, 36, 84, 126, 126, 84, 36, 9, 1, 9, 45, 120, 210, 252, 210, 120, 45, 10, 1, 10, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

Gary W. Adamson, Jan 03 2009

Row sums = A095121: (1, 2, 6, 14, 30, 62, 126,...).
Triangle T(n,k), 0<=k<=n, read by rows, given by [1,1,-1,1,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009
A123110*A007318 as infinite lower triangular matrices. - Philippe Deléham, Jan 06 2009
A153861 is the fusion of polynomial sequences p(n,x)=x^n+x^(n-1)+...+x+1 and q(n,x)=(x+1)^n; see A193722 for the definition of fusion. - Clark Kimberling, Aug 06 2011

			First few rows of the triangle are:
1;
1, 1;
2, 3, 1;
3, 6, 4, 1;
4, 10, 10, 5, 1;
5, 15, 20, 15, 6, 1;
6, 21, 35, 35, 21, 7, 1;
7, 28, 56, 70, 56, 28, 8, 1;
8, 36, 84, 126, 126, 84, 36, 9, 1;
9, 45, 120, 210, 252, 210, 120, 45, 10, 1;
...

G. C. Greubel, Table of n, a(n) for the first 46 rows

Cf. A153860, A095121, A130405.

This is A137396 without the initial column and without signs.

z = 10; c = 1; d = 1;
p[0, x_] := 1
p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;
q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193815 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A153861 *)
(* Clark Kimberling, Aug 06 2011 *)

Triangle read by rows, A007318 * A153860. Remove left two columns of Pascal's triangle and append (1, 1, 2, 3, 4, 5,...).
As a recursive operation by way of example, row 3 = (3, 6, 4, 1) =
[1, 1, 1, 0] * (flipped Pascal's triangle matrix) = [1, 3, 3, 1]
[1, 2, 1, 0]
[1, 1, 0, 0]
[1, 0, 0, 0].
(Cf. analogous operation in A130405, but in A153861 the linear multiplier = [1,1,1,...,0].)
T(n,k) = 2*T(n-1,k)+T(n-1,k-1)-T(n-2,k)-T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0)=2, T(2,1)=3, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 15 2013
G.f.: (1-x+x^2+x^2*y)/((x-1)*(-1+x+x*y)). - R. J. Mathar, Aug 11 2015

A154251 Expansion of (1-x+7x^2)/((1-x)(1-2x)).

Original entry on oeis.org

1, 2, 11, 29, 65, 137, 281, 569, 1145, 2297, 4601, 9209, 18425, 36857, 73721, 147449, 294905, 589817, 1179641, 2359289, 4718585, 9437177, 18874361, 37748729, 75497465, 150994937, 301989881, 603979769, 1207959545, 2415919097

Offset: 0

Philippe Deléham, Jan 05 2009

Binomial transform of 1,1,8,1,8,1,8,1,8,1,8,1,8,1,8,...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf.: A094373, A000079, A083329, A095121, A154117, A131128, A154118, A131130

Join[{1},LinearRecurrence[{3,-2},{2,11}, 25]] (* or *) Join[{1},Table[9*2^(n-1) - 7, {n,1,25}]] (* G. C. Greubel, Sep 08 2016 *)

Vec((1-x+7*x^2)/((1-x)*(1-2*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 3*a(n-1) - 2*a(n-2), n>2, with a(0)=1, a(1)=2, a(2)=11.
a(n) = 9*2^(n-1) - 7, n>0, with a(0)=1.
a(n) = 2*a(n-1) + 7, n>1, with a(0)=1, a(1)=2.
From G. C. Greubel, Sep 08 2016: (Start)
a(n) = 9*2^(n-1) - 7 for n >= 1.
E.g.f.: (1/2)*(9*exp(2*x) - 14*exp(x) + 7). (End)

A131108 T(n,k) = 2*A007318(n,k) - A097806(n,k).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 2, 6, 5, 1, 2, 8, 12, 7, 1, 2, 10, 20, 20, 9, 1, 2, 12, 30, 40, 30, 11, 1, 2, 14, 42, 70, 70, 42, 13, 1, 2, 16, 56, 112, 140, 112, 56, 15, 1, 2, 18, 72, 168, 252, 252, 168, 72, 17, 1, 2, 20, 90, 240, 420, 504, 420, 240, 90, 19, 1

Offset: 0

Gary W. Adamson, Jun 15 2007

Row sums give A095121.

Triangle T(n,k), 0 <= k <= n, read by rows given by [1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 18 2007

			First few rows of the triangle are:
  1;
  1,  1;
  2,  3,  1;
  2,  6,  5,  1;
  2,  8, 12,  7, 1;
  2, 10, 20, 20, 9, 1;
...

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

Cf. A007318, A095121, A097806.

function T(n,k)
  if k eq n-1 then return 2*n-1;
  elif k eq n then return 1;
  else return 2*Binomial(n,k);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 18 2019

seq(seq( `if`(k=n-1, 2*n-1, `if`(k=n, 1, 2*binomial(n,k))), k=0..n), n=0..12); # G. C. Greubel, Nov 18 2019

Table[If[k==n-1, 2*n-1, If[k==n, 1, 2*Binomial[n, k]]], {n,0,12}, {k,0, n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k) = if(k==n-1, 2*n-1, if(k==n, 1, 2*binomial(n,k))); \\ G. C. Greubel, Nov 18 2019

@CachedFunction
def T(n, k):
    if (k==n-1): return 2*n-1
    elif (k==n): return 1
    else: return 2*binomial(n,k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 18 2019

Twice Pascal's triangle minus A097806, the pairwise operator.

G.f.: (1-x*y+x^2+x^2*y)/((-1+x+x*y)*(x*y-1)). - R. J. Mathar, Aug 11 2015

Corrected by Philippe Deléham, Dec 17 2007

More terms added and data corrected by G. C. Greubel, Nov 18 2019

A154252 Expansion of (1-x+8x^2)/((1-x)(1-2x)) .

Original entry on oeis.org

1, 2, 12, 32, 72, 152, 312, 632, 1272, 2552, 5112, 10232, 20472, 40952, 81912, 163832, 327672, 655352, 1310712, 2621432, 5242872, 10485752, 20971512, 41943032, 83886072, 167772152, 335544312, 671088632, 1342177272, 2684354552, 5368709112, 10737418232

Offset: 0

Philippe Deléham, Jan 05 2009

Binomial transform of 1,1,9,1,9,1,9,1,9,1,9,1,9,1,9,...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf.: A094373, A000079, A083329, A095121, A154117, A131128, A154118, A131130, A154251

Join[{1},LinearRecurrence[{3,-2},{2,12},40]]  (* Harvey P. Dale, Dec 30 2014 *)
Join[{1},Table[5*2^n - 8, {n,1,25}]] (* G. C. Greubel, Sep 08 2016 *)

Vec((1-x+8*x^2)/((1-x)*(1-2*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 3*a(n-1) - 2*a(n-2), n>2, with a(0)=1, a(1)=2, a(2)=12.
a(n) = 2*a(n-1) + 8, n>1, with a(0)=1, a(1)=2.
a(n) = 10*2^(n-1) - 8, n>=1, with a(0)=1.
E.g.f.: 5*exp(2*x) - 8*exp(x) + 4. - G. C. Greubel, Sep 08 2016

Two terms corrected by Johannes W. Meijer, May 26 2011

A131129 3A007318 - 2A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 3, 9, 7, 1, 3, 12, 18, 10, 1, 3, 15, 30, 30, 13, 1, 3, 18, 45, 60, 45, 16, 1, 3, 21, 63, 105, 105, 63, 19, 1, 3, 24, 84, 168, 210, 168, 84, 22, 1

Offset: 0

Gary W. Adamson, Jun 16 2007

Row sums = A131128: (1, 2, 8, 20, 44, 92, 188, 380, ...), the binomial transform of (1, 1, 5, 1, 5, 1, 5, ...). Triangle A131108 has row sums (1, 2, 6, 14, 30, 62, ...), the binomial transform of (1, 1, 3, 1, 3, 1, ...). Generalization: Given triangles generated from N*A007318 - (N-1)*A097806, row sums are binomial transforms of (1, 1, (2N-1), 1, (2N-1), 1, ...).

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,2,-3,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 18 2007

			First few rows of the triangle:
  1;
  1,  1;
  3,  4,  1;
  3,  9,  7,  1;
  3, 12, 18, 10,  1;
  3, 15, 30, 30, 13,  1;
  ...

Cf. A097806, A131128, A095121.

G.f.: (1-x*y+2*x^2+2*x^2*y)/((-1+x+x*y)*(x*y-1)). - R. J. Mathar, Aug 12 2015

A296965 Expansion of x(1 - x + 2x^2) / ((1 - x)(1 - 2x)).

Original entry on oeis.org

0, 1, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590, 17179869182

Offset: 0

J. Devillet, Dec 22 2017

a(n) = A000225(n)-1, a(0)=0, a(1)=1. Number of quasilinear weak orderings R on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1<...

Essentially the same as A095121 and A000918. - R. J. Mathar, Jan 02 2018

Colin Barker, Table of n, a(n) for n = 0..1000
J. Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA], 2017-2018.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A000918, A095121, A134067.

CoefficientList[Series[x (1 - x + 2 x^2)/((1 - x) (1 - 2 x)), {x, 0, 33}], x] (* or *)
LinearRecurrence[{3, -2}, {0, 1, 2, 6}, 34] (* Michael De Vlieger, Dec 22 2017 *)

concat(0, Vec(x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Dec 22 2017

From Colin Barker, Dec 22 2017: (Start)
G.f.: x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)).
a(n) = 2^n - 2 for n>1.
a(n) = 3*a(n-1) - 2*a(n-2) for n>3. (End)
a(n) = A134067(n-2) for n >= 3. - Georg Fischer, Oct 30 2018
E.g.f.: 1 + exp(x)*(exp(x) - 2) + x. - Stefano Spezia, May 07 2023

A307457 Longest path length in the n-Apollonian network.

Original entry on oeis.org

3, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590

Offset: 1

Eric W. Weisstein, Apr 08 2019

Eric Weisstein's World of Mathematics, Apollonian Network.
Eric Weisstein's World of Mathematics, Longest Path.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Essentially the same as A000918 shifted, and A095121.

Cf. A307549.

a(n) = 2^(n+1) - 2 = A000918(n+1) for n > 1. - Andrew Howroyd, Jun 09 2025

a(5) onwards from Andrew Howroyd, Jun 09 2025

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cp's OEIS Frontend

A193815 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = x^n + x^(n-1) + ... + x+1 and q(n,x)=(x+1)^n.

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A002783 a(n) = 2*(3^n - 2^n) + 1.

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A134067 Row sums of triangle A134066.

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A153861 Triangle read by rows, binomial transform of triangle A153860.

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A154251 Expansion of (1-x+7x^2)/((1-x)(1-2x)).

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A131108 T(n,k) = 2*A007318(n,k) - A097806(n,k).

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A154252 Expansion of (1-x+8x^2)/((1-x)(1-2x)) .

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A131129 3A007318 - 2A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.

A296965 Expansion of x(1 - x + 2x^2) / ((1 - x)(1 - 2x)).