A185963 -id:A185963 - cp's OEIS frontend

A034943 Binomial transform of Padovan sequence A000931.

1, 1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217, 20330163, 47261895, 109870576, 255418101, 593775046, 1380359512, 3208946545

Offset: 0

N. J. A. Sloane

Trisection of the Padovan sequence: a(n) = A000931(3n). - Paul Barry, Jul 06 2004
a(n+1) gives diagonal sums of Riordan array (1/(1-x),x/(1-x)^3). - Paul Barry, Oct 11 2005
a(n+2) is the sum, over all Boolean n-strings, of the product of the lengths of the runs of 1. For example, the Boolean 7-string (0,1,1,0,1,1,1) has two runs of 1s. Their lengths, 2 and 3, contribute a product of 6 to a(9). The 8 Boolean 3-strings contribute to a(5) as follows: 000 (empty product), 001, 010, 100, 101 all contribute 1, 011 and 110 contribute 2, 111 contributes 3. - David Callan, Nov 29 2007
[a(n), a(n+1), a(n+2)], n > 0, = [0,1,0; 0,0,1; 1,-2,3]^n * [1,1,1]. - Gary W. Adamson, Mar 27 2008
Without the initial 1 and 1: 1, 2, 5, 12, 28, this is also the transform of 1 by the T_{1,0} transformation; see Choulet link. - Richard Choulet, Apr 11 2009
Without the first 1: transform of 1 by T_{0,0} transformation (see Choulet link). - Richard Choulet, Apr 11 2009
Starting (1, 2, 5, 12, ...) = INVERT transform of (1, 1, 2, 3, 4, 5, ...) and row sums of triangle A159974. - Gary W. Adamson, Apr 28 2009
a(n+1) is also the number of 321-avoiding separable permutations. (A permutation is separable if it avoids both 2413 and 3142.) - Vince Vatter, Sep 21 2009
a(n+1) is an eigensequence of the sequence array for (1,1,2,3,4,5,...). - Paul Barry, Nov 03 2010
Equals the INVERTi transform of A055588: (1, 2, 4, 9, 22, 56, ...) - Gary W. Adamson, Apr 01 2011
The Ca3 sums, see A180662, of triangle A194005 equal the terms of this sequence without a(0) and a(1). - Johannes W. Meijer, Aug 16 2011
Without the initial 1, a(n) = row sums of A182097(n)*A007318(n,k); i.e., a Triangular array T(n,k) multiplying the binomial (Pascal's) triangle by the Padovan sequence where a(0) = 1, a(1) = 0 and a(2) = 1. - Bob Selcoe, Jun 28 2013
a(n+1) is the top left entry of the n-th power of any of the 3 X 3 matrices [1, 1, 1; 0, 1, 1; 1, 0, 1] or [1, 1, 0; 1, 1, 1; 1, 0, 1] or [1, 1, 1; 1, 1, 0; 0, 1, 1] or [1, 0, 1; 1, 1, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 0, 1; 1, 1, 1; 0, 1, 1] or of the 3 X 3 matrix [1, 1, 0; 0, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
Number of sequences (e(1), ..., e(n-1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) < e(k) and e(i) <= e(k). [Martinez and Savage, 2.8] - Eric M. Schmidt, Jul 17 2017
a(n+1) is the number of words of length n over the alphabet {0,1,2} that do not contain the substrings 01 or 12 and do not start with a 2 and do not end with a 0. - Yiseth K. Rodríguez C., Sep 11 2020

			G.f. = 1 + x + x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 28*x^6 + 65*x^7 + 151*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Miklos Bona and Rebecca Smith, Pattern avoidance in permutations and their squares, arXiv:1901.00026 [math.CO], 2018. See H(z), Ex. 4.1.
Richard Choulet, Curtz like Transformation
Michael Dairyko, Samantha Tyner, Lara Pudwell, and Casey Wynn, Non-contiguous pattern avoidance in binary trees, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. - From _N. J. A. Sloane_, Feb 01 2013
Stoyan Dimitrov, Sorting by shuffling methods and a queue, arXiv:2103.04332 [math.CO], 2021.
Phan Thuan Do, Thi Thu Huong Tran, and Vincent Vajnovszki, Exhaustive generation for permutations avoiding a (colored) regular sets of patterns, arXiv:1809.00742 [cs.DM], 2018.
Brian Hopkins and Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.
Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 18.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 904
H. Magnusson and H. Ulfarsson, Algorithms for discovering and proving theorems about permutation patterns, arXiv preprint arXiv:1211.7110 [math.CO], 2012.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016
Vincent Vatter, Finding regular insertion encodings for permutation classes, arXiv:0911.2683 [math.CO], 2009.
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

First differences of A052921.
Cf. A000931, A007318, A055588, A095263, A097550.
Cf. A137531, A159974, A182097, A185963, A194005.

[n le 3 select 1 else 3*Self(n-1)-2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 14 2012

A034943 := proc(n): add(binomial(n+k-1, 3*k), k=0..floor(n/2)) end: seq(A034943(n), n=0..28); # Johannes W. Meijer, Aug 16 2011

LinearRecurrence[{3,-2,1},{1,1,1},30] (* Harvey P. Dale, Aug 11 2017 *)

{a(n) = if( n<1, n = 0-n; polcoeff( (1 - x + x^2) / (1 - 2*x + 3*x^2 - x^3) + x * O(x^n), n), n = n-1; polcoeff( (1 - x + x^2) / (1 - 3*x + 2*x^2 - x^3) + x * O(x^n), n))} /* Michael Somos, Mar 31 2012 */

@CachedFunction
def a(n): # a = A034943
    if (n<3): return 1
    else: return 3*a(n-1) - 2*a(n-2) + a(n-3)
[a(n) for n in range(51)] # G. C. Greubel, Apr 22 2023

a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1, 3*k). - Paul Barry, Jul 06 2004
G.f.: (1 - 2*x)/(1 - 3*x + 2*x^2 - x^3). - Paul Barry, Jul 06 2005
G.f.: 1 + x / (1 - x / (1 - x / (1 - x / (1 + x / (1 - x))))). - Michael Somos, Mar 31 2012
a(-1 - n) = A185963(n). - Michael Somos, Mar 31 2012
a(n) = A095263(n) - 2*A095263(n-1). - G. C. Greubel, Apr 22 2023

Edited by Charles R Greathouse IV, Apr 20 2010

A185962 Riordan array ((1-x)^2/(1-x+x^2), x(1-x)^2/(1-x+x^2)).

Original entry on oeis.org

1, -1, 1, -1, -2, 1, 0, -1, -3, 1, 1, 2, 0, -4, 1, 1, 3, 5, 2, -5, 1, 0, 0, 3, 8, 5, -6, 1, -1, -4, -6, -1, 10, 9, -7, 1, -1, -4, -10, -16, -10, 10, 14, -8, 1, 0, 1, 0, -10, -26, -24, 7, 20, -9, 1, 1, 6, 15, 20, 5, -30, -42, 0, 27, -10, 1

Offset: 0

Paul Barry, Feb 07 2011

Riordan array (g(x),xg(x)) where g(x)=(1-x)(1-x^2)(1-x^3)/(1-x^6).
Inverse is A185967. Row sums are A185963.
Diagonal sums are A185964. Central coefficients are A185965.
Subtriangle of the triangle given by (0, -1, 2, -1/2, 1/2, 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. - Philippe Deléham, Mar 19 2012

			Triangle begins:
   1;
  -1,  1;
  -1, -2,   1;
   0, -1,  -3,   1;
   1,  2,   0,  -4,   1;
   1,  3,   5,   2,  -5,   1;
   0,  0,   3,   8,   5,  -6,   1;
  -1, -4,  -6,  -1,  10,   9,  -7,  1;
  -1, -4, -10, -16, -10,  10,  14, -8,  1;
   0,  1,   0, -10, -26, -24,   7, 20, -9,   1;
   1,  6,  15,  20,   5, -30, -42,  0, 27, -10, 1;
  ...
From _Philippe Deléham_, Mar 19 2012: (Start)
(0, -1, 2, -1/2, 1/2, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0, -1,  1;
  0, -1, -2,  1;
  0,  0, -1, -3,  1;
  0,  1,  2,  0, -4,  1;
  0,  1,  3,  5,  2, -5,  1;
  ... (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A195339, A195350.

CoefficientList[CoefficientList[Series[1/(1 - y*x + x/(1 - x)^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jul 23 2017 *)

T(n,k) = Sum_{i=0..(2*k+2)} C(2*k+2,i)*Sum_{j=0..(n-k-i)} C(k+j,j)*C(j,n-k-i-j)*(-1)^(n-k-j).
G.f.: 1/(1-y*x+x/(1-x)^2). - Philippe Deléham, Feb 07 2012
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k-1), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,0) = -1, T(2,1) = -2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham , Nov 11 2013

A109955 Number triangle binomial(n+2k,3k).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 7, 1, 1, 20, 28, 10, 1, 1, 35, 84, 55, 13, 1, 1, 56, 210, 220, 91, 16, 1, 1, 84, 462, 715, 455, 136, 19, 1, 1, 120, 924, 2002, 1820, 816, 190, 22, 1, 1, 165, 1716, 5005, 6188, 3876, 1330, 253, 25, 1, 1, 220, 3003, 11440, 18564, 15504, 7315, 2024

Offset: 0

Paul Barry, Jul 06 2005

Riordan array (1/(1-x),x/(1-x)^3).
Inverse array is A109956. Row sums are A052544.
Diagonal sums are A034943(n+1).

			Rows begin
1;
1,1;
1,4,1;
1,10,7,1;
1,20,28,10,1;
1,35,84,55,13,1;

tabl(nn) = {my(m = matrix(nn, nn, n, k, if (nMichel Marcus, Nov 20 2015

Number triangle T(n, k) = binomial(n+2k, 3k).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k) + T(n-1,k-1). - Philippe Deléham, Feb 18 2012
G.f.: (1-x)^2/((1-x)^3-y*x). - Philippe Deléham, Feb 18 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A185963(n), A000012(n), A052544(n), A049086(n) for x = -1, 0, 1, 2 respectively. - Philippe Deléham, Feb 18 2012

A181984 INVERT transform of A028310.

Original entry on oeis.org

1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217, 20330163, 47261895, 109870576, 255418101, 593775046, 1380359512, 3208946545, 7459895657, 17342153393, 40315615410, 93722435101

Offset: 0

Michael Somos, Apr 04 2012

nonn

			G.f. = 1 + 2*x + 5*x^2 + 12*x^3 + 28*x^4 + 65*x^5 + 151*x^6 + 351*x^7 + 816*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Miklos Bona, Rebecca Smith, Pattern avoidance in permutations and their squares, arXiv:1901.00026 [math.CO], 2018. See H(z), Ex. 4.1.
Index entries for linear recurrences with constant coefficients, signature (3, -2, 1).

Cf. A000931, A028310, A034943, A185963.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+x^2)/(1-3*x+2*x^2-x^3))); // G. C. Greubel, Aug 12 2018

CoefficientList[Series[(1-x+x^2)/(1-3*x+2*x^2-x^3), {x, 0, 50}], x] (* G. C. Greubel, Aug 12 2018 *)

{a(n) = if( n<0, n = -1-n; polcoeff( (1 - x + x^2) / (1 - 2*x + 3*x^2 - x^3) + x * O(x^n), n), polcoeff( (1 - x + x^2) / (1 - 3*x + 2*x^2 - x^3) + x * O(x^n), n))}

x='x+O('x^50); Vec((1-x+x^2)/(1-3*x+2*x^2-x^3)) \\ G. C. Greubel, Aug 12 2018

G.f.: (1 - x + x^2) / (1 - 3*x + 2*x^2 - x^3).
G.f.: 1 / (1 - 2*x / (1 - x / (2 + x / (1 - 2*x / (1 + x))))).
a(n) = A034943(n + 2) = A185963(-2 - n).
a(n) = 3*a(n-1) + 2*a(n-2) - a(n-3).
a(n) satisfies 1 = f(a(n-2), a(n-1), a(n)) where f(u, v, w) = u^3 - 5*v^3 + w^3 + u*v * (7*v -4*u) + u*w * (3*u + 2*w) + v*w * (11*v - 6*w) - 9*u*v*w.
a(n) = A000931(3*n + 6). - Michael Somos, Sep 18 2012

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A034943 Binomial transform of Padovan sequence A000931.

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A185962 Riordan array ((1-x)^2/(1-x+x^2), x(1-x)^2/(1-x+x^2)).

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A109955 Number triangle binomial(n+2k,3k).

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A181984 INVERT transform of A028310.

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