A008971 -id:A008971 - cp's OEIS frontend

A000340 a(0)=1, a(n) = 3*a(n-1) + n + 1.

1, 5, 18, 58, 179, 543, 1636, 4916, 14757, 44281, 132854, 398574, 1195735, 3587219, 10761672, 32285032, 96855113, 290565357, 871696090, 2615088290, 7845264891, 23535794695, 70607384108, 211822152348, 635466457069

Offset: 0

N. J. A. Sloane, Simon Plouffe

From Johannes W. Meijer, Feb 20 2009: (Start)
Second right hand column (n-m=1) of the A156920 triangle.
The generating function of this sequence enabled the analysis of the polynomials A156921 and A156925.
(End)
Partial sums of A003462, and thus the second partial sums of A000244 (3^n). Also column k=2 of A106516. - John Keith, Jan 04 2022

			G.f. = 1 + 5*x + 18*x^2 + 58*x^3 + 179*x^4 + 543*x^5 + 1636*x^6 + ...

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
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
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See pp. 9, 18.
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See p. 7.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 389
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
László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. Mentions this sequence. See also Proc. Amer. Math. Soc. 148 (2020), 461-469.
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

From Johannes W. Meijer, Feb 20 2009: (Start)
Cf. A156921, A156925, A156927, A156933. Other columns A156922, A156923, A156924.
Equals A156920 second right hand column.
Equals A142963 second right hand column divided by 2^n.
Equals A156919 second right hand column divided by 2.
(End)
Cf. A014915.
Equals column k=1 of A008971 (shifted). - Jeremy Dover, Jul 11 2021
Cf. A000340, A003462 (first differences), A106516.

[(3^(n+2)-2*n-5)/4: n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-3*a[n-2]+1 od: seq(a[n],n=0..50); # Miklos Kristof, Mar 09 2005
A000340:=-1/(3*z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

a[ n_] := MatrixPower[ {{1, 0, 0}, {1, 1, 0}, {1, 1, 3}}, n + 1][[3, 1]]; (* Michael Somos, May 28 2014 *)
RecurrenceTable[{a[0]==1,a[n]==3a[n-1]+n+1},a,{n,30}] (* or *) LinearRecurrence[{5,-7,3},{1,5,18},30] (* Harvey P. Dale, Jan 31 2017 *)

G.f.: 1/((1-3*x)*(1-x)^2).
a(n) = (3^(n+2) - 2*n - 5)/4.
a(n) = Sum_{k=0..n+1} (n-k+1)*3^k = Sum_{k=0..n+1} k*3^(n-k+1). - Paul Barry, Jul 30 2004
a(n) = Sum_{k=0..n} binomial(n+2, k+2)*2^k. - Paul Barry, Jul 30 2004
a(-1)=0, a(0)=1, a(n) = 4*a(n-1) - 3*a(n-2) + 1. - Miklos Kristof, Mar 09 2005
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3). - Johannes W. Meijer, Feb 20 2009
a(-2 - n) = 3^-n * A014915(n). - Michael Somos, May 28 2014
E.g.f.: exp(x)*(9*exp(2*x) - 2*x - 5)/4. - Stefano Spezia, Nov 09 2024

A000363 Number of permutations of [n] with exactly 2 increasing runs of length at least 2.

Original entry on oeis.org

5, 61, 479, 3111, 18270, 101166, 540242, 2819266, 14494859, 73802835, 373398489, 1881341265, 9453340172, 47417364268, 237571096820, 1189405165908, 5951965440609, 29775517732665, 148927275340835, 744793282001995

Offset: 4

N. J. A. Sloane

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
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 = 4..1000
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See p. 7.
Index entries for linear recurrences with constant coefficients, signature (14,-75,196,-263,174,-45).

Contribution from Johannes W. Meijer, May 24 2009: (Start)
The a(n) sequence equals the third left hand column of A008971.
The a(2*n) sequence equals the third left hand column of A160486.
(End)

[(5^n-(2*n-1)*3^n+2*n^2-2*n-2)/16: n in [4..30]]; // Vincenzo Librandi, May 03 2013

Table[(5^n-(2*n-1)*3^n+2*n^2-2*n-2)/16,{n,4,20}] (* Vaclav Kotesovec, Nov 19 2012 *)

From Vaclav Kotesovec, Nov 19 2012: (Start)
a(n) = (5^n-(2*n-1)*3^n+2*n^2-2*n-2)/16.
G.f.: -x^4*(9*x-5)/((x-1)^3*(3*x-1)^2*(5*x-1)). (End)
E.g.f.: exp(x)*(exp(4*x) + exp(2*x)*(1 - 6*x) - 2*(1 - x^2))/16. - Stefano Spezia, Nov 09 2024

More terms and better definition from Jon E. Schoenfield, Mar 25 2010

A160486 Triangle of polynomial coefficients related to the o.g.f.s. of the RBS1 polynomials.

Original entry on oeis.org

1, 1, 1, 1, 18, 5, 1, 179, 479, 61, 1, 1636, 18270, 19028, 1385, 1, 14757, 540242, 1949762, 1073517, 50521, 1, 132854, 14494859, 137963364, 241595239, 82112518, 2702765

Offset: 1

Johannes W. Meijer, May 24 2009, Sep 19 2012

As we showed in A160485 the n-th term of the coefficients of matrix row BS1[1-2*m,n] for m = 1 , 2, 3, .. , can be generated with the RBS1(1-2*m,n) polynomials.
We define the o.g.f.s. of these polynomials by GFRBS1(z,1-2*m) = sum(RBS1(1-2*m,n)*z^(n-1), n=1..infinity) for m = 1, 2, 3, .. . The general expression of these o.g.f.s. is GFRBS1(z,1-2*m) = (-1)*RB(z,1-2*m)/(z-1)^m.
The RB(z,1-2*m) polynomials lead to a triangle that is a subtriangle of the 'double triangle' A008971. The even rows of the latter triangle are identical to the rows of our triangle.
The Maple program given below is derived from the one given in A008971.

			The first few rows of the triangle are:
[1]
[1, 1]
[1, 18, 5]
[1, 179, 479, 61]
[1, 1636, 18270, 19028, 1385]
The first few RB(z,1-2*m) polynomials are:
RB(z,-1) = 1
RB(z,-3) = z+1
RB(z,-5) = z^2+18*z+5
RB(z,-7) = z^3+179*z^2+479*z+61
The first few GFRBS1(z,1-2*m) are:
GFRBS1(z,-1) = (-1)*(1)/(z-1)
GFRBS1(z,-3) = (-1)*(z+1)/(z-1)^2
GFRBS1(z,-5) = (-1)*(z^2+18*z+5)/(z-1)^3
GFRBS1(z,-7) = (-1)*(z^3+179*z^2+479*z+61)/(z-1)^4

Cf. A160480 and A160485.
The row sums equal A010050.
This triangle is a sub-triangle of A008971.
A000340(2*n-2), A000363(2*n+2) and A000507(2*n+4) equal the second, third and fourth left hand columns.
The first right hand column equals the Euler numbers A000364.

nmax:=15; G := sqrt(1-t)/(sqrt(1-t)*cosh(x*sqrt(1-t))-sinh(x*sqrt(1-t))): Gser := simplify(series(G, x=0, nmax+1)): for m from 0 to nmax do P[m] := sort(expand(m!* coeff(Gser, x, m))) od: nmx := floor(nmax/2); for n from 0 to nmx do for k from 0 to nmx-1 do A(n+1, n+1-k) := coeff(P[2*n], t, n-k) od: od: seq(seq(A(n,m), m=1..n), n=1..nmx);

A000507 Number of permutations of [n] with exactly 3 increasing runs of length at least 2.

Original entry on oeis.org

61, 1385, 19028, 206276, 1949762, 16889786, 137963364, 1081702420, 8236142455, 61386982075, 450403628440, 3266265481144, 23480284103492, 167687984079924, 1191656966048088, 8436830209386360, 59563995267159825, 419628657826253805

Offset: 6

N. J. A. Sloane

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
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).

Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See p. 7.
Index entries for linear recurrences with constant coefficients, signature (30, -385, 2776, -12418, 35908, -67818, 82552, -62109, 26190, -4725).

From Johannes W. Meijer, May 24 2009: (Start)
The a(n) sequence equals the fourth left hand column of A008971.
The a(2*n) sequence equals the fourth left hand column of A160486.
(End)

[(3*7^n-(6*n-9)*5^n+(6*n^2-18*n+3)*3^n-4*n^3+18*n^2-8*n-15)/192: n in [6..30]]; // Vincenzo Librandi, Feb 09 2016

t[n_, 0] = 1; t[n_, k_] /; k > n/2 = 0; t[n_, k_] /; k <= n/2 := t[n, k] = (2k+1) t[n-1, k] + (n-2k+1) t[n-1, k-1]; a[n_] := t[n, 3]; Table[a[n], {n, 6, 23}] (* Jean-François Alcover, Feb 09 2016 *)

a(n) = (3*7^n-(6*n-9)*5^n+(6*n^2-18*n+3)*3^n-4*n^3+18*n^2-8*n-15)/192.

G.f.: x^6*(61 - 445*x + 963*x^2 - 675*x^3)/((1 - 5*x)^2*(1 - x)^4*(1 - 3*x)^3*(1 - 7*x)). - Stefano Spezia, Nov 09 2024

Definition changed for clarity and for consistency with A008971, and formula and additional terms added by Jon E. Schoenfield, Mar 26 2010

A333273 Irregular triangle read by rows: coefficients of q-Eulerian polynomials of Type B.

Original entry on oeis.org

1, 1, 4, 1, 20, 1, 72, 80, 1, 232, 976, 1, 716, 7664, 3904, 1, 2172, 49776, 88640, 1, 6544, 292320, 1217792, 354560, 1, 19664, 1618656, 13201664, 12933376, 1, 59028, 8643872, 124784768, 274820352, 51733504, 1, 177124, 45108256, 1080946304, 4469939456, 2767631360

Offset: 1

N. J. A. Sloane, Mar 14 2020

For Type A see A101280.

			Triangle begins:
  1;
  1,    4;
  1,   20;
  1,   72,   80;
  1,  232,  976;
  1,  716, 7664, 3904;
  ...

Guoniu Han, Frédéric Jouhet, and Jiang Zeng, Two new triangles of q-integers via q-Eulerian polynomials of type A and B, preprint.
Guoniu Han, Frédéric Jouhet, and Jiang Zeng, Two new triangles of q-integers via q-Eulerian polynomials of type A and B, Ramanujan J (2013) 31:115-127.

Cf. A101280, A008971. Row sums are A182825.

T(n, k) = A008971(n, k) * 4^k. [Han et al.] - Andrey Zabolotskiy, Feb 15 2025

Term T(6, 2) corrected (Table 1 in the Han et al. reference has a typo) by Robert S. Maier, Feb 15 2025

Rows 7-11 from Andrey Zabolotskiy, Mar 03 2025

cp's OEIS Frontend

A000340 a(0)=1, a(n) = 3*a(n-1) + n + 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A000363 Number of permutations of [n] with exactly 2 increasing runs of length at least 2.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A160486 Triangle of polynomial coefficients related to the o.g.f.s. of the RBS1 polynomials.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

A000507 Number of permutations of [n] with exactly 3 increasing runs of length at least 2.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A333273 Irregular triangle read by rows: coefficients of q-Eulerian polynomials of Type B.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions