A008518 -id:A008518 - cp's OEIS frontend

A000325 a(n) = 2^n - n.

1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556, 2097131, 4194282, 8388585, 16777192, 33554407, 67108838, 134217701, 268435428, 536870883, 1073741794, 2147483617

Offset: 0

Rosario Salamone (Rosario.Salamone(AT)risc.uni-linz.ac.at)

Number of permutations of degree n with at most one fall; called "Grassmannian permutations" by Lascoux and Schützenberger. - Axel Kohnert (Axel.Kohnert(AT)uni-bayreuth.de)
Number of different permutations of a deck of n cards that can be produced by a single shuffle. [DeSario]
Number of Dyck paths of semilength n having at most one long ascent (i.e., ascent of length at least two). Example: a(4)=12 because among the 14 Dyck paths of semilength 4, the only paths that have more than one long ascent are UUDDUUDD and UUDUUDDD (each with two long ascents). Here U = (1, 1) and D = (1, -1). Also number of ordered trees with n edges having at most one branch node (i.e., vertex of outdegree at least two). - Emeric Deutsch, Feb 22 2004
Number of {12,1*2*,21*}-avoiding signed permutations in the hyperoctahedral group.
Number of 1342-avoiding circular permutations on [n+1].
2^n - n is the number of ways to partition {1, 2, ..., n} into arithmetic progressions, where in each partition all the progressions have the same common difference and have lengths at least 1. - Marty Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu), May 21 2005
if b(0) = x and b(n) = b(n-1) + b(n-1)^2*x^(n-2) for n > 0, then b(n) is a polynomial of degree a(n). - Michael Somos, Nov 04 2006
The chromatic invariant of the Mobius ladder graph M_n for n >= 2. - Jonathan Vos Post, Aug 29 2008
Dimension sequence of the dual alternative operad (i.e., associative and satisfying the identity xyz + yxz + zxy + xzy + yzx + zyx = 0) over the field of characteristic 3. - Pasha Zusmanovich, Jun 09 2009
An elephant sequence, see A175654. For the corner squares six A[5] vectors, with decimal values between 26 and 176, lead to this sequence (without the first leading 1). For the central square these vectors lead to the companion sequence A168604. - Johannes W. Meijer, Aug 15 2010
a(n+1) is also the number of order-preserving and order-decreasing partial isometries (of an n-chain). - Abdullahi Umar, Jan 13 2011
A040001(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
A130103(n+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
The number of labeled graphs with n vertices whose vertex set can be partitioned into a clique and a set of isolated points. - Alex J. Best, Nov 20 2012
For n > 0, a(n) is a B_2 sequence. - Thomas Ordowski, Sep 23 2014
See coefficients of the linear terms of the polynomials of the table on p. 10 of the Getzler link. - Tom Copeland, Mar 24 2016
Consider n points lying on a circle, then for n>=2 a(n-1) is the maximum number of ways to connect two points with non-intersecting chords. - Anton Zakharov, Dec 31 2016
Also the number of cliques in the (n-1)-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
From Eric M. Schmidt, Jul 17 2017: (Start)
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) < e(k). [Martinez and Savage, 2.7]
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i), e(j), e(k) pairwise distinct. [Martinez and Savage, 2.7]
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(j) >= e(k) and e(i) != e(k) pairwise distinct. [Martinez and Savage, 2.7]
(End)
Number of F-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are F-equivalent iff the positions of pattern F are identical in these paths. -  Sergey Kirgizov, Apr 08 2018
From Gus Wiseman, Feb 10 2019: (Start)
Also the number of connected partitions of an n-cycle. For example, the a(1) = 1 through a(4) = 12 connected partitions are:
  {{1}}  {{12}}    {{123}}      {{1234}}
         {{1}{2}}  {{1}{23}}    {{1}{234}}
                   {{12}{3}}    {{12}{34}}
                   {{13}{2}}    {{123}{4}}
                   {{1}{2}{3}}  {{124}{3}}
                                {{134}{2}}
                                {{14}{23}}
                                {{1}{2}{34}}
                                {{1}{23}{4}}
                                {{12}{3}{4}}
                                {{14}{2}{3}}
                                {{1}{2}{3}{4}}
(End)
Number of subsets of n-set without the single-element subsets. - Yuchun Ji, Jul 16 2019
For every prime p, there are infinitely many terms of this sequence that are divisible by p (see IMO Compendium link and Doob reference). Corresponding indices n are: for p = 2, even numbers A299174; for p = 3, A047257; for p = 5, A349767. - Bernard Schott, Dec 10 2021
Primes are in A081296 and corresponding indices in A048744. - Bernard Schott, Dec 12 2021

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 27*x^5 + 58*x^6 + 121*x^7 + ...

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 4, 1983, page 158, 1993.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Kassie Archer and Aaron Geary, Descents in powers of permutations, arXiv:2406.09369 [math.CO], 2024.
Cory B. H. Ball, The Apprentices' Tower of Hanoi, Electronic Theses and Dissertations, East Tennessee State University, Paper 2512, 2015.
Jean-Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178;
Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.
David Callan, Pattern avoidance in circular permutations, arXiv:math/0210014 [math.CO], 2002.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
Robert DeSario and LeRoy Wenstrom, Invertible shuffles, Problem 10931, Amer. Math. Monthly, 111 (No. 2, 2004), 169-170.
Askar Dzhumadil'daev and Pasha Zusmanovich, The alternative operad is not Koszul, arXiv:0906.1272 [math.RA], 2009-2013.
Marty Getz, Dixon Jones and Ken Dutch, Partitioning by Arithmetic Progressions: Problem 11005, American Math. Monthly, Vol. 112, 2005, p. 89. (The published solution is incomplete. Letting d be the common difference of the arithmetic progressions, the solver's expression q_1(n,d)=2^(n-d) must be summed over all d=1,...,n and duplicate partitions must be removed.)
E. Getzler, The semi-classical approximation for modular operads, arXiv:alg-geom/9612005, 1996.
Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, arXiv:2112.03338 [math.CO], 2021.
Juan B. Gil and Michael D. Weiner, On pattern-avoiding Fishburn permutations, arXiv:1812.01682 [math.CO], 2018.
The IMO Compendium, Problem 4, 15th Canadian Mathematical Olympiad 1983.
R. Kehinde, S. O. Makanjuola and A. Umar, On the semigroup of order-decreasing partial isometries of a finite chain, arXiv:1101.2558 [math.GR], 2011.
Alain Lascoux and Marcel-Paul Schützenberger, Schubert polynomials and the Littlewood Richardson rule, Letters in Math. Physics 10 (1985) 111-124.
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Eric Weisstein's World of Mathematics, Chromatic Invariant
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Moebius Ladder
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 (4,-5,2).
Index to sequences related to Olympiads.

Column 1 of triangle A008518.
Row sum of triangles A184049 and A184050.
Cf. A000108, A002064, A133116, A160692, A005803, A006127, A186947.
Cf. A001610, A066982, A323950, A323951.
Cf. A047257, A299174, A349767.
Cf.  A048744, A081296.

a000325 n = 2 ^ n - n
a000325_list = zipWith (-) a000079_list [0..]
-- Reinhard Zumkeller, Jul 17 2012

[2^n - n: n in [0..35]]; // Vincenzo Librandi, May 13 2011

A000325 := proc(n) option remember; if n <=1 then n+1 else 2*A000325(n-1)+n-1; fi; end;
g:=1/(1-2*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-n, n=0..31); # Zerinvary Lajos, Jan 09 2009

Table[2^n - n, {n, 0, 39}] (* Alonso del Arte, Sep 15 2014 *)
LinearRecurrence[{4, -5, 2}, {1, 2, 5}, {0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)

{a(n) = 2^n - n}; /* Michael Somos, Nov 04 2006 */

def A000325(n): return (1<Chai Wah Wu, Jan 11 2023

a(n+1) = 2*a(n) + n - 1, a(0) = 1. - Reinhard Zumkeller, Apr 12 2003
Binomial transform of 1, 0, 1, 1, 1, .... The sequence starting 1, 2, 5, ... has a(n) = 1 + n + 2*Sum_{k=2..n} binomial(n, k) = 2^(n+1) - n - 1. This is the binomial transform of 1, 1, 2, 2, 2, 2, .... a(n) = 1 + Sum_{k=2..n} C(n, k). - Paul Barry, Jun 06 2003
G.f.: (1-3x+3x^2)/((1-2x)*(1-x)^2). - Emeric Deutsch, Feb 22 2004
A107907(a(n+2)) = A000051(n+2) for n > 0. - Reinhard Zumkeller, May 28 2005
a(n+1) = sum of n-th row of the triangle in A109128. - Reinhard Zumkeller, Jun 20 2005
Row sums of triangle A133116. - Gary W. Adamson, Sep 14 2007
G.f.: 1 / (1 - x / (1 - x / ( 1 - x / (1 + x / (1 - 2*x))))). - Michael Somos, May 12 2012
First difference is A000225. PSUM transform is A084634. - Michael Somos, May 12 2012
a(n) = [x^n](B(x)^n-B(x)^(n-1)), n>0, a(0)=1, where B(x) = (1+2*x+sqrt(1+4*x^2))/2. - Vladimir Kruchinin, Mar 07 2014
E.g.f.: (exp(x) - x)*exp(x). - Ilya Gutkovskiy, Aug 07 2016
a(n) = A125128(n) - A000225(n) + 1. - Miquel Cerda, Aug 12 2016
a(n) = 2*A125128(n) - A095151(n) + 1. - Miquel Cerda, Aug 12 2016
a(n) = A079583(n-1) - A000225(n-1). - Miquel Cerda, Aug 15 2016
a(n)^2 - 4*a(n-1)^2 = (n-2)*(a(n)+2*a(n-1)). - Yuchun Ji, Jul 13 2018
a(n) = 2^(-n) * A186947(n) = 2^n * A002064(-n) for all n in Z. - Michael Somos, Jul 18 2018
a(2^n) = (2^a(n) - 1)*2^n. - Lorenzo Sauras Altuzarra, Feb 01 2022

A098558 Expansion of e.g.f. (1+x)/(1-x).

Original entry on oeis.org

1, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600, 174356582400, 2615348736000, 41845579776000, 711374856192000, 12804747411456000, 243290200817664000, 4865804016353280000, 102181884343418880000, 2248001455555215360000

Offset: 0

Paul Barry, Sep 14 2004

Essentially the same as A052849: a(0)=0 and a(n) = A052849(n) for n>=1.
Equals row sums (unsigned) of triangle A158471. - Gary W. Adamson, Mar 20 2009
Also the number of graceful labelings of the star graph on n+1 nodes. - Eric W. Weisstein, Mar 31 2020

G. C. Greubel, Table of n, a(n) for n = 0..445
Eric Weisstein's World of Mathematics, Graceful Labeling
Eric Weisstein's World of Mathematics, Star Graph

Cf. A000007, A000142, A008290.
Row sums of A008518 and of A128564.
Cf. A158471.

[1] cat [2*Factorial(n): n in [1..30]]; // G. C. Greubel, Jan 17 2018

Join[{1}, 2*Range[30]!] (* G. C. Greubel, Jan 17 2018 *)
With[{nn=30},CoefficientList[Series[(1+x)/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 05 2021 *)
a[n_] := Hypergeometric2F1Regularized[1, -n, 2 - n, -1];
Table[a[n], {n, 0, 22}]  (* Peter Luschny, Apr 26 2024 *)

concat([1], vector(30, n, 2*n!)) \\ G. C. Greubel, Jan 17 2018

CF = ComplexBallField(100)
def a(n):
    return Integer(CF(-1).hypergeometric([1, -n], [2 - n], regularized=True))
print([a(n) for n in range(23)]) # Peter Luschny, Apr 26 2024

a(n) = 2*n! - 0^n.
a(n) = Sum_{k=0..n} (k+1) * A008290(n,k). - Alois P. Heinz, Mar 11 2022
Sum_{n>=0} 1/a(n) = (e+1)/2. - Amiram Eldar, Feb 02 2023
a(n) = HypergeomRegularized([1, -n], [2 - n], -1). - Peter Luschny, Apr 26 2024

A177042 Eulerian version of the Catalan numbers, a(n) = A008292(2*n+1,n+1)/(n+1).

Original entry on oeis.org

1, 2, 22, 604, 31238, 2620708, 325024572, 55942352184, 12765597850950, 3730771315561300, 1359124435588313876, 603916464771468176392, 321511316149669476991132, 202039976682357297272094824, 147980747895225006590333244088, 124963193751534047864734415925360

Offset: 0

Roger L. Bagula, May 01 2010

nonn

According to the Bidkhori and Sullivant reference's abstract, authors show "that the Eulerian-Catalan numbers enumerate Dyck permutations, [providing] two proofs for this fact, the first using the geometry of alcoved polytopes and the second a direct combinatorial proof via an Eulerian-Catalan analog of the Chung-Feller theorem." - Jonathan Vos Post, Jan 07 2011

Twice the number of permutations of {1,2,...,2n} with n ascents. - Peter Luschny, Jan 11 2011

Alois P. Heinz, Table of n, a(n) for n = 0..220
F. Ardila, The Catalan matroid, J. Combin. Theory Ser. A 104 (2003) 49-62.
Hoda Bidkhori and Seth Sullivant, Eulerian-Catalan Numbers, arXiv:1101.1108 [math.CO], 2011.
Digital Library of Mathematical Functions, Table 26.14.1

Cf. A000108, A008518, A025585, A180056.
Bisection (odd part) of A303287.
Row sums of A316728.

A177042:=func< n | n eq 0 select 1 else 2*(&+[(-1)^k*Binomial(2*n+1,k)*(n-k+1)^(2*n): k in [0..n]]) >;
[A177042(n): n in [0..40]]; // G. C. Greubel, Jun 18 2024

A177042 := proc(n) A008292(2*n+1,n+1)/(n+1) ; end proc:
seq(A177042(n),n=0..10) ; # R. J. Mathar, Jan 08 2011
A177042 := n -> A025585(n+1)/(n+1):
A177042 := n -> `if`(n=0,1,2*A180056(n)):
# The A173018-based recursion below needs no division!
A := proc(n, k) option remember;
       if n = 0 and k = 0 then 1
     elif k > n  or k < 0 then 0
     else (n-k) *A(n-1, k-1) +(k+1) *A(n-1, k)
       fi
     end:
A177042 := n-> `if`(n=0, 1, 2*A(2*n, n)):
seq(A177042(n), n=0..30);
# Peter Luschny, Jan 11 2011

<< DiscreteMath`Combinatorica`
Table[(Eulerian[2*n + 1, n])/(n + 1), {n, 0, 20}]
(* Second program: *)
A[n_, k_] := A[n, k] = Which[n == 0 && k == 0, 1, k > n || k < 0, 0, True, (n - k)*A[n - 1, k - 1] + (k + 1)*A[n - 1, k]]; A177042[n_] := If[n == 0, 1, 2*A[2*n, n]]; Table[A177042[n], {n, 0, 30}] (* Jean-François Alcover, Jul 13 2017, after Peter Luschny *)

def A177042(n): return 2*sum((-1)^k*binomial(2*n+1,k)*(n-k+1)^(2*n) for k in range(n+1)) - int(n==0)
[A177042(n) for n in range(41)] # G. C. Greubel, Jun 18 2024

a(n) = 2*A180056(n) for n > 0, A180056 the central Eulerian numbers in the sense of A173018.
a(n) = A025585(n+1)/(n+1), A025585 the central Eulerian numbers in the sense of A008292.
a(n) = 2 Sum_{k=0..n} (-1)^k binomial(2n+1,k) (n-k+1)^(2n).
a(n) = (n+1)^(-1) Sum_{k=0..n} (-1)^k binomial(2n+2,k)(n+1-k)^(2n+1). - Peter Luschny, Jan 11 2011
a(n) = A008518(2n,n). - Alois P. Heinz, Jun 12 2017
From Alois P. Heinz, Jul 21 2018: (Start)
a(n) = (2n)! * [x^(2n) y^n] (exp(x)-y*exp(y*x))/(exp(y*x)-y*exp(x)).
a(n) = (2n+1)!/(n+1) * [x^(2n+1) y^(n+1)] (1-y)/(1-y*exp((1-y)*x)). (End)

Edited by Alois P. Heinz, Jan 14 2011

A174599 Triangle T(n, k) = A154646(n,k) - A154646(n,0) + 1, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 22, 1, 1, 145, 145, 1, 1, 780, 2246, 780, 1, 1, 3919, 25144, 25144, 3919, 1, 1, 19202, 243047, 524812, 243047, 19202, 1, 1, 93349, 2168107, 8760511, 8760511, 2168107, 93349, 1, 1, 453592, 18445564, 127880680, 235517062, 127880680, 18445564, 453592, 1

Offset: 0

Roger L. Bagula, Mar 23 2010

The first and last element of each row of A154646 are reduced to 1 by subtracting a constant from each row.

			Triangle begins as:
  1;
  1,     1;
  1,    22,       1;
  1,   145,     145,       1;
  1,   780,    2246,     780,       1;
  1,  3919,   25144,   25144,    3919,       1;
  1, 19202,  243047,  524812,  243047,   19202,     1;
  1, 93349, 2168107, 8760511, 8760511, 2168107, 93349,    1;

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

Related triangles dependent on q: A008518 (q=1), A176198 (q=2), this sequence (q=3), A176199 (q=4).

Cf. A154646.

m:=13;
R:=PowerSeriesRing(Integers(), m+2);
p:= func< x,n,q | (1-x)^(n+1)*(&+[((q*j+1)^n + (q*(j+1)-1)^n)*x^j: j in [0..m+2]]) >;
f:= func< n,k,q | Coefficient(R!( p(x,n,q) ), k) >;
T:= func< n,k,q | f(n,k,q) -f(n,0,q) +1 >; // T = A174599
[T(n,k,3): k in [0..n], n in [0..m]]; // G. C. Greubel, Jun 18 2024

m:=13;
p[x_,n_,q_]:= (1-x)^(n+1)*Sum[((q*j+1)^n+(q*(j+1)-1)^n)*x^j, {j,0,m+2}];
f[n_,k_,q_]:= Coefficient[Series[p[x,n,q], {x,0,m+2}], x, k];
T[n_,k_,q_]:= f[n,k,q] -f[n,0,q] +1;
Table[T[n,k,3], {n,0,m}, {k,0,n}]//Flatten

m=13
def p(x,n,q): return (1-x)^(n+1)*sum(((q*j+1)^n + (q*(j+1)-1)^n)*x^j for j in range(m+3))
def f(n,k,q): return ( p(x,n,q) ).series(x, n+1).list()[k]
def T(n,k,q): return f(n,k,q) - f(n,0,q) + 1 # T = A174599
flatten([[T(n,k,3) for k in range(n+1)] for n in (0..m)]) # G. C. Greubel, Jun 18 2024

From G. C. Greubel, Jun 18 2024: (Start)
T(n, k) = f(n, k, q) - f(n, 0, q) + 1, where f(n, k, q) = [x^k]( p(x, n, q) ), p(x, n, q) = (1-x)^(n+1) * Sum_{k >= 0} ( (q*k + 1)^n + (q*(k+1) - 1)^n )*x^k, and q = 3.
T(n, k) f(n, k, q) - f(n, 0, q) + 1, where f(n, k, q) = [x^k]( p(x, n, q) ), p(x, n, q) = q^n * (1-x)^(n+1) * ( LerchPhi(x, -n, 1/q) + LerchPhi(x, -n, (q-1)/q) ), and q = 3.
T(n, k) = A154646(n,k) - A154646(n,0) + 1.
T(n, n-k) = T(n, k). (End)

A176198 Triangle, read by rows, T(n, k) = f(n,k,q) - f(n,0,q) + 1, where f(n, k, q) = [x^k](p(x,n,q)), p(x, n, q) = (1-x)^(n+1)Sum_{k >= 0} ( (qk+1)^n + (q(k+1)-1)^n )x^k, and q = 2.

Original entry on oeis.org

1, 1, 1, 1, 11, 1, 1, 45, 45, 1, 1, 151, 459, 151, 1, 1, 473, 3363, 3363, 473, 1, 1, 1443, 21085, 47095, 21085, 1443, 1, 1, 4357, 121313, 519445, 519445, 121313, 4357, 1, 1, 13103, 663223, 4970575, 9350027, 4970575, 663223, 13103, 1, 1, 39345, 3512679, 43415943, 138826587, 138826587, 43415943, 3512679, 39345, 1

Offset: 0

Roger L. Bagula, Apr 11 2010

			Triangle begins as:
  1;
  1,     1;
  1,    11,      1;
  1,    45,     45,       1;
  1,   151,    459,     151,       1;
  1,   473,   3363,    3363,     473,       1;
  1,  1443,  21085,   47095,   21085,    1443,      1;
  1,  4357, 121313,  519445,  519445,  121313,   4357,     1;
  1, 13103, 663223, 4970575, 9350027, 4970575, 663223, 13103,     1;

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

Related triangles dependent on q: A008518 (q=1), this sequence (q=2), A174599 (q=3), A176199 (q=4).

m:=13;
R:=PowerSeriesRing(Integers(), m+2);
p:= func< x,n,q | (1-x)^(n+1)*(&+[((q*j+1)^n + (q*(j+1)-1)^n)*x^j: j in [0..m+2]]) >;
f:= func< n,k,q | Coefficient(R!( p(x,n,q) ), k) >;
T:= func< n,k,q | f(n,k,q) - f(n,0,q) + 1 >; // T = A176198
[T(n,k,2): k in [0..n], n in [0..m]]; // G. C. Greubel, Jun 18 2024

m:=13;
p[x_,n_,q_]:= (1-x)^(n+1)*Sum[((q*j+1)^n+(q*(j+1)-1)^n)*x^j, {j,0,m+ 2}];
f[n_,k_,q_]:= Coefficient[Series[p[x,n,q], {x,0,m+2}], x, k];
T[n_,k_,q_]:= f[n,k,q] - f[n,0,q] + 1; (* T = A176198 *)
Table[T[n,k,2], {n,0,m}, {k,0,n}]//Flatten

m=13
def p(x,n,q): return (1-x)^(n+1)*sum(((q*j+1)^n + (q*(j+1)-1)^n)*x^j for j in range(m+3))
def f(n,k,q): return ( p(x,n,q) ).series(x, n+1).list()[k]
def T(n,k,q): return f(n,k,q) - f(n,0,q) + 1 # T = A176198
flatten([[T(n,k,2) for k in range(n+1)] for n in (0..m)]) # G. C. Greubel, Jun 18 2024

T(n, k) = f(n, k, q) - f(n, 0, q) + 1, where f(n, k, q) = [x^k]( p(x, n, q) ), p(x, n, q) = (1-x)^(n+1) * Sum_{k >= 0} ( (q*k + 1)^n + (q*(k+1) - 1)^n )*x^k, and q = 2.
T(n, k) f(n, k, q) - f(n, 0, q) + 1, where f(n, k, q) = [x^k]( p(x, n, q) ), p(x, n, q) = q^n * (1-x)^(n+1) * ( LerchPhi(x, -n, 1/q) + LerchPhi(x, -n, (q-1)/q) ), and q = 2.
T(n, n-k) = T(n, k).

Edited by G. C. Greubel, Jun 19 2024

A176199 Triangle, read by rows, T(n, k) = f(n,k,q) - f(n,0,q) + 1, where f(n, k, q) = [x^k](p(x,n,q)), p(x, n, q) = (1-x)^(n+1)Sum_{k >= 0} ( (qk+1)^n + (q(k+1)-1)^n )x^k, and q = 4.

Original entry on oeis.org

1, 1, 1, 1, 35, 1, 1, 329, 329, 1, 1, 2535, 6811, 2535, 1, 1, 18225, 103925, 103925, 18225, 1, 1, 127435, 1384685, 2868895, 1384685, 127435, 1, 1, 881977, 17115873, 64568761, 64568761, 17115873, 881977, 1, 1, 6089807, 202236439, 1283008495, 2302094507, 1283008495, 202236439, 6089807, 1

Offset: 0

Roger L. Bagula, Apr 11 2010

			Triangle begins as:
  1;
  1,      1;
  1,     35,        1;
  1,    329,      329,        1;
  1,   2535,     6811,     2535,        1;
  1,  18225,   103925,   103925,    18225,        1;
  1, 127435,  1384685,  2868895,  1384685,   127435,      1;
  1, 881977, 17115873, 64568761, 64568761, 17115873, 881977,     1;

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

Related triangles dependent on q: A008518 (q=1), A176198 (q=2), A174599 (q=3), this sequence (q=4).

m:=13;
R:=PowerSeriesRing(Integers(), m+2);
p:= func< x,n,q | (1-x)^(n+1)*(&+[((q*j+1)^n + (q*(j+1)-1)^n)*x^j: j in [0..m+2]]) >;
f:= func< n,k,q | Coefficient(R!( p(x,n,q) ), k) >;
T:= func< n,k,q | f(n,k,q) - f(n,0,q) + 1 >; // T = A176199
[T(n,k,4): k in [0..n], n in [0..m]]; // G. C. Greubel, Jun 18 2024

m:=13;
p[x_,n_,q_]:= (1-x)^(n+1)*Sum[((q*j+1)^n+(q*(j+1)-1)^n)*x^j, {j,0,m+ 2}];
f[n_,k_,q_]:= Coefficient[Series[p[x,n,q], {x,0,m+2}], x, k];
T[n_,k_,q_]:= f[n,k,q] - f[n,0,q] + 1;
Table[T[n,k,4], {n,0,m}, {k,0,n}]//Flatten

m=13
def p(x,n,q): return (1-x)^(n+1)*sum(((q*j+1)^n + (q*(j+1)-1)^n)*x^j for j in range(m+3))
def f(n,k,q): return ( p(x,n,q) ).series(x, n+1).list()[k]
def T(n,k,q): return f(n,k,q) - f(n,0,q) + 1 # T = A176199
flatten([[T(n,k,4) for k in range(n+1)] for n in (0..m)]) # G. C. Greubel, Jun 18 2024

T(n, k) = f(n, k, q) - f(n, 0, q) + 1, where f(n, k, q) = [x^k]( p(x, n, q) ), p(x, n, q) = (1-x)^(n+1) * Sum_{k >= 0} ( (q*k + 1)^n + (q*(k+1) - 1)^n )*x^k, and q = 4.
T(n, k) f(n, k, q) - f(n, 0, q) + 1, where f(n, k, q) = [x^k]( p(x, n, q) ), p(x, n, q) = q^n * (1-x)^(n+1) * ( LerchPhi(x, -n, 1/q) + LerchPhi(x, -n, (q-1)/q) ), and q = 4.
T(n, n-k) = T(n, k).

Edited by G. C. Greubel, Jun 18 2024

cp's OEIS Frontend

A168291 T(n,k) = 4*A046802(n+1,k+1) - 2*A008518(n,k) - A007318(n,k), triangle read by rows (0 <= k <= n).

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A168292 T(n,k) = 24*A046802(n+1,k+1) - 9*A008518(n,k) - 8*A007318(n,k), triangle read by rows (0 <= k <= n).

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A168293 T(n,k) = 12*A046802(n+1,k+1) - 9*A008518(n,k) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).

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A000325 a(n) = 2^n - n.

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Haskell

Magma

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A098558 Expansion of e.g.f. (1+x)/(1-x).

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A177042 Eulerian version of the Catalan numbers, a(n) = A008292(2*n+1,n+1)/(n+1).

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A174599 Triangle T(n, k) = A154646(n,k) - A154646(n,0) + 1, 0 <= k <= n.

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A168291 T(n,k) = 4A046802(n+1,k+1) - 2A008518(n,k) - A007318(n,k), triangle read by rows (0 <= k <= n).

A168292 T(n,k) = 24A046802(n+1,k+1) - 9A008518(n,k) - 8*A007318(n,k), triangle read by rows (0 <= k <= n).

A168293 T(n,k) = 12A046802(n+1,k+1) - 9A008518(n,k) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).

A176198 Triangle, read by rows, T(n, k) = f(n,k,q) - f(n,0,q) + 1, where f(n, k, q) = [x^k](p(x,n,q)), p(x, n, q) = (1-x)^(n+1)Sum_{k >= 0} ( (qk+1)^n + (q(k+1)-1)^n )x^k, and q = 2.

A176199 Triangle, read by rows, T(n, k) = f(n,k,q) - f(n,0,q) + 1, where f(n, k, q) = [x^k](p(x,n,q)), p(x, n, q) = (1-x)^(n+1)Sum_{k >= 0} ( (qk+1)^n + (q(k+1)-1)^n )x^k, and q = 4.