A160232 -id:A160232 - cp's OEIS frontend

A011782 Coefficients of expansion of (1-x)/(1-2*x) in powers of x.

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Lee D. Killough (killough(AT)wagner.convex.com)

Apart from initial term, same as A000079 (powers of 2).
Number of compositions (ordered partitions) of n. - Toby Bartels, Aug 27 2003
Number of ways of putting n unlabeled items into (any number of) labeled boxes where every box contains at least one item. Also "unimodal permutations of n items", i.e., those which rise then fall. (E.g., for three items: ABC, ACB, BCA and CBA are unimodal.) - Henry Bottomley, Jan 17 2001
Number of permutations in S_n avoiding the patterns 213 and 312. - Tuwani Albert Tshifhumulo, Apr 20 2001. More generally (see Simion and Schmidt), the number of permutations in S_n avoiding (i) the 123 and 132 patterns; (ii) the 123 and 213 patterns; (iii) the 132 and 213 patterns; (iv) the 132 and 231 patterns; (v) the 132 and 312 patterns; (vi) the 213 and 231 patterns; (vii) the 213 and 312 patterns; (viii) the 231 and 312 patterns; (ix) the 231 and 321 patterns; (x) the 312 and 321 patterns.
a(n+2) is the number of distinct Boolean functions of n variables under action of symmetric group.
Number of unlabeled (1+2)-free posets. - Detlef Pauly, May 25 2003
Image of the central binomial coefficients A000984 under the Riordan array ((1-x), x*(1-x)). - Paul Barry, Mar 18 2005
Binomial transform of (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...); inverse binomial transform of A007051. - Philippe Deléham, Jul 04 2005
Also, number of rationals in [0, 1) whose binary expansions terminate after n bits. - Brad Chalfan, May 29 2006
Equals row sums of triangle A144157. - Gary W. Adamson, Sep 12 2008
Prepend A089067 with a 1, getting (1, 1, 3, 5, 13, 23, 51, ...) as polcoeff A(x); then (1, 1, 2, 4, 8, 16, ...) = A(x)/A(x^2). - Gary W. Adamson, Feb 18 2010
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 2, 8, 32 and 128, lead to this sequence. For the corner squares these vectors lead to the companion sequence A094373. - Johannes W. Meijer, Aug 15 2010
From Paul Curtz, Jul 20 2011: (Start)
Array T(m,n) = 2*T(m,n-1) + T(m-1,n):
  1, 1,  2,   4,   8,   16, ... = a(n)
  1, 3,  8,  20,  48,  112, ... = A001792,
  1, 5, 18,  56, 160,  432, ... = A001793,
  1, 7, 32, 120, 400, 1232, ... = A001794,
  1, 9, 50, 220, 840, 2912, ... = A006974, followed with A006975, A006976, gives nonzero coefficients of Chebyshev polynomials of first kind A039991 =
  1,
  1, 0,
  2, 0, -1,
  4, 0, -3, 0,
  8, 0, -8, 0, 1.
T(m,n) third vertical: 2*n^2, n positive (A001105).
Fourth vertical appears in Janet table even rows, last vertical (A168342 array, A138509, rank 3, 13, = A166911)). (End)
A131577(n) and differences are:
  0, 1, 2, 4, 8, 16,
  1, 1, 2, 4, 8, 16,  = a(n),
  0, 1, 2, 4, 8, 16,
  1, 1, 2, 4, 8, 16.
Number of 2-color necklaces of length 2n equal to their complemented reversal. For length 2n+1, the number is 0. - David W. Wilson, Jan 01 2012
Edges and also central terms of triangle A198069: a(0) = A198069(0,0) and for n > 0: a(n) = A198069(n,0) = A198069(n,2^n) = A198069(n,2^(n-1)). - Reinhard Zumkeller, May 26 2013
These could be called the composition numbers (see the second comment) since the equivalent sequence for partitions is A000041, the partition numbers. - Omar E. Pol, Aug 28 2013
Number of self conjugate integer partitions with exactly n parts for n>=1. - David Christopher, Aug 18 2014
The sequence is the INVERT transform of (1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...). - Gary W. Adamson, Jul 16 2015
Number of threshold graphs on n nodes [Hougardy]. - Falk Hüffner, Dec 03 2015
Number of ternary words of length n in which binary subwords appear in the form 10...0. - Milan Janjic, Jan 25 2017
a(n) is the number of words of length n over an alphabet of two letters, of which one letter appears an even number of times (the empty word of length 0 is included). See the analogous odd number case in A131577, and the Balakrishnan reference in A006516 (the 4-letter odd case), pp. 68-69, problems 2.66, 2.67 and 2.68. - Wolfdieter Lang, Jul 17 2017
Number of D-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are D-equivalent iff the positions of pattern D are identical in these paths. - Sergey Kirgizov, Apr 08 2018
Number of color patterns (set partitions) for an oriented row of length n using two or fewer colors (subsets). Two color patterns are equivalent if we permute the colors. For a(4)=8, the 4 achiral patterns are AAAA, AABB, ABAB, and ABBA; the 4 chiral patterns are the 2 pairs AAAB-ABBB and AABA-ABAA. - Robert A. Russell, Oct 30 2018
The determinant of the symmetric n X n matrix M defined by M(i,j) = (-1)^max(i,j) for 1 <= i,j <= n is equal to a(n) * (-1)^(n*(n+1)/2). - Bernard Schott, Dec 29 2018
For n>=1, a(n) is the number of permutations of length n whose cyclic representations can be written in such a way that when the cycle parentheses are removed what remains is 1 through n in natural order. For example, a(4)=8 since there are exactly 8 permutations of this form, namely, (1 2 3 4), (1)(2 3 4), (1 2)(3 4), (1 2 3)(4), (1)(2)(3 4), (1)(2 3)(4), (1 2)(3)(4), and (1)(2)(3)(4). Our result follows readily by conditioning on k, the number of parentheses pairs of the form ")(" in the cyclic representation. Since there are C(n-1,k) ways to insert these in the cyclic representation and since k runs from 0 to n-1, we obtain a(n) = Sum_{k=0..n-1} C(n-1,k) = 2^(n-1). - Dennis P. Walsh, May 23 2020
Maximum number of preimages that a permutation of length n + 1 can have under the consecutive-231-avoiding stack-sorting map. - Colin Defant, Aug 28 2020
a(n) is the number of occurrences of the empty set {} in the von Neumann ordinals from 0 up to n. Each ordinal k is defined as the set of all smaller ordinals: 0 = {}, 1 = {0}, 2 = {0,1}, etc. Since {} is the foundational element of all ordinals, the total number of times it appears grows as powers of 2. - Kyle Wyonch, Mar 30 2025

			G.f. = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 32*x^6 + 64*x^7 + 128*x^8 + ...
    ( -1   1  -1)
det (  1   1   1)  = 4
    ( -1  -1  -1)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.
S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7
Xavier Merlin, Methodix Algèbre, Ellipses, 1995, p. 153.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Christopher Bao, Yunseo Choi, Katelyn Gan, and Owen Zhang, On a Conjecture by Baril, Cerbai, Khalil, and Vajnovszki on Two Restricted Stacks, arXiv:2308.09344 [math.CO], 2023.
Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, arXiv:1803.06706 [math.CO], 2018.
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Christian Bean, Bjarki Gudmundsson, and Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.
Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, Pattern-avoiding stabilized-interval-free permutations, arXiv:2306.03155 [math.CO], 2023.
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 14.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015.
Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
Colin Defant and Kai Zheng, Stack-sorting with consecutive-pattern-avoiding stacks, arXiv:2008.12297 [math.CO], 2020.
John B. Dobson, A matrix variation on Ramus's identity for lacunary sums of binomial coefficients, arXiv preprint arXiv:1610.09361 [math.NT], 2016.
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Remark 10.
Juan B. Gil and Jessica A. Tomasko, Fibonacci colored compositions and applications, arXiv:2108.06462 [math.CO], 2021.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 25.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Mats Granvik, Alternating powers of 2 as convoluted divisor recurrence
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Nickolas Hein and Jia Huang, Modular Catalan Numbers, arXiv:1508.01688 [math.CO], 2015-2016. See Table 1.1 p. 2.
S. Heubach and T. Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (Corollary 3, case k=2, pages 10-11). - From _N. J. A. Sloane_, May 09 2012
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. Also on arXiv, arXiv:1302.2274 [math.CO], 2013.
Olivia Nabawanda and Fanja Rakotondrajao, The sets of flattened partitions with forbidden patterns, arXiv:2011.07304 [math.CO], 2020.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
L. Pudwell, Pattern avoidance in trees (slides from a talk, mentions many sequences), 2012. - From _N. J. A. Sloane_, Jan 03 2013
Santiago Rojas-Rojas, Camila Muñoz, Edgar Barriga, Pablo Solano, Aldo Delgado, and Carla Hermann-Avigliano, Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation, arXiv:2310.12366 [quant-ph], 2023. See p. 12.
R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6, 383-406, 1985, see pp. 392-393.
Christoph Wernhard and Wolfgang Bibel, Learning from Łukasiewicz and Meredith: Investigations into Proof Structures (Extended Version), arXiv:2104.13645 [cs.AI], 2021.
Yan X. Zhang, Four Variations on Graded Posets, arXiv preprint arXiv:1508.00318 [math.CO], 2015.
Index entries for sequences related to Boolean functions
Index to divisibility sequences
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (2).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000670, A051486, A053120, A057711, A080929, A080961, A082138.
Cf. A082139, A082140, A082141, A089067, A144157, A131577, A211216.
Sequences with g.f.'s of the form ((1-x)/(1-2*x))^k: this sequence (k=1), A045623 (k=2), A058396 (k=3), A062109 (k=4), A169792 (k=5), A169793 (k=6), A169794 (k=7), A169795 (k=8), A169796 (k=9), A169797 (k=10).
Cf. A005418 (unoriented), A122746(n-3) (chiral), A016116 (achiral).
Row sums of triangle A100257.
A row of A160232.
Row 2 of A278984.

a011782 n = a011782_list !! n
a011782_list = 1 : scanl1 (+) a011782_list
-- Reinhard Zumkeller, Jul 21 2013

[Floor((1+2^n)/2): n in [0..35]]; // Vincenzo Librandi, Aug 21 2011

A011782:= n-> ceil(2^(n-1)): seq(A011782(n), n=0..50); # Wesley Ivan Hurt, Feb 21 2015
with(PolynomialTools):  A011782:=seq(coeftayl((1-x)/(1-2*x), x = 0, k),k=0..10^2); # Muniru A Asiru, Sep 26 2017

f[s_] := Append[s, Ceiling[Plus @@ s]]; Nest[f, {1}, 32] (* Robert G. Wilson v, Jul 07 2006 *)
CoefficientList[ Series[(1-x)/(1-2x), {x, 0, 32}], x] (* Robert G. Wilson v, Jul 07 2006 *)
Table[Sum[StirlingS2[n, k], {k,0,2}], {n, 0, 30}] (* Robert A. Russell, Apr 25 2018 *)
Join[{1},NestList[2#&,1,40]] (* Harvey P. Dale, Dec 06 2018 *)

PARI
```
{a(n) = if( n<1, n==0, 2^(n-1))};
```

Vec((1-x)/(1-2*x) + O(x^30)) \\ Altug Alkan, Oct 31 2015

def A011782(n): return 1 if n == 0 else 2**(n-1) # Chai Wah Wu, May 11 2022

[sum(stirling_number2(n,j) for j in (0..2)) for n in (0..35)] # G. C. Greubel, Jun 02 2020

a(0) = 1, a(n) = 2^(n-1).
G.f.: (1 - x) / (1 - 2*x) = 1 / (1 - x / (1 - x)). - Michael Somos, Apr 18 2012
E.g.f.: cosh(z)*exp(z) = (exp(2*z) + 1)/2.
a(0) = 1 and for n>0, a(n) = sum of all previous terms.
a(n) = Sum_{k=0..n} binomial(n, 2*k). - Paul Barry, Feb 25 2003
a(n) = Sum_{k=0..n} binomial(n,k)*(1+(-1)^k)/2. - Paul Barry, May 27 2003
a(n) = floor((1+2^n)/2). - Toby Bartels (toby+sloane(AT)math.ucr.edu), Aug 27 2003
G.f.: Sum_{i>=0} x^i/(1-x)^i. - Jon Perry, Jul 10 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k+1, n-k)*binomial(2*k, k). - Paul Barry, Mar 18 2005
a(n) = Sum_{k=0..floor(n/2)} A055830(n-k, k). - Philippe Deléham, Oct 22 2006
a(n) = Sum_{k=0..n} A098158(n,k). - Philippe Deléham, Dec 04 2006
G.f.: 1/(1 - (x + x^2 + x^3 + ...)). - Geoffrey Critzer, Aug 30 2008
a(n) = A000079(n) - A131577(n).
a(n) = A173921(A000079(n)). - Reinhard Zumkeller, Mar 04 2010
a(n) = Sum_{k=2^n..2^(n+1)-1} A093873(k)/A093875(k), sums of rows of the full tree of Kepler's harmonic fractions. - Reinhard Zumkeller, Oct 17 2010
E.g.f.: (exp(2*x)+1)/2 = (G(0) + 1)/2; G(k) = 1 + 2*x/(2*k+1 - x*(2*k+1)/(x + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2011
A051049(n) = 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, Apr 18 2012
A008619(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, Apr 18 2012
INVERT transform is A122367. MOBIUS transform is A123707. EULER transform of A059966. PSUM transform is A000079. PSUMSIGN transform is A078008. BINOMIAL transform is A007051. REVERT transform is A105523. A002866(n) = a(n)*n!. - Michael Somos, Apr 18 2012
G.f.: U(0), where U(k) = 1 + x*(k+3) - x*(k+2)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012
a(n) = A000041(n) + A056823(n). - Omar E. Pol, Aug 31 2013
E.g.f.: E(0), where E(k) = 1 + x/( 2*k+1 - x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 25 2013
G.f.: 1 + x/(1 + x)*( 1 + 3*x/(1 + 3*x)*( 1 + 5*x/(1 + 5*x)*( 1 + 7*x/(1 + 7*x)*( 1 + ... )))). - Peter Bala, May 27 2017
a(n) = Sum_{k=0..2} stirling2(n, k).
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=2. - Robert A. Russell, Apr 25 2018
a(n) = A053120(n, n), n >= 0, (main diagonal of triangle of Chebyshev's T polynomials). - Wolfdieter Lang, Nov 26 2019

Additional comments from Emeric Deutsch, May 14 2001

Typo corrected by Philippe Deléham, Oct 25 2008

A208510 Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 7, 9, 5, 1, 1, 9, 16, 14, 6, 1, 1, 11, 25, 30, 20, 7, 1, 1, 13, 36, 55, 50, 27, 8, 1, 1, 15, 49, 91, 105, 77, 35, 9, 1, 1, 17, 64, 140, 196, 182, 112, 44, 10, 1, 1, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 1, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1

Offset: 1

Clark Kimberling, Feb 28 2012

Row sums:  A083329
Alternating row sums:  1,0,-1,-1,-1,-1,-1,-1,-1,-1,...
Antidiagonal sums: A000071 (-1+Fibonacci numbers)
col 1:  A000012
col 2:  A005408
col 3:  A000290
col 4:  A000330
col 5:  A002415
col 6:  A005585
col 7:  A040977
col 8:  A050486
col 9:  A053347
col 10:  A054333
col 11:  A054334
col 12:  A057788
col 2n-1 of A208510 is column n of A208508
col 2n of A208510 is column n of A208509.
...
GENERAL DISCUSSION:
A208510 typifies arrays generated by paired recurrence equations of the following form:
u(n,x)=a(n,x)*u(n-1,x)+b(n,x)*v(n-1,x)+c(n,x)
v(n,x)=d(n,x)*u(n-1,x)+e(n,x)*v(n-1,x)+f(n,x).
...
These first-order recurrences imply separate second-order recurrences.  In order to show them, the six functions a(n,x),...,f(n,x) are abbreviated as a,b,c,d,e,f.
Then, starting with initial values u(1,x)=1 and u(2,x)=a+b+c: u(n,x) = (a+e)u(n-1,x) + (bd-ae)u(n-2,x) + bf-ce+c.
With initial values v(1,x)=1 and v(2,x)=d+e+f: v(n,x) = (a+e)v(n-1,x) + (bd-ae)v(n-2,x) + cd-af+f.
...
In the guide below, the last column codes certain sequences that occur in one of these ways: row, column, edge, row sum, alternating row sum.  Coding:
A: 1,-1,1,-1,1,-1,1.... A033999
B: 1,2,4,8,16,32,64,... powers of 2
C: 1,1,1,1,1,1,1,1,.... A000012
D: 2,2,2,2,2,2,2,2,.... A007395
E: 2,4,6,8,10,12,14,... even numbers
F: 1,1,2,3,5,8,13,21,.. Fibonacci numbers
N: 1,2,3,4,5,6,7,8,.... A000027
O: 1,3,5,7,9,11,13,.... odd numbers
P: 1,3,9,27,81,243,.... powers of 3
S: 1,4,9,16,25,36,49,.. squares
T: 1,3,6,10,15,21,38,.. triangular numbers
Z: 1,0,0,0,0,0,0,0,0,.. A000007
*: (eventually) periodic alternating row sums
^: has a limiting row; i.e., the polynomials "approach" a power series
This coding includes indirect and repeated occurrences; e.g. F occurs thrice at A094441: in column 1 directly as Fibonacci numbers, in row sums as odd-indexed Fibonacci numbers, and in alternating row sums as signed Fibonacci numbers.
......... a....b....c....d....e....f....code
A034839 u 1....1....0....1....x....0....CCOT
A034867 v 1....1....0....1....x....0....CEN
A210221 u 1....1....0....1....2x...0....BBFF
A210596 v 1....1....0....1....2x...0....BBFF
A105070 v 1....2x...0....1....1....0....BN
A207605 u 1....1....0....1....x+1..0....BCFFN
A106195 v 1....1....0....1....x+1..0....BCFFN
A207606 u 1....1....0....x....x+1..0....DNT
A207607 v 1....1....0....x....x+1..0....DNT
A207608 u 1....1....0....2x...x+1..0....N
A207609 v 1....1....0....2x...x+1..0....C
A207610 u 1....1....0....1....x....1....CF
A207611 v 1....1....0....1....x....1....BCF
A207612 u 1....1....0....1....2x...1....BF
A207613 v 1....1....0....1....2x...1....BF
A207614 u 1....1....0....1....x+1..1....CN
A207615 v 1....1....0....1....x+1..1....CFN
A207616 u 1....1....0....x....1....1....CE
A207617 v 1....1....0....x....1....1....CNO
A029638 u 1....1....0....x....x....1....CDNO
A029635 v 1....1....0....x....x....1....CDNOZ
A207618 u 1....1....0....x....2x...1....N
A207619 v 1....1....0....x....2x...1....CFN
A207620 u 1....1....0....x....x+1..1....DET
A207621 v 1....1....0....x....x+1..1....DNO
A207622 u 1....1....0....2x...1....1....BT
A207623 v 1....1....0....2x...1....1....BN
A207624 u 1....1....0....2x...x....1....N
A102662 v 1....1....0....2x...x....1....CO
A207625 u 1....1....0....2x...x+1..1....T
A207626 v 1....1....0....2x...x+1..1....N
A207627 u 1....1....0....2x...2x...1....BN
A207628 v 1....1....0....2x...2x...1....BCE
A207629 u 1....1....0....x+1..1....1....CET
A207630 v 1....1....0....x+1..1....1....CO
A207631 u 1....1....0....x+1..x....1....DF
A207632 v 1....1....0....x+1..x....1....DEF
A207633 u 1....1....0....x+1..2x...1....F
A207634 v 1....1....0....x+1..2x...1....F
A207635 u 1....1....0....x+1..x+1..1....DN
A207636 v 1....1....0....x+1..x+1..1....CD
A160232 u 1....x....0....1....2x...0....BCFN
A208341 v 1....x....0....1....2x...0....BCFFN
A085478 u 1....x....0....1....x+1..0....CCOFT*
A078812 v 1....x....0....1....x+1..0....CEFN*
A208342 u 1....x....0....x....x....0....CCFNO
A208343 v 1....x....0....x....x....0....BBCDFZ
A208344 u 1....x....0....x....2x...0....CCFN
A208345 v 1....x....0....x....2x...0....CFZ
A094436 u 1....x....0....x....x+1..0....CFFN
A094437 v 1....x....0....x....x+1..0....CEFF
A117919 u 1....x....0....2x...1....0....BCNT
A135837 v 1....x....0....2x...1....0....BCET
A208328 u 1....x....0....2x...x....0....CCOP
A208329 v 1....x....0....2x...x....0....DPZ
A208330 u 1....x....0....2x...x+1..0....CNPT
A208331 v 1....x....0....2x...x+1..0....CN
A208332 u 1....x....0....2x...2x...0....CCE
A208333 v 1....x....0....2x...2x...0....DZ
A208334 u 1....x....0....x+1..1....0....CCNT
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A210601 v 2x...1....1....x+1...1....1...BFN*
A210603 u 2x...1....1....x+1...x+1..1...BF
A210738 v 2x...1....1....x+1...x+1..1...CBF*
A210739 u 2x...x....1....x+1...x....1...CF^
A210740 v 2x...x....1....x+1...x....1...DF*^
A210741 u 2x...x....1....x+1...x+1..1...BCFO
A210742 v 2x...x....1....x+1...x+1..1...CFO*
A210743 u 2x...x+1..1....x+1...1....1...F
A210744 v 2x...x+1..1....x+1...1....1...FN
A210747 u 2x...x+1..1....x+1...x+1..1...FF
A210748 v 2x...x+1..1....x+1...x+1..1...CFF*
A210749 u x+1..1....1....x+1...2x...1...BCF
A210750 v x+1..1....1....x+1...2x...1...BF
A210751 u x+1..x....1....x+1...2x...1...FNT
A210752 v x+1..x....1....x+1...2x...1...FN
A210753 u x+1..x....1....x+1...x+1..1...BNZ*
A210754 v x+1..x....1....x+1...x+1..1...BCT*
A210755 u x+1..2x...1....x+1...x+1..1...N*
A210756 v x+1..2x...1....x+1...x+1..1...CT*
A210789 u 1....x....0....x+2...x-1..0...CFFN
A210790 v 1....x....0....x+2...x-1..0...CEFF
A210791 u 1....x....0....x-1...x+2..0...CFNP
A210792 v 1....x....0....x-1...x+2..0...CF
A210793 u 1....x+1..0....x+2...x-1..0...CFNP
A210794 v 1....x+1..0....x+2...x-1..0...FPP
A210795 u 1....x....1....x+2...x-1..0...FN
A210796 v 1....x....1....x+2...x-1..0...FO
A210797 u 1....x....0....x+2...x-1..1...CF
A210798 v 1....x....0....x+2...x-1..1...F
A210799 u 1....x+1..1....x+2...x-1..0...FN
A210800 v 1....x+1..1....x+2...x-1..0...F
A210801 u 1....x+1..1....x+2...x-1..1...FN
A210802 v 1....x+1..1....x+2...x-1..1...F
A210803 u 1....x....0....x-1...x+3..0...F*
A210804 v 1....x....0....x-1...x+3..0...F*
A210805 u 1....x....0....x+2...x-1.-1...CFFN
A210806 v 1....x....0....x+2...x-1.-1...FF
A210858 u 1....x....0....x+n...x....0...CFT*
A210859 v 1....x....0....x+n...x....0...FN*
A210860 u 1....x+1..0....x+n...x....0...F
A210861 v 1....x+1..0....x+n...x....0...F*
A210862 u 1....x....1....x+n-1.x....0...FN
A210863 v 1....x....1....x+n-1.x....0...FS
A210864 u 1....x....1....x+n...x....0...FN
A210865 v 1....x....1....x+n...x....0...FT
A210866 u 1....x....0....x+n...x...-x...CFT
A210867 v 1....x....0....x+n...x...-x...FN
A210868 u 1....x....0....x+1...x-1..0...BCFN
A210869 v 1....x....0....x+1...x-1..0...BBCFNZ
A210870 u 1....x....0....x+1...x-1..1...CFFN
A210871 v 1....x....0....x+1...x-1..1...CFF
A210872 u x....1...-1....x.....x....1...BDFZ^
A210873 v x....1...-1....x.....x....1...BCFN^
A210876 u x....1....1....x.....x....x...BCCF^
A210877 v x....1....1....x.....x....x...BDFNZ^
A210878 u x....2x...0....x+1...x....1...DFZ^
A210879 v x....2x...0....x+1...x....1...FC*^
Some of these triangles have irregular row lengths, making it difficult to retrieve individual rows/columns/diagonals without actually computing the recurrence. - Georg Fischer, Sep 04 2021

			First five rows:
1
1...1
1...3...1
1...5...4...1
1...7...9...5...1
First five polynomials u(n,x):
1
1 + x
1 + 3x + x^2
1 + 5x + 4x^2 + x^3
1 + 7x + 9x^2 + 5x^3 + x^4

Cf. A029653, A208508, A208509.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A208510 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A029653 *)

from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Also, u(n,x)=(x+1)*u(n-1,x)+x for n>2, with u(n,2)=x+1.

Corrected by Philippe Deléham, Apr 10 2012
Corrections and additions by Clark Kimberling, May 09 2012
Corrections in the overview by Georg Fischer, Sep 04 2021

A169792 Expansion of ((1-x)/(1-2x))^5.

Original entry on oeis.org

1, 5, 20, 70, 225, 681, 1970, 5500, 14920, 39520, 102592, 261760, 657920, 1632000, 4001280, 9708544, 23336960, 55623680, 131563520, 309002240, 721092608, 1672806400, 3859415040, 8859156480, 20240138240, 46038777856, 104291368960, 235342397440, 529153392640

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 4 parts equal to 0. - Milan Janjic, Jun 27 2010

Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^5; see A291000. - Clark Kimberling, Aug 24 2017

Muniru A Asiru, Table of n, a(n) for n = 0..500
Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

((1-x)/(1-2x))^k: A011782, A045623, A058396, A062109, A169792-A169797; a row of A160232.

Concatenation([1],List([1..30],n->2^n*(n+4)*(n^3+26*n^2+171*n+186)/768)); # Muniru A Asiru, Aug 22 2018

seq(coeff(series(((1-x)/(1-2*x))^5, x,n+1),x,n),n=0..30); # Muniru A Asiru, Aug 22 2018

CoefficientList[Series[((1 - x)/(1 - 2 x))^5, {x, 0, 28}], x] (* Michael De Vlieger, Oct 15 2018 *)

Vec(((1-x)/(1-2*x))^5+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5), n >= 6. - Vincenzo Librandi, Mar 14 2011

a(n) = 2^n*(n+4)*(n^3 + 26*n^2 + 171*n + 186)/768, n > 0. - R. J. Mathar, Mar 14 2011

A169797 Expansion of ((1-x)/(1-2x))^10.

Original entry on oeis.org

1, 10, 65, 340, 1550, 6412, 24650, 89440, 309605, 1030490, 3317445, 10377180, 31655820, 94451520, 276313200, 794169792, 2246410560, 6262748160, 17230138880, 46831339520, 125870737408, 334826700800, 882159984640, 2303540756480, 5965195018240, 15327324667904

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 9 parts equal to 0. - Milan Janjic, Jun 27 2010

Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (20,-180,960,-3360,8064,-13440,15360,-11520,5120,-1024)

((1-x)/(1-2x))^k: A011782, A045623, A058396, A062109, A169792-A169797; a row of A160232.

CoefficientList[Series[((1-x)/(1-2x))^10,{x,0,30}],x] (* or *) Join[ {1}, LinearRecurrence[{20,-180,960,-3360,8064,-13440,15360,-11520,5120,-1024},{10,65,340,1550,6412,24650,89440,309605,1030490,3317445},30]] (* Harvey P. Dale, Aug 21 2014 *)

Vec(((1-x)/(1-2*x))^10+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

a(n) = 2^(n-17)*(n+11) *(n^8 + 124*n^7 + 5986*n^6 + 143944*n^5 + 1836529*n^4 + 12358156*n^3 + 42005484*n^2 + 64730736*n + 33747840)/2835, n > 0. - R. J. Mathar, Mar 14 2011

A208341 Triangle read by rows, T(n,k) = hypergeometric_2F1([n-k+1, -k], [1], -1) for n>=0 and k>=0.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 8, 8, 1, 5, 13, 20, 16, 1, 6, 19, 38, 48, 32, 1, 7, 26, 63, 104, 112, 64, 1, 8, 34, 96, 192, 272, 256, 128, 1, 9, 43, 138, 321, 552, 688, 576, 256, 1, 10, 53, 190, 501, 1002, 1520, 1696, 1280, 512, 1, 11, 64, 253, 743, 1683, 2972, 4048

Offset: 0

Clark Kimberling, Feb 25 2012

Previous name was: Triangle of coefficients of polynomials v(n,x) jointly generated with A160232; see the Formula section.
Row sums: (1,3,8,...), even-indexed Fibonacci numbers.
Alt. row sums: (1,-1,2,-3,...), signed Fibonacci numbers.
v(n,2) = A107839(n), v(n,n) = 2^(n-1), v(n+1,n) = A001792(n),
v(n+2,n) = A049611, v(n+3,n) = A049612.
Subtriangle of the triangle T(n,k) given by (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 12 2012
Essentially triangle in A049600. - Philippe Deléham, Mar 23 2012

			First five rows:
  1;
  1, 2;
  1, 3,  4;
  1, 4,  8,  8;
  1, 5, 13, 20, 16;
First five polynomials v(n,x):
  1
  1 + 2x
  1 + 3x +  4x^2
  1 + 4x +  8x^2 +  8x^3
  1 + 5x + 13x^2 + 20x^3 + 16x^4
(1, 0, -1/2, 1/2, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:
  1;
  1, 0;
  1, 2,  0;
  1, 3,  4,  0;
  1, 4,  8,  8,  0;
  1, 5, 13, 20, 16,  0;
  1, 6, 19, 38, 48, 32, 0;
Triangle in A049600 begins:
  0;
  0, 1;
  0, 1, 2;
  0, 1, 3,  4;
  0, 1, 4,  8,  8;
  0, 1, 5, 13, 20, 16;
  0, 1, 6, 19, 38, 48, 32;
  ... - _Philippe Deléham_, Mar 23 2012

Reinhard Zumkeller, Rows n = 0..124 of triangle, flattened

Cf. A160232, A000045, A049600, A106195.

a208341 n k = a208341_tabl !! (n-1) !! (k-1)
a208341_row n = a208341_tabl !! (n-1)
a208341_tabl = map reverse a106195_tabl
-- Reinhard Zumkeller, Dec 16 2013

T := (n,k) -> hypergeom([n-k+1, -k],[1],-1):
seq(lprint(seq(simplify(T(n,k)),k=0..n)),n=0..7); # Peter Luschny, May 20 2015

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2*x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A160232 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A208341 *)

T(n,k) = sum(i = 0, k, 2^(k-i)*binomial(n-k,i)*binomial(k,i));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print();); \\ Michel Marcus, Aug 14 2015

u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = u(n-1,x) + 2x*v(n-1,x), where u(1,x) = 1, v(1,x) = 1.
As DELTA-triangle with 0 <= k <= n: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 12 2012
G.f.: (1-2*y*x+y*x^2)/(1-x-2*y*x+y*x^2). - Philippe Deléham, Mar 12 2012
T(n,k) = A106195(n-1,n-k), k = 1..n. - Reinhard Zumkeller, Dec 16 2013
From Peter Bala, Aug 11 2015: (Start)
The following remarks assume the row and column indexing start at 0.
T(n,k) = Sum_{i = 0..k} 2^(k-i)*binomial(n-k,i)*binomial(k,i) = Sum_{i = 0..k} binomial(n-k+i,i)*binomial(k,i).
Riordan array (1/(1 - x), x*(2 - x)/(1 - x)).
O.g.f. 1/(1 - (2*t + 1)*x + t*x^2) = 1 + (1 + 2*t)*x + (1 + 3*t + 4*t^2)*x^2 + ....
Read as a square array, this equals P * transpose(P^2), where P denotes Pascal's triangle A007318. (End)
For kGlen Whitney, Aug 17 2021

New name from Peter Luschny, May 20 2015

Offset corrected by Joerg Arndt, Aug 12 2015

A062110 A(n,k) is the coefficient of x^k in (1-x)^n/(1-2*x)^n for n, k >= 0; Table A read by descending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 5, 3, 1, 0, 8, 12, 9, 4, 1, 0, 16, 28, 25, 14, 5, 1, 0, 32, 64, 66, 44, 20, 6, 1, 0, 64, 144, 168, 129, 70, 27, 7, 1, 0, 128, 320, 416, 360, 225, 104, 35, 8, 1, 0, 256, 704, 1008, 968, 681, 363, 147, 44, 9, 1, 0, 512, 1536, 2400, 2528, 1970

Offset: 0

Henry Bottomley, May 30 2001

The triangular version of this square array is defined by T(n,k) = A(k,n-k) for 0 <= k <= n. Conversely, A(n,k) = T(n+k,n) for n,k >= 0. We have [o.g.f of T](x,y) = [o.g.f. of A](x*y, x) and [o.g.f. of A](x,y) = [o.g.f. of T](y,x/y). - Petros Hadjicostas, Feb 11 2021
From Paul Barry, Nov 10 2008: (Start)
As number triangle, Riordan array (1, x(1-x)/(1-2x)). A062110*A007318 is A147703.
[0,1,1,0,0,0,....] DELTA [1,0,0,0,.....]. (Philippe Deléham's DELTA is defined in A084938.) (End)
Modulo 2, this triangle T becomes triangle A106344. - Philippe Deléham, Dec 18 2008

			Table A(n,k) (with rows n >= 0 and columns k >= 0) begins:
  1, 0,  0,   0,   0,    0,    0,     0,     0,     0, ...
  1, 1,  2,   4,   8,   16,   32,    64,   128,   256, ...
  1, 2,  5,  12,  28,   64,  144,   320,   704,  1536, ...
  1, 3,  9,  25,  66,  168,  416,  1008,  2400,  5632, ...
  1, 4, 14,  44, 129,  360,  968,  2528,  6448, 16128, ...
  1, 5, 20,  70, 225,  681, 1970,  5500, 14920, 39520, ...
  1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, ...
  ... - _Petros Hadjicostas_, Feb 15 2021
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,   1;
  0,   4,   5,   3,   1;
  0,   8,  12,   9,   4,   1;
  0,  16,  28,  25,  14,   5,   1;
  0,  32,  64,  66,  44,  20,   6,   1;
  0,  64, 144, 168, 129,  70,  27,   7,   1;
  0, 128, 320, 416, 360, 225, 104,  35,   8,   1;
  ... - _Philippe Deléham_, Nov 30 2008

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega, J. Rafael Sendra, and Juana Sendra, Upper Hessenberg and Toeplitz Bohemians, arXiv:1907.10677 [cs.SC], 2019.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq., 21 (2018), #18.1.4.

Rows of A include A000007, A011782, A045623, A058396, A062109, A169792, A169793, A169794, A169795, A169796, A169797.
Columns of A include A000012, A001477, A000096, A000297.
Main diagonal of A is A002002.
Table A(n, k) is a multiple of 2^(k-n); dividing by this gives a table similar to A050143 except at the edges.
Essentially the same array as A105306, A160232.

t[n_, n_] = 1; t[n_, k_] := 2^(n-2*k)*k*Hypergeometric2F1[1-k, n-k+1, 2, -1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 30 2013, after Philippe Deléham + symbolic sum *)

a(i,j)=if(i<0 || j<0,0,polcoeff(((1-x)/(1-2*x)+x*O(x^j))^i,j))

Formulas for the square array (A(n,k): n,k >= 0):
A(n, k) = A(n-1, k) + Sum_{0 <= j < k} A(n, j) for n >= 1 and k >= 0 with A(0, k) = 0^k for k >= 0.
G.f.: 1/(1-x*(1-y)/(1-2*y)) = Sum_{i, j >= 0} A(i, j) x^i*y^j.
From Petros Hadjicostas, Feb 15 2021: (Start)
A(n,k) = 2^(k-n)*n*hypergeom([1-n, k+1], [2], -1) for n >= 0 and k >= 1.
A(n,k) = 2*A(n,k-1) + A(n-1,k) - A(n-1,k-1) for n,k >= 1 with A(n,0) = 1 for n >= 0 and A(0,k) = 0 for k >= 1. (End)
Formulas for the triangle (T(n,k): 0 <= k <= n):
From Philippe Deléham, Aug 01 2006: (Start)
T(n,k) = A121462(n+1,k+1)*2^(n-2*k) for 0 <= k < n.
T(n,k) = 2^(n-2*k)*k*hypergeom([1-k, n-k+1], [2], -1) for 0 <= k < n. (End)
Sum_{k=0..n} T(n,k)*x^k = A152239(n), A152223(n), A152185(n), A152174(n), A152167(n), A152166(n), A152163(n), A000007(n), A001519(n), A006012(n), A081704(n), A082761(n), A147837(n), A147838(n), A147839(n), A147840(n), A147841(n), for x = -7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Dec 09 2008
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 1 <= k <= n-1 with T(0,0) = T(1,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, and T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 30 2013
G.f.: Sum_{n.k>=0} T(n,k)*x^n*y^k = (1 - 2*x)/(x^2*y - x*y - 2*x + 1). - Petros Hadjicostas, Feb 15 2021

Various sections edited by Petros Hadjicostas, Feb 15 2021

A210554 Triangle of coefficients of polynomials v(n,x) jointly generated with A208341; see the Formula section.

Original entry on oeis.org

1, 2, 2, 3, 5, 4, 4, 9, 12, 8, 5, 14, 25, 28, 16, 6, 20, 44, 66, 64, 32, 7, 27, 70, 129, 168, 144, 64, 8, 35, 104, 225, 360, 416, 320, 128, 9, 44, 147, 363, 681, 968, 1008, 704, 256, 10, 54, 200, 553, 1182, 1970, 2528, 2400, 1536, 512

Offset: 1

Clark Kimberling, Mar 22 2012

For a discussion and guide to related arrays, see A208510.

Also the number of multisets of size k that fit within some normal multiset of size n. A multiset is normal if it spans an initial interval of positive integers. - Andrew Howroyd, Sep 18 2018

			Triangle begins:
  1;
  2,  2;
  3,  5,   4;
  4,  9,  12,   8;
  5, 14,  25,  28,  16;
  6, 20,  44,  66,  64,  32;
  7, 27,  70, 129, 168, 144, 64;
  ...
First three polynomials v(n,x): 1, 2 + 2x , 3 + 5x + 4x^2.
The T(3, 1) = 3 multisets: (1), (2), (3).
The T(3, 2) = 5 multisets: (11), (12), (13), (22), (23).
The T(3, 3) = 4 multisets: (111), (112), (122), (123).

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A027941.

Cf. A160232, A208341, A208510, A303974.

T := (n,k) -> simplify((n + 1 - k)*hypergeom([1 - k, -k + n + 2], [2], -1)):
seq(seq(T(n,k), k=1..n), n=1..10); # Peter Luschny, Sep 18 2018

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A208341 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210554 *)

T(n,k)={sum(i=1, k, binomial(k-1, i-1)*binomial(n-k+i, i))} \\ Andrew Howroyd, Sep 18 2018

u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
T(n,k) = Sum_{i=1..k} binomial(k-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018
T(n,k) = (n - k + 1)*hypergeom([1 - k, n - k + 2], [2], -1). - Peter Luschny, Sep 18 2018

Example corrected by Philippe Deléham, Mar 23 2012

A169793 Expansion of ((1-x)/(1-2*x))^6.

Original entry on oeis.org

1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, 236640, 632448, 1661056, 4296192, 10961664, 27630592, 68889600, 170065920, 416071680, 1009582080, 2431254528, 5814222848, 13815054336, 32629850112, 76640681984, 179080003584, 416412598272, 963876225024

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 5 parts equal to 0. - Milan Janjic, Jun 27 2010

Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^6; see A291000. - Clark Kimberling, Aug 24 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).

Cf. for ((1-x)/(1-2x))^k: A011782, A045623, A058396, A062109, A169792-A169797; a row of A160232.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^6)); // G. C. Greubel, Oct 16 2018

seq(coeff(series(((1-x)/(1-2*x))^6,x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 16 2018

CoefficientList[Series[((1 - x)/(1 - 2 x))^6, {x, 0, 27}], x] (* Michael De Vlieger, Oct 15 2018 *)

x='x+O('x^30); Vec(((1-x)/(1-2*x))^6) \\ G. C. Greubel, Oct 16 2018

G.f.: ((1-x)/(1-2*x))^6.

For n > 0, a(n) = 2^(n-9)*(n+7)*(n^4 + 38*n^3 + 419*n^2 + 1342*n + 1080)/15. - Bruno Berselli, Aug 07 2011

A169794 Expansion of ((1-x)/(1-2*x))^7.

Original entry on oeis.org

1, 7, 35, 147, 553, 1925, 6321, 19825, 59906, 175504, 500864, 1397536, 3823680, 10282496, 27230464, 71129856, 183518720, 468213760, 1182433280, 2958376960, 7338426368, 18059821056, 44120473600, 107055742976, 258122317824, 618683957248, 1474700509184

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 6 parts equal to 0. - Milan Janjic, Jun 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (14, -84, 280, -560, 672, -448, 128).

Cf. for ((1-x)/(1-2x))^k: A011782, A045623, A058396, A062109, A169792-A169797; a row of A160232.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^7)); // G. C. Greubel, Oct 16 2018

seq(coeff(series(((1-x)/(1-2*x))^7,x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 16 2018

CoefficientList[Series[((1 - x)/(1 - 2 x))^7, {x, 0, 26}], x] (* Michael De Vlieger, Oct 15 2018 *)

x='x+O('x^30); Vec(((1-x)/(1-2*x))^7) \\ G. C. Greubel, Oct 16 2018

G.f.: ((1-x)/(1-2*x))^7.

For n > 0, a(n) = 2^(n-11)*(n+3)*(n+6)*(n^4 + 54*n^3 + 931*n^2 + 5454*n + 5080)/45. - Bruno Berselli, Aug 07 2011

A169795 Expansion of ((1-x)/(1-2x))^8.

Original entry on oeis.org

1, 8, 44, 200, 806, 2984, 10364, 34232, 108545, 332688, 990736, 2878144, 8182432, 22823680, 62595328, 169090048, 450568960, 1185832960, 3085885440, 7947714560, 20275478528, 51272351744, 128605356032, 320145981440, 791358537728, 1943278714880, 4742573981696

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 7 parts equal to 0. - Milan Janjic, Jun 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (16, -112, 448, -1120, 1792, -1792, 1024, -256).

Cf. for ((1-x)/(1-2x))^k: A011782, A045623, A058396, A062109, A169792-A169797; a row of A160232.

CoefficientList[Series[((1-x)/(1-2x))^8,{x,0,30}],x] (* Harvey P. Dale, Nov 24 2016 *)

G.f.: ((1-x)/(1-2*x))^8.

For n > 0, a(n) = 2^(n-12)*(n+9) * (n^6 + 75*n^5 + 1999*n^4 + 23169*n^3 + 115768*n^2 + 232284*n + 142800)/315. - Bruno Berselli, Aug 07 2011

cp's OEIS Frontend

A011782 Coefficients of expansion of (1-x)/(1-2*x) in powers of x.

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A208510 Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.

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A169792 Expansion of ((1-x)/(1-2x))^5.

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A169797 Expansion of ((1-x)/(1-2x))^10.

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A208341 Triangle read by rows, T(n,k) = hypergeometric_2F1([n-k+1, -k], [1], -1) for n>=0 and k>=0.

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A062110 A(n,k) is the coefficient of x^k in (1-x)^n/(1-2*x)^n for n, k >= 0; Table A read by descending antidiagonals.

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