A006976 -id:A006976 - 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

A053120 Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of x in increasing order).

Original entry on oeis.org

1, 0, 1, -1, 0, 2, 0, -3, 0, 4, 1, 0, -8, 0, 8, 0, 5, 0, -20, 0, 16, -1, 0, 18, 0, -48, 0, 32, 0, -7, 0, 56, 0, -112, 0, 64, 1, 0, -32, 0, 160, 0, -256, 0, 128, 0, 9, 0, -120, 0, 432, 0, -576, 0, 256, -1, 0, 50, 0, -400, 0, 1120, 0, -1280, 0, 512, 0, -11, 0, 220, 0, -1232, 0, 2816, 0, -2816, 0, 1024

Offset: 0

Wolfdieter Lang

Row sums (signed triangle): A000012 (powers of 1). Row sums (unsigned triangle): A001333(n).
From Wolfdieter Lang, Oct 21 2013: (Start)
The row polynomials T(n,x) equal (S(n,2*x) - S(n-2,2*x))/2, n >= 0, with the row polynomials S from A049310, with S(-1,x) = 0, and S(-2,x) = -1.
The zeros of T(n,x) are x(n,k) = cos((2*k+1)*Pi/(2*n)), k = 0, 1, ..., n-1, n >= 1. (End)
From Wolfdieter Lang, Jan 03 2020 and Paul Weisenhorn: (Start)
The (sub)diagonal sequences {D_{2*k}(m)}{m >= 0}, for k >= 0, have o.g.f. GD{2*k}(x) = (-1)^k*(1-x)/(1-2*x)^(k+1), for k >= 0, and  GD_{2*k+1}(x) = 0, for k >= 0. This follows from their o.g.f. GGD(z, x) := Sum_{k>=0} GD_k(x)*z^n which is obtained from the o.g.f. of the T-triangle GT(z, x) = (1-x*z)/(1 - 2*x + z^2) (see the formula section) by GGD(z, x) = GT(z, x/z).
The explicit form is then D_{2*k}(m) = (-1)^k, for m = 0, and
  (-1)^k*(2*k+m)*2^(m-1)*risefac(k+1, m-1)/m!, for m >= 1, with the rising factorial risefac(x, n). (End)

			The triangle a(n,m) begins:
n\m  0  1   2    3     4    5     6     7      8    9   10...
0:   1
1:   0  1
2:  -1  0   2
3:   0 -3   0    4
4:   1  0  -8    0     8
5:   0  5   0  -20     0   16
6:  -1  0  18    0   -48    0    32
7:   0 -7   0   56     0 -112     0    64
8:   1  0 -32    0   160    0  -256     0    128
9:   0  9   0 -120     0  432     0  -576      0  256
10: -1  0  50    0  -400    0  1120     0  -1280    0  512
... Reformatted and extended - _Wolfdieter Lang_, Oct 21 2013
E.g., the fourth row (n=3) corresponds to the polynomial T(3,x) = -3*x + 4*x^3.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available), p. 795.
F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 pp. 77, 105.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 22, page 196.
TableCurve 2D, Automated curve fitting and equation discovery, Version 5.01 for Windows, User's Manual, Chebyshev Series Polynomials and Rationals, pages 12-21 - 12-24, SYSTAT Software, Inc., Richmond, WA, 2002.

T. D. Noe, Rows 0 to 100 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [scanned copy], p. 795.
Paul Barry and A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, section 5.
Tom Copeland, Addendum to Elliptic Lie Triad.
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014. - From _Tom Copeland_, Oct 11 2014
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Wolfdieter Lang, Rows n = 0..20.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 16.
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Cf. A000007, A000012, A001333, A036909, A039991.

The first nonzero (sub)diagonal sequences are A011782, -A001792, A001793(n+1), -A001794, A006974, -A006975, A006976, -A209404.

using Nemo
function A053120Row(n)
    R, x = PolynomialRing(ZZ, "x")
    p = chebyshev_t(n, x)
    [coeff(p, j) for j in 0:n] end
for n in 0:6 A053120Row(n) |> println end # Peter Luschny, Mar 13 2018

&cat[ Coefficients(ChebyshevT(n)): n in [0..11] ]; // Klaus Brockhaus, Mar 08 2008

with(orthopoly) ;
A053120 := proc(n,k)
    T(n,x) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Jun 30 2013
T := (n, x) -> `if`(n = 0, 1, add((-1)^(n - k) * (n/(2*k))*binomial(k, n - k) *(2*x)^(2*k - n), k = 1 ..n)):
seq(seq(coeff(T(n, x), x, k), k = 0..n), n = 0..11); # Peter Luschny, Sep 20 2022

t[n_, k_] := Coefficient[ ChebyshevT[n, x], x, k]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* Jean-François Alcover, Jan 16 2012 *)

for(n=0,5,P=polchebyshev(n);for(k=0,n,print1(polcoeff(P,k)", "))) \\ Charles R Greathouse IV, Jan 16 2012

def f(n,k): # f = A039991
    if (n<2 and k==0): return 1
    elif (k<0 or k>n): return 0
    else: return 2*f(n-1, k) - f(n-2, k-2)
def A053120(n,k): return f(n, n-k)
flatten([[A053120(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 10 2022

T(n, m) = A039991(n, n-m).
G.f. for row polynomials T(n,x) (signed triangle): (1-x*z)/(1-2*x*z+z^2). If unsigned: (1-x*z)/(1-2*x*z-z^2).
T(n, m) := 0 if n < m or n+m odd; T(n, m) = (-1)^(n/2) if m=0 (n even); otherwise T(n, m) = ((-1)^((n+m)/2 + m))*(2^(m-1))*n*binomial((n+m)/2-1, m-1)/m.
Recursion for n >= 2: T(n, m) = T*a(n-1, m-1) - T(n-2, m), T(n, m)=0 if n < m, T(n, -1) := 0, T(0, 0) = T(1, 1) = 1.
G.f. for m-th column (signed triangle): 1/(1+x^2) if m=0, otherwise (2^(m-1))*(x^m)*(1-x^2)/(1+x^2)^(m+1).
From G. C. Greubel, Aug 10 2022: (Start)
Sum_{k=0..floor(n/2)} T(n-k, k) = A000007(n).
T(2*n, n) = i^n * A036909(n/2) * (1+(-1)^n)/2 + [n=0]/3. (End)
T(n, k) = [x^k] T(n, x) for n >= 1, where T(n, x) = Sum_{k=1..n}(-1)^(n - k)*(n/ (2*k))*binomial(k, n - k)*(2*x)^(2*k - n). - Peter Luschny, Sep 20 2022

A001793 a(n) = n(n+3)2^(n-3).

Original entry on oeis.org

1, 5, 18, 56, 160, 432, 1120, 2816, 6912, 16640, 39424, 92160, 212992, 487424, 1105920, 2490368, 5570560, 12386304, 27394048, 60293120, 132120576, 288358400, 627048448, 1358954496, 2936012800, 6325010432, 13589544960, 29125246976

Offset: 1

N. J. A. Sloane and Simon Plouffe

Coefficients of Chebyshev T polynomials: the subdiagonal A053120(n+3, n-1), for n > = 1. [rewritten by Wolfdieter Lang, Nov 25 2019]
Number of 132-avoiding permutations of [n+3] containing exactly two 123 patterns. - Emeric Deutsch, Jul 13 2001
Number of Dyck paths of semilength n+2 having pyramid weight n+1 (for pyramid weight see Denise and Simion). Example: a(2)=5 because the Dyck paths of semilength 4 having pyramid weight 3 are: (ud)u(ud)(ud)d, u(ud)(ud)d(ud), u(ud)(ud)(ud)d, u(ud)(uudd)d and u(uudd)(ud)d [here u=(1,1), d=(1,-1) and the maximal pyramids, of total length 3, are shown between parentheses]. - Emeric Deutsch, Mar 10 2004
a(n) is the number of dissections of a regular (n+3)-gon using n-1 noncrossing diagonals such that every piece of the dissection contains at least one non-base side of the (n+3)-gon. (One side of the (n+3)-gon is designated the base.) - David Callan, Mar 23 2004
If X_1,X_2,...,X_n are 2-blocks of a (2n+1)-set X then a(n) is the number of (n+2)-subsets of X intersecting each X_i, (i=1..n). - Milan Janjic, Nov 18 2007
The second corrector line for transforming 2^n offset 0 with a leading 1 into the Fibonacci sequence. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009
Sum of all nodes of all integer compositions of n, see example. - Olivier Gérard, Oct 22 2011
Number of compositions of 2n with exactly two odd summands (see example). - Mamuka Jibladze, Sep 04 2013
4*a(n) is the number of North-East paths from (0,0) to (n+2,n+2) with exactly two east steps below y = x-1 or above y = x+1. It is related to paired pattern P_1 and P_6 in Pan and Remmel's link. - Ran Pan, Feb 04 2016
From Paul Weisenhorn, Oct 18 2019: (Start)
The polynomials S(n,x)= Sum_(k>=1) b(n,k)*x^k has the recurrence relation S(n+2,x)=2*S(n+1,x))-x*S(n) with S(1,x)=1, S(2,x)=2-x and are generated by the coefficients b(n,k). b(n,k) is defined by b(n,k)=Sum_(j=1..k) binomials(k+1,j)*b(n-j,k) or by b(n,k)=((n-2+k)!*(n-1+2k)*2^n)/(4*(n-1)!*k!). b(n,1)=A001792, b(n,2)=A001793, b(n,3)=A001794, b(n,4)=A006974, b(n,5)=A006975, b(n,6)=A006976, b(n,7)=A209404.
The general formula for the sequences with k>=1: a(n)=((n-2+k)!*(n-1+2k)*2^n)/(4*(n-1)!*k!) with n >= 1. (End) [See a comment in A053120 on subdiagonal sequences. - Wolfdieter Lang, Jan 03 2020]

			a(2)=5 since 32415, 32451, 34125, 42135 and 52134 are the only 132-avoiding permutations of 12345 containing exactly two increasing subsequences of length 3.
a(4)=56: the compositions of 4 are 4, 3+1, 1+3, 2+2, 2+1+1, 1+2+1, 1+1+2, 1+1+1+1, the corresponding nodes (partial sums) are {0, 4}, {0, 3, 4}, {0, 1, 4}, {0, 2, 4}, {0, 2, 3, 4}, {0, 1, 3, 4}, {0, 1, 2, 4}, {0, 1, 2, 3, 4}, with individual sums {4, 7, 5, 6, 9, 8, 7, 10} and total 56. - _Olivier Gérard_, Oct 22 2011
The a(3)=18 compositions of 2*3=6 with two odd summands are 5+1, 1+5, 3+3, 4+1+1, 1+4+1, 1+1+4, 3+2+1, 3+1+2, 2+3+1, 2+1+3, 1+3+2, 1+2+3, 2+2+1+1, 2+1+2+1, 2+1+1+2, 1+2+2+1, 1+2+1+2, 1+1+2+2. - _Mamuka Jibladze_, Sep 04 2013

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
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).

T. D. Noe, Table of n, a(n) for n = 1..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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.
David Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Alain Denise and Rodica Simion, Two combinatorial statistics on Dyck paths, Discrete Math., Vol. 137, No. 1-3 (1995), pp. 155-176.
Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
Milan Janjic, Two Enumerative Functions.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
C. W. Jones, J. C. P. Miller, J. F. C. Conn and R. C. Pankhurst, Tables of Chebyshev polynomials Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.
Igor Makhlin, Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties, arXiv:2003.02916 [math.CO], 2020.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
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.
Lara Pudwell, Connor Scholten, Tyler Schrock and Alexa Serrato, Noncontiguous pattern containment in binary trees, ISRN Comb. 2014, Article ID 316535, 8 p. (2014), tr_{t4}(n,2).
Aaron Robertson, Herbert S. Wilf and Doron Zeilberger, Permutation patterns and continued fractions, Electr. J. Combin., Vol. 6 (1999), Article R38.
I. Tasoulas, K. Manes, and A. Sapounakis, Hamiltonian intervals in the lattice of binary paths, Elect. J. Comb. (2024) Vol. 31, Issue 1, P1.39.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

a(n) = A039991(n+3, 4) = A055252(n, 1).
Cf. A058396, A058645.
Cf. A053120.

List([1..30], n-> 2^(n-3)*n*(n+3) ); # G. C. Greubel, Nov 06 2019

[2^(n-3)*n*(n+3): n in [1..30]]; // G. C. Greubel, Nov 06 2019

A001793 := n*(n+3)*2^(n-3); seq(A001793(n), n=1..40);
A001793:=(-1+z)/(2*z-1)**3; # Simon Plouffe in his 1992 dissertation.

Table[n(n+3)*2^(n-3), {n, 28}] (* or *) Rest@ CoefficientList[Series[x(1-x)/(1-2x)^3, {x, 0, 28}], x] (* Michael De Vlieger, Sep 21 2017 *)

a(n)=n*(n+3)<<(n-3) \\ Charles R Greathouse IV, Sep 24 2015

[2^(n-3)*n*(n+3) for n in (1..30)] # G. C. Greubel, Nov 06 2019

G.f.: x*(1-x)/(1-2*x)^3. Binomial transform of squares [1, 4, 9, ...].
a(n) = Sum_{k=0..floor((n+4)/2)} C(n+4, 2k)*C(k, 2). - Paul Barry, May 15 2003
With two leading zeros, binomial transform of quarter-squares A002620. - Paul Barry, May 27 2003
a(n) = Sum_{k=0..n+2} C(n+2, k) * floor(k^2/4). - Paul Barry, May 27 2003
a(n) = Sum_{i=0..j} binomial(i+1, 2)*binomial(j, i). - Jon Perry, Feb 26 2004
With one leading zero, binomial transform of triangular numbers A000217. - Philippe Deléham, Aug 02 2005
a(n) = Sum_{k=0..n+1} (-1)^(n-k+1)*C(k, n-k+1)*k*C(2k, k)/2. - Paul Barry, Oct 07 2005
Left-shifted sequence is binomial transform of left-shifted squares (A000290). - Franklin T. Adams-Watters, Nov 29 2006
Binomial transform of a(n) = n^2 offset 1. a(3)=18. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009
a(n) = (1/n) * Sum_{k=0..n} binomial(n,k)*k^3. - Gary Detlefs, Nov 26 2011
For n > 1, a(n) = Sum_{k=0..n-1} Sum_{i=0..n} (k+2) * C(n-2,i). - Wesley Ivan Hurt, Sep 20 2017
a(n) = a(-3-n)*2^(2*n+3), a(n)*(n+3) = -A058645(-3-n)*2^(2*n+3) for all n in Z. - Michael Somos, Apr 19 2019
E.g.f.: (1/2)*exp(2*x)*x*(2 + x). - Stefano Spezia, Aug 17 2019
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=1} 1/a(n) = 128/9 - 56*log(2)/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 24*log(3/2) - 80/9. (End)

A008310 Triangle of coefficients of Chebyshev polynomials T_n(x).

Original entry on oeis.org

1, 1, -1, 2, -3, 4, 1, -8, 8, 5, -20, 16, -1, 18, -48, 32, -7, 56, -112, 64, 1, -32, 160, -256, 128, 9, -120, 432, -576, 256, -1, 50, -400, 1120, -1280, 512, -11, 220, -1232, 2816, -2816, 1024, 1, -72, 840, -3584, 6912, -6144, 2048, 13, -364, 2912, -9984, 16640, -13312, 4096

Offset: 0

N. J. A. Sloane

The row length sequence of this irregular array is A008619(n), n >= 0. Even or odd powers appear in increasing order starting with 1 or x for even or odd row numbers n, respectively. This is the standard triangle A053120 with 0 deleted. - Wolfdieter Lang, Aug 02 2014

Let T* denote the triangle obtained by replacing each number in this triangle by its absolute value. Then T* gives the coefficients for cos(nx) as a polynomial in cos x. - Clark Kimberling, Aug 04 2024

			Rows are: (1), (1), (-1,2), (-3,4), (1,-8,8), (5,-20,16) etc., since if c = cos(x): cos(0x) = 1, cos(1x) = 1c; cos(2x) = -1+2c^2; cos(3x) = -3c+4c^3, cos(4x) = 1-8c^2+8c^4, cos(5x) = 5c-20c^3+16c^5, etc.
From _Wolfdieter Lang_, Aug 02 2014: (Start)
This irregular triangle a(n,k) begins:
  n\k   0    1     2      3      4      5      6      7 ...
  0:    1
  1:    1
  2:   -1    2
  3:   -3    4
  4:    1   -8     8
  5:    5  -20    16
  6:   -1   18   -48     32
  7:   -7   56  -112     64
  8:    1  -32   160   -256    128
  9:    9 -120   432   -576    256
 10:   -1   50  -400   1120  -1280    512
 11:  -11  220 -1232   2816  -2816   1024
 12:    1  -72   840  -3584   6912  -6144   2048
 13:   13 -364  2912  -9984  16640 -13312   4096
 14:   -1   98 -1568   9408 -26880  39424 -28672   8192
 15:  -15  560 -6048  28800 -70400  92160 -61440  16384
  ...
T(4,x) = 1 - 8*x^2 + 8*x^4, T(5,x) = 5*x - 20*x^3 +16*x^5.
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
E. A. Guilleman, Synthesis of Passive Networks, Wiley, 1957, p. 593.
Yaroslav Zolotaryuk, J. Chris Eilbeck, "Analytical approach to the Davydov-Scott theory with on-site potential", Physical Review B 63, p543402, Jan. 2001. The authors write, "Since the algebra of these is 'hyperbolic', contrary to the usual Chebyshev polynomials defined on the interval 0 <= x <= 1, we call the set of functions (21) the hyperbolic Chebyshev polynomials." (This refers to the triangle T* described in Comments.)

R. J. Mathar, Table of n, a(n) for n = 0..2600
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Renato Ferreira Pinto Jr. and Nathaniel Harms, Testing Support Size More Efficiently Than Learning Histograms, arXiv:2410.18915 [cs.DS], 2024. See p. 40.
D. Foata and G.-N. Han, Nombres de Fibonacci et polynomes orthogonaux
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
I. Rivin, Growth in free groups (and other stories), arXiv:math/9911076 [math.CO], 1999.
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind.
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

A039991 is a row reversed version, but has zeros which enable the triangle to be seen. Columns/diagonals are A011782, A001792, A001793, A001794, A006974, A006975, A006976 etc.
Reflection of A028297. Cf. A008312, A053112.
Row sums are one. Polynomial evaluations include A001075 (x=2), A001541 (x=3), A001091, A001079, A023038, A011943, A001081, A023039, A001085, A077422, A077424, A097308, A097310, A068203.
Cf. A053120.

A008310 := proc(n,m) local x ; coeftayl(simplify(ChebyshevT(n,x),'ChebyshevT'),x=0,m) ; end: i := 0 : for n from 0 to 100 do for m from n mod 2 to n by 2 do printf("%d %d ",i,A008310(n,m)) ; i := i+1 ; od ; od ; # R. J. Mathar, Apr 20 2007
# second Maple program:
b:= proc(n) b(n):= `if`(n<2, 1, expand(2*b(n-1)-x*b(n-2))) end:
T:= n-> (p-> (d-> seq(coeff(p, x, d-i), i=0..d))(degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Sep 04 2019

Flatten[{1, Table[CoefficientList[ChebyshevT[n, x], x], {n, 1, 13}]}]//DeleteCases[#, 0, Infinity]& (* or *) Flatten[{1, Table[Table[((-1)^k*2^(n-2 k-1)*n*Binomial[n-k, k])/(n-k), {k, Floor[n/2], 0, -1}], {n, 1, 13}]}] (* Eugeniy Sokol, Sep 04 2019 *)

a(n,m) = 2^(m-1) * n * (-1)^((n-m)/2) * ((n+m)/2-1)! / (((n-m)/2)! * m!) if n>0. - R. J. Mathar, Apr 20 2007
From Paul Weisenhorn, Oct 02 2019: (Start)
T_n(x) = 2*x*T_(n-1)(x) - T_(n-2)(x), T_0(x) = 1, T_1(x) = x.
T_n(x) = ((x+sqrt(x^2-1))^n + (x-sqrt(x^2-1))^n)/2. (End)
From Peter Bala, Aug 15 2022: (Start)
T(n,x) = [z^n] ( z*x + sqrt(1 + z^2*(x^2 - 1)) )^n.
Sum_{k = 0..2*n} binomial(2*n,k)*T(k,x) = (2^n)*(1 + x)^n*T(n,x).
exp( Sum_{n >= 1} T(n,x)*t^n/n ) = Sum_{n >= 0} P(n,x)*t^n, where P(n,x) denotes the n-th Legendre polynomial. (End)

Additional comments and more terms from Henry Bottomley, Dec 13 2000

Edited: Corrected Cf. A039991 statement. Cf. A053120 added. - Wolfdieter Lang, Aug 06 2014

A002409 a(n) = 2^n*C(n+6,6). Number of 6D hypercubes in an (n+6)-dimensional hypercube.

Original entry on oeis.org

1, 14, 112, 672, 3360, 14784, 59136, 219648, 768768, 2562560, 8200192, 25346048, 76038144, 222265344, 635043840, 1778122752, 4889837568, 13231325184, 35283533824, 92851404800, 241413652480, 620777963520, 1580162088960

Offset: 0

N. J. A. Sloane

If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>5, a(n-6) is equal to the number of (n+6)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

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..400
Herbert Izbicki, Über Unterbaeume eines Baumes, Monatshefte für Mathematik, Vol. 74 (1970), pp. 56-62.
Herbert Izbicki, Über Unterbaeume eines Baumes, Monatshefte für Mathematik, Vol. 74 (1970), pp. 56-62.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
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 (14,-84,280,-560,672,-448,128).

First differences are in A006976.
Cf. A000079, A001787, A001788, A001789, A003472, A054849, A054851, A082140.
a(n) = A038207(n+6,6).

[2^n*Binomial(n+6, 6): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011

A002409:=-1/(2*z-1)**7; # Simon Plouffe in his 1992 dissertation
seq(binomial(n+6,6)*2^n,n=0..22); # Zerinvary Lajos, Jun 16 2008

CoefficientList[Series[1/(1-2x)^7,{x,0,40}],x] (* or *) LinearRecurrence[ {14,-84,280,-560,672,-448,128},{1,14,112,672,3360,14784,59136},40] (* Harvey P. Dale, Jan 24 2022 *)

G.f.: 1/(1-2*x)^7.
a(n) = 2*a(n-1) + A054849(n-1).
For n>0, a(n) = 2*A082140(n).
a(n) = Sum_{i=6..n+6} binomial(i,6)*binomial(n+6,i). Example: for n=5, a(5) = 1*462 + 7*330 + 28*165 + 84*55 + 210*11 + 462*1 = 14784. - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 47/5 - 12*log(2).
Sum_{n>=0} (-1)^n/a(n) = 2916*log(3/2) - 5907/5. (End)
n*a(n) +2*(-n-6)*a(n-1)=0. - R. J. Mathar, Jul 22 2025

More terms from Henry Bottomley and James Sellers, Apr 15 2000

Typo in definition corrected by Zerinvary Lajos, Jun 16 2008

A124182 A skewed version of triangular array A081277.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 3, 4, 0, 0, 1, 8, 8, 0, 0, 0, 5, 20, 16, 0, 0, 0, 1, 18, 48, 32, 0, 0, 0, 0, 7, 56, 112, 64, 0, 0, 0, 0, 1, 32, 160, 256, 128, 0, 0, 0, 0, 0, 9, 120, 432, 576, 256, 0, 0, 0, 0, 0, 1, 50, 400, 1120, 1280, 512

Offset: 0

Philippe Deléham, Dec 05 2006

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0,...] where DELTA is the operator defined in A084938. Falling diagonal sums in A052980.

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 3, 4;
  0, 0, 1, 8,  8;
  0, 0, 0, 5, 20, 16;
  0, 0, 0, 1, 18, 48,  32;
  0, 0, 0, 0,  7, 56, 112,  64;
  0, 0, 0, 0,  1, 32, 160, 256,  128;
  0, 0, 0, 0,  0,  9, 120, 432,  576,  256;
  0, 0, 0, 0,  0,  1,  50, 400, 1120, 1280, 512;

Cf. A025192 (column sums). Diagonals include A011782, A001792, A001793, A001794, A006974, A006975, A006976.

T(0,0)=T(1,1)=1, T(n,k)=0 if n < k or if k < 0, T(n,k) = T(n-2,k-1) + 2*T(n-1,k-1).
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A090965(n), (-1)^n*A084120(n), (-1)^n*A006012(n), A033999(n), A000007(n), A001333(n), A084059(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively.
Sum_{k=0..floor(n/2)} T(n-k,k) = Fibonacci(n-1) = A000045(n-1).
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 respectively. - Philippe Deléham, Dec 26 2007
Sum_{k=0..n} T(n,k)*(-x)^(n-k) = A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x= 0,1,2,3,4,5,6 respectively. - Philippe Deléham, Nov 14 2008
G.f.: (1-y*x)/(1-2y*x-y*x^2). - Philippe Deléham, Dec 04 2011
Sum_{k=0..n} T(n,k)^2 = A002002(n) for n > 0. - Philippe Deléham, Dec 04 2011

A200139 Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 8, 5, 1, 8, 20, 18, 7, 1, 16, 48, 56, 32, 9, 1, 32, 112, 160, 120, 50, 11, 1, 64, 256, 432, 400, 220, 72, 13, 1, 128, 576, 1120, 1232, 840, 364, 98, 15, 1, 256, 1280, 2816, 3584, 2912, 1568, 560, 128, 17, 1, 512, 2816, 6912, 9984, 9408, 6048, 2688, 816, 162, 19, 1

Offset: 0

Philippe Deléham, Nov 13 2011

Riordan array ((1-x)/(1-2x),x/(1-2x)).
Product A097805*A007318 as infinite lower triangular arrays.
Product A193723*A130595 as infinite lower triangular arrays.
T(n,k) is the number of ways to place n unlabeled objects into any number of labeled bins (with at least one object in each bin) and then designate k of the bins. - Geoffrey Critzer, Nov 18 2012
Apparently, rows of this array are unsigned diagonals of A028297. - Tom Copeland, Oct 11 2014
Unsigned A118800, so my conjecture above is true. - Tom Copeland, Nov 14 2016

			Triangle begins:
   1
   1,   1
   2,   3,   1
   4,   8,   5,   1
   8,  20,  18,   7,   1
  16,  48,  56,  32,   9,   1
  32, 112, 160, 120,  50,  11,   1

Cf. A118800 (signed version), A081277, A039991, A001333 (antidiagonal sums), A025192 (row sums); diagonals: A000012, A005408, A001105, A002492, A072819l; columns: A011782, A001792, A001793, A001794, A006974, A006975, A006976.

Cf. A007318, A028297, A059260, A097805, A118801, A130595.

nn=15;f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[(1-x)/(1-2x-y x) ,{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Nov 18 2012 *)

T(n,k) = 2*T(n-1,k)+T(n-1,k-1) with T(0,0)=T(1,0)=T(1,1)=1 and T(n,k)=0 for k<0 or for n

T(n,k) = A011782(n-k)*A135226(n,k) = 2^(n-k)*(binomial(n,k)+binomial(n-1,k-1))/2.

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A011782(n), A025192(n), A002001(n), A005054(n), A052934(n), A055272(n), A055274(n), A055275(n), A052268(n), A055276(n), A196731(n) for n=-1,0,1,2,3,4,5,6,7,8,9,10 respectively.

G.f.: (1-x)/(1-(2+y)*x).

T(n,k) = Sum_j>=0 T(n-1-j,k-1)*2^j.

T = A007318*A059260, so the row polynomials of this entry are given umbrally by p_n(x) = (1 + q.(x))^n, where q_n(x) are the row polynomials of A059260 and (q.(x))^k = q_k(x). Consequently, the e.g.f. is exp[tp.(x)] = exp[t(1+q.(x))] = e^t exp(tq.(x)) = [1 + (x+1)e^((x+2)t)]/(x+2), and p_n(x) = (x+1)(x+2)^(n-1) for n > 0. - Tom Copeland, Nov 15 2016

T^(-1) = A130595*(padded A130595), differently signed A118801. Cf. A097805. - Tom Copeland, Nov 17 2016

The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 + x)/(1 + 2*x) * (1 + 2*x)^n about 0. For example, for n = 4, (1 + x)/(1 + 2*x) * (1 + 2*x)^4 = (8*x^4 + 20*x*3 + 18*x^2 + 7*x + 1) + O(x^5). - Peter Bala, Feb 24 2018

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 3, 0, 0, 8, 8, 1, 0, 0, 16, 20, 5, 0, 0, 0, 32, 48, 18, 1, 0, 0, 0, 64, 112, 56, 7, 0, 0, 0, 0, 128, 256, 160, 32, 1, 0, 0, 0, 0, 256, 576, 432, 120, 9, 0, 0, 0, 0, 0, 512, 1280, 1120, 400, 50, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 03 2011

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
Skewed version of triangle in A200139.
Triangle without zeros: A207537.
For the version with negative odd numbered columns, which is Riordan ((1-x)/(1-2*x), -x^2/(1-2*x)) see comments on A028297 and A039991. - Wolfdieter Lang, Aug 06 2014
This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - Peter Bala, Jul 14 2015

			The triangle T(n,k) begins:
  n\k      0     1     2     3     4    5   6  7 8 9 10 11 ...
  0:       1
  1:       1     0
  2:       2     1     0
  3:       4     3     0     0
  4:       8     8     1     0     0
  5:      16    20     5     0     0    0
  6:      32    48    18     1     0    0   0
  7:      64   112    56     7     0    0   0  0
  8:     128   256   160    32     1    0   0  0 0
  9:     256   576   432   120     9    0   0  0 0 0
  10:    512  1280  1120   400    50    1   0  0 0 0  0
  11:   1024  2816  2816  1232   220   11   0  0 0 0  0  0
  ...  reformatted and extended. - _Wolfdieter Lang_, Aug 06 2014

C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Columns include A011782, A001792, A001793, A001794, A006974, A006975, A006976.
Diagonals sums are in A052980.
Cf. A028297, A081265, A124182, A131577, A039991 (zero-columns deleted, unsigned and zeros appended).
Cf. A098158, A200139, A207537.
Cf. A028297 (signed version, zeros deleted). Cf. A034839.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - #)/(1 - 2 #)&, #^2/(1 - 2 #)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 for k<0 or for n

Sum_{k=0..n} T(n,k)^2 = A002002(n) for n>0.

Sum_{k=0..n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n), A087455(n), A146559(n), A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 respectively.

G.f.: (1-x)/(1-2*x-y*x^2). - Philippe Deléham, Mar 03 2012

From Peter Bala, Jul 14 2015: (Start)

Factorizes as A034839 * A007318 = (1/(1 - x), x^2/(1 - x)^2) * (1/(1 - x), x/(1 - x)) as a product of Riordan arrays.

T(n,k) = Sum_{i = k..floor(n/2)} binomial(n,2*i) *binomial(i,k). (End)

Name changed, keyword:easy added, crossrefs A028297 and A039991 added, and g.f. corrected by Wolfdieter Lang, Aug 06 2014

A207537 Triangle of coefficients of polynomials u(n,x) jointly generated with A207538; see Formula section.

Original entry on oeis.org

1, 2, 1, 4, 3, 8, 8, 1, 16, 20, 5, 32, 48, 18, 1, 64, 112, 56, 7, 128, 256, 160, 32, 1, 256, 576, 432, 120, 9, 512, 1280, 1120, 400, 50, 1, 1024, 2816, 2816, 1232, 220, 11, 2048, 6144, 6912, 3584, 840, 72, 1, 4096, 13312, 16640, 9984, 2912, 364, 13

Offset: 1

Clark Kimberling, Feb 18 2012

Another version in A201701. - Philippe Deléham, Mar 03 2012
Subtriangle of the triangle given by (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012
Columns: A011782, A001792, A001793, A001794, A006974, A006975, A006976. - Philippe Deléham, Mar 03 2012
Diagonal sums: A052980. - Philippe Deléham, Mar 03 2012

			First seven rows:
   1;
   2,   1;
   4,   3;
   8,   8,  1;
  16,  20,  5,
  32,  48, 18, 1;
  64, 112, 56, 7;
From _Philippe Deléham_, Mar 03 2012: (Start)
Triangle A201701 begins:
   1;
   1,   0;
   2,   1,  0;
   4,   3,  0, 0;
   8,   8,  1, 0, 0;
  16,  20,  5, 0, 0, 0;
  32,  48, 18, 1, 0, 0, 0;
  64, 112, 56, 7, 0, 0, 0, 0;
  ... (End)

Cf. A028297, A207538, A133156.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207537, |A028297| *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207538, |A133156| *)
(* Prepending 1 and with offset 0: *)
Tpoly[n_] := HypergeometricPFQ[{-n/2, -n/2 + 1/2}, {1/2}, x + 1];
Table[CoefficientList[Tpoly[n], x], {n, 0, 12}] // Flatten (* Peter Luschny, Feb 03 2021 *)

u(n,x) = u(n-1,x) + (x+1)*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x), where u(1,x)=1, v(1,x)=1.  Also, A207537 = |A028297|.
T(n,k) = 2*T(n-1,k) + T(n-2,k-1). - Philippe Deléham, Mar 03 2012
G.f.: -(1+x*y)*x*y/(-1+2*x+x^2*y). - R. J. Mathar, Aug 11 2015
T(n, k) = [x^k] hypergeom([-n/2, -n/2 + 1/2], [1/2], x + 1) provided offset is set to 0 and 1 prepended. - Peter Luschny, Feb 03 2021

A209404 Negated coefficients of Chebyshev T polynomials: a(n) = -A053120(n+14, n), n >= 0.

Original entry on oeis.org

1, 15, 128, 816, 4320, 20064, 84480, 329472, 1208064, 4209920, 14057472, 45260800, 141213696, 428654592, 1270087680, 3683254272, 10478223360, 29297934336, 80648077312, 218864025600, 586290298880, 1551944908800, 4063273943040

Offset: 0

Brad Clardy, Mar 08 2012

The MAGMA program provided calculates the coefficients of order one Chebyshev polynomials, for any arbitrary level. For example, setting Rn to 0 produces A001792, 1 produces A001793, 2 produces A001794, 3 produces A006974, 4 produces A006975, and 5 produces A006976. This sequence is produced with an Rn of 6.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Cf. A001792, A001793, A001794, A006974, A006975, A006976, A053120.

List([0..30], n-> 2^(n-1)*(n+14)*Binomial(n+6,6)/7); # G. C. Greubel, Oct 18 2019

Rn:=6; [2^(n-1)/(Rn+1)*Binomial(n+Rn,Rn)*(n+(Rn+1)*2) : n in [0..22]];

R:=PowerSeriesRing(Integers(), 23); Coefficients(R!( (1-x)/(1-2*x)^8 )); // Marius A. Burtea, Oct 17 2019

seq(2^(n-1)*(n+14)*binomial(n+6,6)/7, n=0..30); # G. C. Greubel, Oct 18 2019

CoefficientList[Series[(1-x)/(1-2*x)^8, {x,0,30}], x] (* or *) Table[2^(n-1)*Binomial[n+6,6]*(n+14)/7, {n,0,30}] (* G. C. Greubel, Jan 03 2018 *)

for(n=0,30, print1(2^(n-1)*binomial(n+6,6)*(n+14)/7, ", ")) \\ G. C. Greubel, Jan 03 2018

a(n) = 2^(n-1)*binomial(n+6, 6)*(n+14)/7 = -A053120(n+14, n), n >= 0. [See a comment in A053120 on subdiagonal sequences. - Wolfdieter Lang, Jan 03 2020]
G.f.: (1-x)/(1-2*x)^8. - Colin Barker, May 31 2013
E.g.f.: (1/315)*exp(2*x)*(315 + 4095*x + 11340*x^2 + 11550*x^3 + 5250*x^4 + 1134*x^5 + 112*x^6 + 4*x^7). - Stefano Spezia, Oct 17 2019

Name made more specific by Wolfdieter Lang, Nov 25 2019

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cp's OEIS Frontend

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

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A053120 Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of x in increasing order).

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A001793 a(n) = n*(n+3)*2^(n-3).

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A008310 Triangle of coefficients of Chebyshev polynomials T_n(x).

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A002409 a(n) = 2^n*C(n+6,6). Number of 6D hypercubes in an (n+6)-dimensional hypercube.

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A001793 a(n) = n(n+3)2^(n-3).

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).