A201701 -id:A201701 - cp's OEIS frontend

A034839 Triangular array formed by taking every other term of each row of Pascal's triangle.

1, 1, 1, 1, 1, 3, 1, 6, 1, 1, 10, 5, 1, 15, 15, 1, 1, 21, 35, 7, 1, 28, 70, 28, 1, 1, 36, 126, 84, 9, 1, 45, 210, 210, 45, 1, 1, 55, 330, 462, 165, 11, 1, 66, 495, 924, 495, 66, 1, 1, 78, 715, 1716, 1287, 286, 13

Offset: 0

N. J. A. Sloane

Number of compositions of n having k parts greater than 1. Example: T(5,2)=5 because we have 3+2, 2+3, 2+2+1, 2+1+2 and 1+2+2. Number of binary words of length n-1 having k runs of consecutive 1's. Example: T(5,2)=5 because we have 1010, 1001, 0101, 1101 and 1011. - Emeric Deutsch, Mar 30 2005
From Gary W. Adamson, Oct 17 2008: (Start)
Received from Herb Conn:
Let T = tan x, then
tan x = T
tan 2x = 2T / (1 - T^2)
tan 3x = (3T - T^3) / (1 - 3T^2)
tan 4x = (4T - 4T^3) / (1 - 6T^2 + T^4)
tan 5x = (5T - 10T^3 + T^5) / (1 - 10T^2 + 5T^4)
tan 6x = (6T - 20T^3 + 6T^5) / (1 - 15T^2 + 15T^4 - T^6)
tan 7x = (7T - 35T^3 + 21T^5 - T^7) / (1 - 21T^2 + 35T^4 - 7T^6)
tan 8x = (8T - 56T^3 + 56T^5 - 8T^7) / (1 - 28T^2 + 70T^4 - 28T^6 + T^8)
tan 9x = (9T - 84T^3 + 126T^5 - 36T^7 + T^9) / (1 - 36 T^2 + 126T^4 - 84T^6 + 9T^8)
... To get the next one in the series, (tan 10x), for the numerator add:
9....84....126....36....1 previous numerator +
1....36....126....84....9 previous denominator =
10..120....252...120...10 = new numerator
For the denominator add:
......9.....84...126...36...1 = previous numerator +
1....36....126....84....9.... = previous denominator =
1....45....210...210...45...1 = new denominator
...where numerators = A034867, denominators = A034839
(End)
Triangle, with zeros omitted, given by (1, 0, 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. - Philippe Deléham, Dec 12 2011
The row (1,66,495,924,495,66,1) plays a role in expansions of powers of the Dedekind eta function. See the Chan link, p. 534. - Tom Copeland, Dec 12 2016
Binomial(n,2k) is also the number of permutations avoiding both 123 and 132 with k ascents, i.e., positions with w[i]Lara Pudwell, Dec 19 2018
Coefficients in expansion of ((x-1)^n+(x+1)^n)/2 or ((x-i)^n+(x+i)^n)/2 with alternating sign. - Eugeniy Sokol, Sep 20 2020
Number of permutations of length n avoiding simultaneously the patterns 213 and 312 with the maximum number of non-overlapping descents equal k (equivalently, with the maximum number of non-overlapping ascents equal k). An ascent (resp., descent) in a permutation a(1)a(2)...a(n) is position i such that a(i) < a(i+1) (resp., a(i) > a(i+1)). - Tian Han, Nov 16 2023

			Triangular array begins:
  1
  1
  1  1
  1  3
  1  6  1
  1 10  5
  1 15 15 1
  ...
cosh(4x) = (cosh x)^5 + 10 (cosh x)^3 (sinh x)^2 + 5 (cosh x) (sinh x)^4, so row 4 is (1,10,5). See Mathematica program. - _Clark Kimberling_, Aug 03 2024

G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened
M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.
H. Chan, S. Cooper, and P. Toh, The 26th power of Dedekind's eta function Advances in Mathematics, 207 (2006) 532-543.
Tom Copeland, Juggling Zeros in the Matrix: Example II, 2020.
C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.
Tian Han and Sergey Kitaev, Joint distributions of statistics over permutations avoiding two patterns of length 3, arXiv:2311.02974 [math.CO], 2023.
S.-M. Ma, On some binomial coefficients related to the evaluation of tan(nx), arXiv preprint arXiv:1205.0735 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 13 2012
K. Oliver and H. Prodinger, The continued fraction expansion of Gauss' hypergeometric function and a new application to the tangent function, Transactions of the Royal Society of South Africa, Vol. 76 (2012), 151-154, [DOI]; [PDF]. - From _N. J. A. Sloane_, Jan 03 2013
Eric Weisstein's World of Mathematics, Tangent [From _Eric W. Weisstein_, Oct 18 2008]

Cf. A007318, A034867, A034870, A080928, A119326, A201701.

Cf. A008619 (row lengths), A086645.

/* As a triangle */ [[Binomial(n,2*k):k in [0..Floor(n/2)]] : n in [0..10]]; // G. C. Greubel, Feb 23 2018

for n from 0 to 13 do seq(binomial(n,2*k),k=0..floor(n/2)) od;# yields sequence in triangular form; # Emeric Deutsch, Mar 30 2005

u[1, x_] := 1; v[1, x_] := 1; z = 12;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]  (* A034839 as a triangle *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]  (* A034867 as a triangle *)
(* Clark Kimberling, Feb 18 2012 *)
Table[Binomial[n, k], {n, 0, 13}, {k, 0, Floor[n, 2], 2}] // Flatten (* Michael De Vlieger, Dec 13 2016 *)
(* The triangle gives coefficients for cosh(nx) as a linear combination of products (cosh(x)^h)*(sinh(x)^k) *)
Column[Table[TrigExpand[Cosh[n  x]], {n, 0, 10}]]
(* Clark Kimberling, Aug 03 2024 *)

for(n=0,15, for(k=0,floor(n/2), print1(binomial(n, 2*k), ", "))) \\ G. C. Greubel, Feb 23 2018

E.g.f.: exp(x)*cosh(x*sqrt(y)). - Vladeta Jovovic, Mar 20 2005
From Emeric Deutsch, Mar 30 2005: (Start)
T(n, k) = binomial(n, 2*k), for n >= 0 and k = 0, 1, ..., floor(n/2).
G.f.: (1-z)/((1-z)^2 - t*z^2). (End)
O.g.f. for column no. k is (1/(1-x))*(x/(1-x))^(2*k), k >= 0 [from the g.f. given in the preceding formula]. - Wolfdieter Lang, Jan 18 2013
From Peter Bala, Jul 14 2015: (Start)
Stretched Riordan array ( 1/(1 - x ), x^2/(1 - x)^2 ) in the terminology of Corsani et al.
Denote this array by P. Then P * A007318 = A201701.
P * transpose(P) is A119326 read as a square array.
Let Q denote the array ( (-1)^k*binomial(2*n,k) )n,k>=0. Q is a signed version of A034870. Then Q*P = the identity matrix, that is, Q is a left-inverse array of P (see Corsani et al., p. 111).
P * A034870 = A080928. (End)
Even rows are A086645. An aerated version of this array is A099174 with each diagonal divided by the first element of the diagonal, the double factorials A001147. - Tom Copeland, Dec 12 2015

A002002 a(n) = Sum_{k=0..n-1} binomial(n,k+1) * binomial(n+k,k).

Original entry on oeis.org

0, 1, 5, 25, 129, 681, 3653, 19825, 108545, 598417, 3317445, 18474633, 103274625, 579168825, 3256957317, 18359266785, 103706427393, 586889743905, 3326741166725, 18885056428537, 107347191941249, 610916200215241

Offset: 0

N. J. A. Sloane, Simon Plouffe

From Benoit Cloitre, Jan 29 2002: (Start)
Array interpretation (first row and column are the natural numbers):
  1 2 3 ..j ... if b(i,j) = b(i-1,j) + b(i-1,j-1) + b(i,j-1) then a(n+1) = b(n,n)
  2 5 .........
  .............
  i........... b(i,j)
(End)
Number of ordered trees with 2n edges, having root of even degree, nonroot nodes of outdegree at most 2 and branches of odd length. - Emeric Deutsch, Aug 02 2002
Coefficient of x^n in ((1-x)/(1-2x))^n, n>0. - Michael Somos, Sep 24 2003
Number of peaks in all Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0). Example: a(2)=5 because HH, HU*D, U*DH, UHD, U*DU*D, UU*DD contain 5 peaks (indicated by *). - Emeric Deutsch, Dec 06 2003
a(n) is the total number of HHs in all Schroeder (n+1)-paths. Example: a(2)=5 because UH*HD, H*H*H, UDH*H, H*HUD contain 5 HHs (indicated by *) and the other 18 Schroeder 3-paths contain no HHs. - David Callan, Jul 03 2006
a(n) is the total number of Hs in all Schroeder n-paths. Example: a(2)=5 as the Schroeder 2-paths are HH, DUH, DHU, HDU, DUDU and DDUU, and there are 5 H's. In general, a(n) is the total number of H..Hs (m+1 H's) in all Schroeder (n+m)-paths. - FUNG Cheok Yin, Jun 19 2021
a(n) is the number of points in Z^(n+1) that are L1 (Manhattan) distance <= n from the origin, or the number of points in Z^n that are L1 distance <= n+1 from the origin. These terms occur in the crystal ball sequences: a(n) here is the n-th term in the sequence for the (n+1)-dimensional cubic lattice as well as the (n+1)-st term in the sequence for the n-dimensional cubic lattice. See A008288 for a list of crystal ball sequences (rows or columns of A008288). - Shel Kaphan, Dec 25 2022 [Edited by Peter Munn, Jan 05 2023]

			G.f. = x + 5*x^2 + 25*x^3 + 129*x^4 + 681*x^5 + 3653*x^6 + 19825*x^7 + 108545*x^8 + ...

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).

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0 through 100 were computed by Vincenzo Librandi)
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, Correlators for the Wigner-Smith time-delay matrix of chaotic cavities, arXiv:1601.06690 [math-ph], 2016.
L. Ericksen, Lattice path combinatorics for multiple product identities, J. of Statistical Planning and Inference 140 (2010) 2113-2226, see p. 2219.
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. 19.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32.

Bisection of A002003, Cf. A047781, A001003.
a(n)=T(n, n+1), array T as in A050143.
a(n)=T(n, n+1), array T as in A064861.
Half the first differences of central Delannoy numbers (A001850).
a(n)=T(n, n+1), array T as in A008288.
Cf. A026002, A190666, A259554.

[&+[Binomial(n,k+1)*Binomial(n+k,k): k in [0..n]]: n in [0..21]];  // Bruno Berselli, May 19 2011

A064861 := proc(n,k) option remember; if n = 1 then 1; elif k = 0 then 0; else A064861(n,k-1)+(3/2-1/2*(-1)^(n+k))*A064861(n-1,k); fi; end; seq(A064861(i,i+1),i=1..40);

CoefficientList[Series[((1-x)/Sqrt[1-6x+x^2]-1)/2, {x,0,30}],x]  (* Harvey P. Dale, Mar 17 2011 *)
a[ n_] := n Hypergeometric2F1[ n + 1, -n + 1, 2, -1] (* Michael Somos, Aug 09 2011 *)
a[ n_] := With[{m = Abs@n}, Sign[n] Sum[ Binomial[ m, k] Binomial[ m + k - 1, m], {k, m}]]; (* Michael Somos, Aug 09 2011 *)

makelist(sum(binomial(n,k+1)*binomial(n+k,k), k, 0, n), n, 0, 21); /* Bruno Berselli, May 19 2011 */

{a(n) = my(m = abs(n)); sign( n) * sum( k=0, m-1, binomial( m, k+1) * binomial( m+k, k))}; /* Michael Somos, Aug 09 2011 */

/* L.g.f.: Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*(1-x)^(-n)/n! */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A=(sum(m=1, n+1, Dx(m-1, x^(2*m-1)/(1-x)^m/m!)+x*O(x^n))); n*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, May 17 2015

a = lambda n: hypergeometric([1-n, -n], [1], 2) if n>0 else 0
[simplify(a(n)) for n in range(22)] # Peter Luschny, Nov 19 2014

G.f.: ((1-x)/sqrt(1-6*x+x^2)-1)/2. - Emeric Deutsch, Aug 02 2002
E.g.f.: exp(3*x)*(BesselI(0, 2*sqrt(2)*x)+sqrt(2)*BesselI(1, 2*sqrt(2)*x)). - Vladeta Jovovic, Mar 28 2004
a(n) = Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k+1). - Paul Barry, Sep 20 2004
a(n) = n * hypergeom([n + 1, -n + 1], [2], -1) = ((n+1)*LegendreP(n+1,3) - (5*n+3)*LegendreP(n,3))/(2*n) for n > 0. - Mark van Hoeij, Jul 12 2010
G.f.: x*d/dx log(1/(1-x*A006318(x))). - Vladimir Kruchinin, Apr 19 2011
a(n) = -a(-n) for all n in Z. - Michael Somos, Aug 09 2011
G.f.: -1 + 1 / ( 1 - x / (1 - 4*x / (1 - x^2 / (1 - 4*x / (1 - x^2 / (1 - 4*x / ...)))))). - Michael Somos, Jan 03 2013
a(n) = Sum_{k=0..n} A201701(n,k)^2 = Sum_{k=0..n} A124182(n,k)^2 for n > 0. - Philippe Deléham, Dec 05 2011
D-finite with recurrence: 2*(6*n^2-12*n+5)*a(n-1)-(n-2)*(2*n-1)*a(n-2)-n*(2*n-3)*a(n)=0. - Vaclav Kotesovec, Oct 04 2012
a(n) ~ (3+2*sqrt(2))^n/(2^(5/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 04 2012
D-finite (an alternative): n*a(n) = (6-n)*a(n-6) + (14*n-72)*a(n-5) + (264-63*n)*a(n-4) + 100*(n-3)*a(n-3) + (114-63*n)*a(n-2) + 2*(7*n-6)*a(n-1), n >= 7. - Fung Lam, Feb 05 2014
a(n) = (-1)^(n-1)*Sum_{k=0..n-1} (-2)^k*binomial(n-1,k)*binomial(n+k,k) and n^3*a(n) = Sum_{k=0..n-1} (4*k^3+4*k^2+4*k+1)*binomial(n-1,k)*binomial(n+k,k). For each of the two equalities, both sides satisfy the same recurrence -- this follows from the Zeilberger algorithm. - Zhi-Wei Sun, Aug 30 2014
a(n) = hypergeom([1-n, -n], [1], 2) for n >= 1. - Peter Luschny, Nov 19 2014
Logarithmic derivative of A001003 (little Schroeder numbers). - Paul D. Hanna, May 17 2015
L.g.f.: L(x) = Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1-x)^(-n) / n! = Sum_{n>=1} a(n)*x^n/n where exp(L(x)) = g.f. of A001003. - Paul D. Hanna, May 17 2015
a(n+1) = (1/2^(n+1)) * Sum_{k >= 0} (1/2^k) * binomial(n + k, n)*binomial(n + k, n + 1). - Peter Bala, Mar 02 2017
2*a(n) = A110170(n), n > 0. - R. J. Mathar, Feb 10 2022
a(n) = (LegendreP(n,3) - LegendreP(n-1,3))/2. - Mark van Hoeij, Jul 14 2022
D-finite with recurrence n*a(n) +(-7*n+5)*a(n-1) +(7*n-16)*a(n-2) +(-n+3)*a(n-3)=0. - R. J. Mathar, Aug 01 2022
From Peter Bala, Nov 08 2022: (Start)
a(n) = (-1)^(n+1)*hypergeom( [n+1, -n+1], [1], 2) for n >= 1.
The Gauss congruences hold: a(n*p^r) == a(n^p^(r-1)) (mod p^r) for all primes p and all positive integers n and r. (End)
From Peter Bala, Apr 18 2024: (Start)
G.f.: Sum_{n >= 1} binomial(2*n-1, n)*x^n/(1 - x)^(2*n) = x + 5*x^2 + 25*x^3 + 129*x^4 + ....
Row sums of A253283. (End)

More terms from Clark Kimberling

A028297 Coefficients of Chebyshev polynomials of the first kind: triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).

Original entry on oeis.org

1, 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

Offset: 0

N. J. A. Sloane

Rows are of lengths 1, 1, 2, 2, 3, 3, ... (A008619).
This triangle is generated from A118800 by shifting down columns to allow for (1, 1, 2, 2, 3, 3, ...) terms in each row. - Gary W. Adamson, Dec 16 2007
Unsigned triangle = A034839 * A007318. - Gary W. Adamson, Nov 28 2008
Triangle, with zeros omitted, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, -1, 1, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 16 2011
From Wolfdieter Lang, Aug 02 2014: (Start)
This irregular triangle is the row reversed version of the Chebyshev T-triangle A053120 given by A039991 with vanishing odd-indexed columns removed.
If zeros are appended in each row n >= 1, in order to obtain a regular triangle (see the Philippe Deléham comment, g.f. and example) this becomes the Riordan triangle (1-x)/(1-2*x), -x^2/(1-2*x). See also the unsigned version A201701 of this regular triangle.
(End)
Apparently, unsigned diagonals of this array are rows of A200139. - Tom Copeland, Oct 11 2014
It appears that the coefficients are generated by the following: Let SM_k = Sum( d_(t_1, t_2)* eM_1^t_1 * eM_2^t_2) summed over all length 2 integer partitions of k, i.e., 1*t_1 + 2*t_2 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 2 data (i.e., SM_k = S_k/2 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(2,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2) form an irregular triangle, with one row for each k value starting with k=1. Thus this procedure and associated OEIS sequences A287768, A288199, A288207, A288211, A288245, A288188 are generalizations of Chebyshev polynomials of the first kind. - Gregory Gerard Wojnar, Jul 01 2017

			Letting c = cos x, we have: cos 0x = 1, cos 1x = 1c; cos 2x = 2c^2-1; cos 3x = 4c^3-3c, cos 4x = 8c^4-8c^2+1, etc.
T4 = 8x^4 - 8x^2 + 1 = 8, -8, +1 = 2^(3) - (4)(2) + [2^(-1)](4)/2.
From _Wolfdieter Lang_, Aug 02 2014: (Start)
The irregular triangle T(n,k) begins:
n\k     1      2     3      4     5     6   7   8 ....
0:      1
1:      1
2:      2     -1
3:      4     -3
4:      8     -8     1
5:     16    -20     5
6:     32    -48    18     -1
7:     64   -112    56     -7
8:    128   -256   160    -32     1
9:    256   -576   432   -120     9
10:   512  -1280  1120   -400    50    -1
11:  1024  -2816  2816  -1232   220   -11
12:  2048  -6144  6912  -3584   840   -72   1
13:  4096 -13312 16640  -9984  2912  -364  13
14:  8192 -28672 39424 -26880  9408 -1568  98  -1
15: 16384 -61440 92160 -70400 28800 -6048 560 -15
...
T(4,x) = 8*x^4 -8*x^2 + 1*x^0, T(5,x) = 16*x^5 - 20*x^3 + 5*x^1, with Chebyshev's T-polynomials (A053120). (End)
From _Philippe Deléham_, Dec 16 2011: (Start)
The triangle (1,1,0,0,0,0,...) DELTA (0,-1,1,0,0,0,0,...) includes zeros and 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; (End)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 1.335, p. 35.
S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 106. [From Rick L. Shepherd, Jul 06 2010]

Alois P. Heinz, Rows n = 0..200, flattened
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] p. 795.
Pantelis A. Damianou, A Beautiful Sine Formula, Amer. Math. Monthly 121 (2014), no. 2, 120--135. MR3149030.
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 172.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 94.
Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=2, p. 22, arXiv:1706.08381 [math.GM], 2017.

Cf. A028298.
Reflection of A008310, the main entry. With zeros: A039991.
Cf. A053120 (row reversed table including zeros).
Cf. A118800, A034839, A081277, A124182.
Cf. A001333 (row sums 1), A001333 (alternating row sums). - Wolfdieter Lang, Aug 02 2014

b:= proc(n) b(n):= `if`(n<2, 1, expand(2*b(n-1)-x*b(n-2))) end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Sep 04 2019

t[n_] := (Cos[n x] // TrigExpand) /. Sin[x]^m_ /; EvenQ[m] -> (1 - Cos[x]^2)^(m/2) // Expand; Flatten[Table[ r = Reverse @ CoefficientList[t[n], Cos[x]]; If[OddQ[Length[r]], AppendTo[r,0]]; Partition[r,2][[All, 1]],{n, 0, 13}] ][[1 ;; 53]] (* Jean-François Alcover, May 06 2011 *)
Tpoly[n_] := HypergeometricPFQ[{(1 - n)/2, -n/2}, {1/2}, 1 - x];
Table[CoefficientList[Tpoly[n], x], {n, 0, 12}] // Flatten (* Peter Luschny, Feb 03 2021 *)

cos(n*x) = 2 * cos((n-1)*x) * cos(x) - cos((n-2)*x) (from CRC's Multiple-angle relations). - Rick L. Shepherd, Jul 06 2010
G.f.: (1-x) / (1-2x+y*x^2). - Philippe Deléham, Dec 16 2011
Sum_{k=0..n} T(n,k)*x^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, Dec 16 2011
T(n,k) = [x^k] hypergeom([1/2 - n/2, -n/2], [1/2], 1 - x). - Peter Luschny, Feb 03 2021
T(n,k) = (-1)^k * 2^(n-1-2*k) * A034807(n,k). - Hoang Xuan Thanh, Jun 21 2025

More terms from David W. Wilson

Row length sequence and link to Abramowitz-Stegun added by Wolfdieter Lang, Aug 02 2014

A039991 Triangle of coefficients of cos(x)^n in polynomial for cos(nx).

Original entry on oeis.org

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

Offset: 0

David W. Wilson

Also triangle of coefficients of Chebyshev polynomials of first kind (T(n,x)) in decreasing order of powers of x. A053120 gives the coefficients in increasing order.

The polynomials R(n,x) := Sum_{m=0..n} a(n,m)*sqrt(x)^m, have g.f. (1-z)/(1 - 2*z + x*z^2) = ((1-z)/(1-2*z))/(1 - x*(-z^2/(1-2*z))) (from the row reversion of the g.f. of A053120 and x^2 -> x). Therefore this triangle becomes the Riordan triangle ((1-z)/(1-2*z), -z^2/(1-2*z)) if the vanishing columns are deleted (see A028297) and zeros are appended in each row numbered n>=1 in order to obtain a triangle. This is then A201701 with negative odd numbered columns. - Wolfdieter Lang, Aug 06 2014

			Letting c = cos x, we have: cos 0x = 1, cos 1x = 1c; cos 2x = 2c^2-1; cos 3x = 4c^3-3c, cos 4x = 8c^4-8c^2+1, etc.
From _Wolfdieter Lang_, Aug 06 2014: (Start)
The triangle a(n,m) begins:
  n\m    0 1     2 3    4 5     6 7   8 9  10 11 ...
  0:     1
  1:     1 0
  2:     2 0    -1
  3:     4 0    -3 0
  4:     8 0    -8 0    1
  5:    16 0   -20 0    5 0
  6:    32 0   -48 0   18 0    -1
  7:    64 0  -112 0   56 0    -7 0
  8:   128 0  -256 0  160 0   -32 0   1
  9:   256 0  -576 0  432 0  -120 0   9 0
  10:  512 0 -1280 0 1120 0  -400 0  50 0  -1
  11: 1024 0 -2816 0 2816 0 -1232 0 220 0 -11  0
  ...
--------------------------------------------------------------------------
Chebyshev T-polynomials (decreasing even or odd powers):
n=3: T(3, x) = 4*x^3 - 3*x^1; n=4: T(4, x) = 8*x^4 - 8*x^2 + 1. (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.
Martin Aigner and Gunter M. Ziegler, Proofs From the Book, Springer 2004. See Chapter 18, Appendix.
E. A. Guilleman, Synthesis of Passive Networks, Wiley, 1957, p. 593.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

T. D. Noe, Table of n, a(n) for n = 0..5150
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].
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 172.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 94.
Index entries for sequences related to Chebyshev polynomials.

Cf. A028297 (without vanishing columns). A008310 (zero columns deleted then rows reversed).
Triangle without zeros: A028297. Without signs: A081265.
Cf. A053120 (increasing powers of x).

function T(n,k) // T = A039991
  if k lt 0 or k gt n then return 0;
  elif n lt 2 and k eq 0 then return 1;
  else return 2*T(n-1, k) - T(n-2, k-2);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 10 2022

seq(seq(coeff(orthopoly[T](i,x),x,i-j),j=0..i),i=0..20); # Robert Israel, Aug 07 2014

row[n_] := CoefficientList[ ChebyshevT[n, x], x] // Reverse; Table[row[n], {n, 0, 11}] // Flatten(* Jean-François Alcover, Sep 14 2012 *)

T(n,m)=(1+(-1)^m)*(binomial(n-m/2,n-m)+binomial(n-1-m/2,n-m))*2^(n-m-2)*(-1)^((m+1-(-1)^m)/2) /* Tani Akinari, Jul 18 2024 */

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

T(n, m) = 0 if n= 2, m >= 0; T(n, -2) = T(n, -1) = 0, T(0, 0) = T(1, 0) = 1.
G.f. for m-th column: 0 if m odd, (1-x)/(1-2*x) if m=0, else ((-1)^(m/2))*(x^m)*(1-x)/(1-2*x)^(m/2+1). For g.f. for row polynomials and row sums, see A053120.
G.f. row polynomials: (1-z)/(1 - 2*z + (x*z)^2). - Wolfdieter Lang, Aug 06 2014
Recurrence for the row polynomials Trev(n, x):= x^n*T(n, 1/x) = Sum_{m=0..n} T(n, m)*x^m; Trev(n, x) = 2*Trev(n-1, x) - x^2*Trev(n-2, x), n >= 1, Trev(-1, x) = 1/x^2 and Trev(0, x) = 1. From the T(n, x) recurrence. Compare this with A081265. - Wolfdieter Lang, Aug 07 2014
T(n,m) = (1+(-1)^m)*(binomial(n-m/2,n-m)+binomial(n-1-m/2,n-m))*2^(n-m-2)*(-1)^((m+1-(-1)^m)/2). - Tani Akinari, Jul 18 2024

Entry improved by comments from Wolfdieter Lang, Jan 11 2000.

Edited: A053120 added in comment and crossrefs. Cfs. A028297 and A008310 specified. - Wolfdieter Lang, Aug 06 2014

A201730 Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 14, 6, 0, 0, 41, 26, 1, 0, 0, 122, 100, 10, 0, 0, 0, 365, 363, 63, 1, 0, 0, 0, 1094, 1274, 322, 14, 0, 0, 0, 0, 3281, 4372, 1462, 116, 1, 0, 0, 0, 0, 9842, 14760, 6156, 744, 18, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 04 2011

Riordan array ((1-2x)/(1-4x+3x^2),x^2/(1-4x+3x^2)).

A007318*A201701 as lower triangular matrices.

			Triangle begins:
1
2, 0
5, 1, 0
14, 6, 0, 0
41, 26, 1, 0, 0
122, 100, 10, 0, 0, 0
365, 363, 63, 1, 0, 0, 0

Cf. A007051 (1st column), A261064 (2nd column).

A201730 := proc(n,k)
    (1-2*x)/(1-4*x+(3-y)*x^2) ;
    coeftayl(%,y=0,k) ;
    coeftayl(%,x=0,n) ;
end proc:
seq(seq(A201730(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 06 2011

m = 13;
(* DELTA is defined in A084938 *)
DELTA[Join[{2, 1/2, 3/2}, Table[0, {m}]], Join[{0, 1/2, -1/2}, Table[0, {m}]], m] // Flatten (* Jean-François Alcover, Feb 19 2020 *)

G.f.: (1-2x)/(1-4x+(3-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A139011(n), A000079(n), A007051(n), A006012(n), A001075(n), A081294(n), A001077(n), A084059(n), A108851(n), A084128(n), A081340(n), A084132(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
Sum_{k, k>+0} T(n+k,k) = A081704(n) .
T(n,k) = 3*T(n-1,k)+ Sum_{j>0} T(n-1-j,k-1).
T(n,k) = 4*T(n-1,k)+ T(n-2,k-1) - 3*T(n-2,k) with T(0,0)=1, T(1,0)= 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if n

A141041 a(n) = ((3 + 2sqrt(3))^n + (3 - 2sqrt(3))^n)/2.

Original entry on oeis.org

1, 3, 21, 135, 873, 5643, 36477, 235791, 1524177, 9852435, 63687141, 411680151, 2661142329, 17201894427, 111194793549, 718774444575, 4646231048097, 30033709622307, 194140950878133, 1254946834135719, 8112103857448713

Offset: 0

Roger L. Bagula, Aug 18 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,3).

Cf. A011943, A034478, A081336, A099842, A201701.

[n le 2 select 3^(n-1) else 6*Self(n-1) +3*Self(n-2): n in [1..31]]; // G. C. Greubel, Oct 10 2022

a[n_]= ((3+2*Sqrt[3])^n + (3-2*Sqrt[3])^n)/2; Table[FullSimplify[a[n]], {n,0,30}]
LinearRecurrence[{6,3},{1,3},30] (* Harvey P. Dale, Aug 25 2014 *)

A141041 = BinaryRecurrenceSequence(6,3,1,3)
[A141041(n) for n in range(31)] # G. C. Greubel, Oct 10 2022

a(n) = 3*abs(A099842(n-1)), for n > 0.
G.f.: (1-3*x)/(1-6*x-3*x^2). - Philippe Deléham, Mar 03 2012
a(n) = 6*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 3. - Philippe Deléham, Mar 03 2012
a(n) = Sum_{k=0..n} A201701(n,k)*3^(n-k). - Philippe Deléham, Mar 03 2012
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(4*k-3)/(x*(4*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
a(n) = (-i*sqrt(3))^n * ChebyshevT(n, i*sqrt(3)). - G. C. Greubel, Oct 10 2022

Edited by N. J. A. Sloane, Aug 24 2008

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

A201509 Irregular triangle read by rows: T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = 0, T(n,0) = T(1,1) = 1 and T(n,k) = 0 if k < 0 or if n < k.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 1, 8, 12, 4, 16, 28, 13, 1, 32, 64, 38, 6, 64, 144, 104, 25, 1, 128, 320, 272, 88, 8, 256, 704, 688, 280, 41, 1, 512, 1536, 1696, 832, 170, 10, 1024, 3328, 4096, 2352, 620, 61, 1, 2048, 7168

Offset: 0

Paul Curtz, Dec 02 2011

This is the pseudo-triangle whose successive lines are of the type T(n,0), T(n,1)+T(n-1,0), T(n,2)+T(n-1,1), ... T(n,k)+T(n-1,k-1), without 0's, with T=A201701. [e-mail, Philippe Deléham, Dec 04 2011]

			Triangle starts:
    1   1
    2   2
    4   5   1
    8  12   4
   16  28  13  1
   32  64  38  6
   64 144 104 25 1
  128 320 272 88 8
  ...
Triangle begins (full version):
    0
    1,   1
    2,   2,   0
    4,   5,   1,  0
    8,  12,   4,  0, 0
   16,  28,  13,  1, 0, 0
   32,  64,  38,  6, 0, 0, 0
   64, 144, 104, 25, 1, 0, 0, 0
  128, 320, 272, 88, 8, 0, 0, 0, 0

Cf. A052542 (row sums).
Cf. A039991, A034867.
Cf. A201780, A263633.

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = 0, T(n,0) = T(1,1) = 1 and T(n,k) = 0 if k < 0 or if n < k. - Philippe Deléham, Dec 05 2011
The n-th row polynomial appears to equal Sum_{k = 1..floor((n+1)/2)} binomial(n,2*k-1)*(1+t)^k. Cf. A034867. - Peter Bala, Sep 10 2012
Aside from the first two rows below, the signed coefficients appear in the expansion (b*x - 1)^2 / (a*b*x^2 - 2a*x + 1) = 1 + (2 a - 2 b)x + (4 a^2 - 5 a b + b^2)x^2 + (8 a^3 - 12 a^2b + 4 ab^2)x^3 + ..., the reciprocal of the derivative of x*(1-a*x) / (1-b*x). This is related to A263633 via the expansion (a*b*x^2 - 2a*x + 1) / (b*x - 1)^2  = 1 + (b - a) (2x + 3b x^2 + 4b^2 x^3 + ...). See also A201780. - Tom Copeland, Oct 30 2023

Edited and new name using Philippe Deléham's formula, Joerg Arndt, Dec 13 2023

Showing 1-8 of 8 results.

cp's OEIS Frontend

A034839 Triangular array formed by taking every other term of each row of Pascal's triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A002002 a(n) = Sum_{k=0..n-1} binomial(n,k+1) * binomial(n+k,k).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

PARI

Sage

Formula

Extensions

A028297 Coefficients of Chebyshev polynomials of the first kind: triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A039991 Triangle of coefficients of cos(x)^n in polynomial for cos(nx).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A201730 Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A141041 a(n) = ((3 + 2*sqrt(3))^n + (3 - 2*sqrt(3))^n)/2.

Views

Author

Keywords

A141041 a(n) = ((3 + 2sqrt(3))^n + (3 - 2sqrt(3))^n)/2.