A002492 -id:A002492 - cp's OEIS frontend

1, 1, -2, 1, -4, 3, 1, -6, 10, -4, 1, -8, 21, -20, 5, 1, -10, 36, -56, 35, -6, 1, -12, 55, -120, 126, -56, 7, 1, -14, 78, -220, 330, -252, 84, -8, 1, -16, 105, -364, 715, -792, 462, -120, 9, 1, -18, 136, -560, 1365, -2002, 1716, -792, 165, -10, 1, -20, 171, -816, 2380, -4368, 5005, -3432, 1287, -220, 11, 1

Offset: 0

Wolfdieter Lang

T(n,m) = A053122(n,n-m).
G.f. for row polynomials and row sums same as in A053122.
Unsigned column sequences are A000012, A005843, A014105, A002492 for m=0..3, resp. and A053126-A053131 for m=4..9.
This is also the coefficient triangle for Chebyshev's U(2*n+1,x) polynomials expanded in decreasing odd powers of (2*x): U(2*n+1,x) = Sum_{m=0..n} T(n,m)*(2*x)^(2*(n-m)+1). See the W. Lang link given in A053125.
Unsigned version is mirror image of A078812. - Philippe Deléham, Dec 02 2008

			Triangle begins:
  1;
  1,  -2;
  1,  -4,  3;
  1,  -6, 10,   -4;
  1,  -8, 21,  -20,   5;
  1, -10, 36,  -56,  35,  -6;
  1, -12, 55, -120, 126, -56, 7; ...
E.g. fourth row (n=3) {1,-6,10,-4} corresponds to polynomial S(3,x-2) = x^3-6*x^2+10*x-4.

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
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
Stephen Barnett, "Matrices: Methods and Applications", Oxford University Press, 1990, p. 132, 343.

T. D. Noe, Rows n=0..50 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 [alternative scanned copy].
Index entries for sequences related to Chebyshev polynomials.

Cf. A053124, A053122, A172431.

Flat(List([0..10], n-> List([0..n], k-> (-1)^k*Binomial(2*n-k+1,k) ))); # G. C. Greubel, Jul 23 2019

[(-1)^k*Binomial(2*n-k+1,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 23 2019

A053123 := proc(n,m)
    (-1)^m*binomial(2*n+1-m,m) ;
end proc: # R. J. Mathar, Sep 08 2013

T[n_, m_]:= (-1)^m*Binomial[2*n+1-m, m]; Table[T[n, m], {n, 0, 11}, {m, 0, n}]//Flatten (* Jean-François Alcover, Mar 05 2014, after R. J. Mathar *)

for(n=0,10, for(k=0,n, print1((-1)^k*binomial(2*n-k+1,k), ", "))) \\ G. C. Greubel, Jul 23 2019

[[(-1)^k*binomial(2*n-k+1,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 23 2019

T(n, m) = 0 if n

T(n, m) = -2*T(n-1, m-1) + T(n-1, m) - T(n-2, m-2), T(n, -2) = 0, T(-2, m) = 0, T(n, -1) = 0 = T(-1, m), T(0, 0) = 1, T(n, m) = 0 if n

G.f. for m-th column (signed triangle): ((-1)^m)*x^m*Po(m+1, x)/(1-x)^(m+1), with Po(k, x) := Sum_{j=0..floor(k/2)} binomial(k, 2*j+1)*x^j.

The n-th degree polynomial is the characteristic equation for an n X n tridiagonal matrix with (diagonal = all 2's, sub and superdiagonals all -1's and the rest 0's), exemplified by the 4X4 matrix M = [2 -1 0 0 / -1 2 -1 0 / 0 -1 2 -1 / 0 0 -1 2]. - Gary W. Adamson, Jan 05 2005

Sum_{m=0..n} T(n,m)*(c(n))^(2*n-2*m) = 1/c(n), where c(n) = 2*cos(Pi/(2*n+3)). - L. Edson Jeffery, Sep 13 2013

A053126 Binomial coefficients binomial(2*n-3,4).

Original entry on oeis.org

5, 35, 126, 330, 715, 1365, 2380, 3876, 5985, 8855, 12650, 17550, 23751, 31465, 40920, 52360, 66045, 82251, 101270, 123410, 148995, 178365, 211876, 249900, 292825, 341055, 395010, 455126, 521855, 595665, 677040, 766480

Offset: 4

Wolfdieter Lang

Number of intersections of diagonals in the interior of regular (2n-3)-gon. - Philippe Deléham, Jun 07 2013

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

Vincenzo Librandi, Table of n, a(n) for n = 4..200
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].
Milan Janjic, Two Enumerative Functions University of Banja Luka (Bosnia and Herzegovina, 2017).
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A053123, A002492.

[Binomial(2*n-3,4): n in [4..40]]; // Vincenzo Librandi, Oct 07 2011

Table[Binomial[2*n-3,4], {n,4,50}] (* G. C. Greubel, Aug 26 2018 *)

for(n=4,50, print1(binomial(2*n-3,4), ", ")) \\ G. C. Greubel, Aug 26 2018

a(n) = binomial(2*n-3, 4) if n >= 4 else 0;
G.f.: (5+10*x+x^2)/(1-x)^5.
a(n) = A053123(n,4), n >= 4; a(n) = 0, n=0..3 (fifth column of shifted Chebyshev's S-triangle, decreasing order).
a(n) = A006561(2n-3). - Philippe Deléham, Jun 07 2013
E.g.f.: (90 - 84*x + 39*x^2 - 12*x^3 + 4*x^4)*exp(x)/6. - G. C. Greubel, Aug 26 2018
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=4} 1/a(n) = 34/3 - 16*log(2).
Sum_{n>=4} (-1)^n/a(n) = 2*Pi - 4*log(2) - 10/3. (End)

A035006 Number of possible rook moves on an n X n chessboard.

Original entry on oeis.org

0, 8, 36, 96, 200, 360, 588, 896, 1296, 1800, 2420, 3168, 4056, 5096, 6300, 7680, 9248, 11016, 12996, 15200, 17640, 20328, 23276, 26496, 30000, 33800, 37908, 42336, 47096, 52200, 57660, 63488, 69696, 76296, 83300, 90720, 98568, 106856

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

Obviously A035005(n) = A002492(n-1) + a(n) since Queen = Bishop + Rook. - Johannes W. Meijer, Feb 04 2010

X values of solutions of the equation: (X-Y)^3-2*X*Y=0. Y values are b(n)=2*n*(n-1)^2 (see A181617). - Mohamed Bouhamida, Jul 06 2023

			On a 3 X 3-board, rook has 9*4 moves, so a(3)=36.

E. Bonsdorff, K. Fabel and O. Riihimaa, Schach und Zahl (Chess and numbers), Walter Rau Verlag, Dusseldorf, 1966.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Alexander M. Haupt, Bijective enumeration of rook walks, arXiv:2007.01018 [math.CO], 2020.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
M. Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Richard P. Stanley, Bijective Proof Problems, Problem 540 p. 63, (2015).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A033586 (King), A035005 (Queen), A035008 (Knight), A002492 (Bishop) and A049450 (Pawn).

Cf. A006002, A006564, A006566.

[(n-1)*2*n^2: n in [1..40]]; // Vincenzo Librandi, Jun 16 2011

Table[(n-1) 2 n^2,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,8,36,96},40] (* Harvey P. Dale, May 12 2012 *)

a(n) = (n-1)*2*n^2.
a(n) = Sum_{j=1..n} ((n+j-1)^2 - (n-j+1)^2). - Zerinvary Lajos, Sep 13 2006
1/a(n+1) = Integral_{x=1/(n+1)..1/n} x*h(x) = Integral_{x=1/(n+1)..1/n} x*(1/x - floor(1/x)) = 1/((2*(n^2+2*n+1))*n) and Sum_{n>=1} 1/((2*(n^2+2*n+1))*n) = 1-Zeta(2)/2 where h(x) is the Gauss (continued fraction) map h(x)={x^-1} and {x} is the fractional part of x. - Stephen Crowley, Jul 24 2009
a(n) = 4 * A006002(n-1). - Johannes W. Meijer, Feb 04 2010
G.f.: 4*x^2*(2+x)/(1-x)^4. - Colin Barker, Mar 11 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(1)=0, a(2)=8, a(3)=36, a(4)=96. - Harvey P. Dale, May 12 2012
a(n) = A006566(n) - A006564(n). - Peter M. Chema, Feb 10 2016
E.g.f.: 2*exp(x)*x^2*(2 + x). - Stefano Spezia, May 10 2022
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=2} 1/a(n) = 1 - Pi^2/12.
Sum_{n>=2} (-1)^n/a(n) = Pi^2/24 + log(2) - 1. (End)

A202076 T(n,k)=Number of arrays of n+2 integers in -k..k with sum zero and the sum of every adjacent pair being odd.

Original entry on oeis.org

2, 4, 4, 10, 20, 2, 14, 56, 26, 0, 24, 120, 78, 0, 6, 30, 220, 264, 0, 96, 12, 44, 364, 504, 0, 1014, 524, 6, 52, 560, 1128, 0, 3752, 5832, 726, 0, 70, 816, 1786, 0, 15010, 34632, 8412, 0, 20, 80, 1140, 3262, 0, 35604, 142692, 80812, 0, 2760, 40, 102, 1540, 4660, 0, 95342

Offset: 1

R. H. Hardin Dec 10 2011

Table starts
..2.....4.....10.......14........24........30.........44..........52
..4....20.....56......120.......220.......364........560.........816
..2....26.....78......264.......504......1128.......1786........3262
..0.....0......0........0.........0.........0..........0...........0
..6....96...1014.....3752.....15010.....35604......95342......181834
.12...524...5832....34632....142692....462436....1264272.....3044496
..6...726...8412....80812....340660...1516410....4213042....12861252
..0.....0......0........0.........0.........0..........0...........0
.20..2760.118560..1201220..10924220..50331332..241384794...755963886
.40.15560.691352.11395632.105606040.670671976.3259289000.12973320840

			Some solutions for n=3 k=3
..0....0....0....2....2....0....2....2....0....0....0...-2....2....0...-2...-2
..1...-1....3....1...-1....1....1...-1....1...-1...-1....3...-3...-3....3....3
.-2...-2...-2...-2....0....2....0....0....0....2....0...-2....2....2...-2....0
..1....1...-3...-3....1...-3...-1...-3...-3...-1....3....3...-3...-1...-1...-3
..0....2....2....2...-2....0...-2....2....2....0...-2...-2....2....2....2....2

R. H. Hardin, Table of n, a(n) for n = 1..9999

Row 1 is A152749

Row 2 is A002492

A128908 Riordan array (1, x/(1-x)^2).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 4, 10, 6, 1, 0, 5, 20, 21, 8, 1, 0, 6, 35, 56, 36, 10, 1, 0, 7, 56, 126, 120, 55, 12, 1, 0, 8, 84, 252, 330, 220, 78, 14, 1, 0, 9, 120, 462, 792, 715, 364, 105, 16, 1, 0, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1

Offset: 0

Philippe Deléham, Apr 22 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,2,-1/2,1/2,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
Row sums give A088305. - Philippe Deléham, Nov 21 2007
Column k is C(n,2k-1) for k > 0. - Philippe Deléham, Jan 20 2012
From R. Bagula's comment in A053122 (cf. Damianou link p. 10), this array gives the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - Tom Copeland, Oct 11 2014
T is the convolution triangle of the positive integers (see A357368). - Peter Luschny, Oct 19 2022

			The triangle T(n,k) begins:
   n\k  0    1    2    3    4    5    6    7    8    9   10
   0:   1
   1:   0    1
   2:   0    2    1
   3:   0    3    4    1
   4:   0    4   10    6    1
   5:   0    5   20   21    8    1
   6:   0    6   35   56   36   10    1
   7:   0    7   56  126  120   55   12    1
   8:   0    8   84  252  330  220   78   14    1
   9:   0    9  120  462  792  715  364  105   16    1
  10:   0   10  165  792 1716 2002 1365  560  136   18    1
  ... reformatted by _Wolfdieter Lang_, Jul 31 2017
From _Peter Luschny_, Mar 06 2022: (Start)
The sequence can also be seen as a square array read by upwards antidiagonals.
   1, 1,   1,    1,    1,     1,     1,      1,      1, ...  A000012
   0, 2,   4,    6,    8,    10,    12,     14,     16, ...  A005843
   0, 3,  10,   21,   36,    55,    78,    105,    136, ...  A014105
   0, 4,  20,   56,  120,   220,   364,    560,    816, ...  A002492
   0, 5,  35,  126,  330,   715,  1365,   2380,   3876, ... (A053126)
   0, 6,  56,  252,  792,  2002,  4368,   8568,  15504, ... (A053127)
   0, 7,  84,  462, 1716,  5005, 12376,  27132,  54264, ... (A053128)
   0, 8, 120,  792, 3432, 11440, 31824,  77520, 170544, ... (A053129)
   0, 9, 165, 1287, 6435, 24310, 75582, 203490, 490314, ... (A053130)
    A27,A292, A389, A580,  A582, A1288, A10966, A10968, A165817       (End)

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv:1110.6620 [math.RT], 2014.

Cf. A002450, A007318, A034008, A053122, A078812, A084938, A088305.
Cf. Columns : A000007, A000027, A000292, A000389, A000580, A000582, A001288, A010966 ..(+2).. A011000, A017713 ..(+2).. A017763.
Cf. A000007, A001352, A008574, A054888, A084099, A084103, A163810, A357368.
Cf. A165817 (the main diagonal of the array).

# Computing the rows of the array representation:
S := proc(n,k) option remember;
if n = k then 1 elif k < 0 or k > n then 0 else
S(n-1, k-1) + 2*S(n-1, k) - S(n-2, k) fi end:
Arow := (n, len) -> seq(S(n+k-1, k-1), k = 0..len-1):
for n from 0 to 8 do Arow(n, 9) od; # Peter Luschny, Mar 06 2022
# Uses function PMatrix from A357368.
PMatrix(10, n -> n); # Peter Luschny, Oct 19 2022

With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - x)^2/(1 - (2 + y)*x + x^2), {x, 0, nmax}, {y, 0, nmax}], x], y]] // Flatten (* G. C. Greubel, Nov 22 2017 *)

for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, if(k==0, 0, binomial(n+k-1,2*k-1))), ", "))) \\ G. C. Greubel, Nov 22 2017

from functools import cache
@cache
def A128908(n, k):
    if n == k: return 1
    if (k <= 0 or k > n): return 0
    return A128908(n-1, k-1) + 2*A128908(n-1, k) - A128908(n-2, k)
for n in range(10):
    print([A128908(n, k) for k in range(n+1)]) # Peter Luschny, Mar 07 2022

@cached_function
def T(k,n):
    if k==n: return 1
    if k==0: return 0
    return sum(i*T(k-1,n-i) for i in (1..n-k+1))
A128908 = lambda n,k: T(k,n)
for n in (0..10): print([A128908(n,k) for k in (0..n)]) # Peter Luschny, Mar 12 2016

T(n,0) = 0^n, T(n,k) = binomial(n+k-1, 2k-1) for k >= 1.
Sum_{k=0..n} T(n,k)*2^(n-k) = A002450(n) = (4^n-1)/3 for n>=1. - Philippe Deléham, Oct 19 2008
G.f.: (1-x)^2/(1-(2+y)*x+x^2). - Philippe Deléham, Jan 20 2012
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A001352(n), (-1)^(n+1)*A054888(n+1), (-1)^n*A008574(n), (-1)^n*A084103(n), (-1)^n*A084099(n), A163810(n), A000007(n), A088305(n) for x = -6, -5, -4, -3, -2, -1, 0, 1 respectively. - Philippe Deléham, Jan 20 2012
Riordan array (1, x/(1-x)^2). - Philippe Deléham, Jan 20 2012

A166464 a(n) = (3 + 2n + 6n^2 + 4*n^3)/3.

Original entry on oeis.org

1, 5, 21, 57, 121, 221, 365, 561, 817, 1141, 1541, 2025, 2601, 3277, 4061, 4961, 5985, 7141, 8437, 9881, 11481, 13245, 15181, 17297, 19601, 22101, 24805, 27721, 30857, 34221, 37821, 41665, 45761, 50117, 54741, 59641, 64825, 70301, 76077, 82161, 88561, 95285, 102341, 109737, 117481, 125581

Offset: 0

Paul Curtz, Oct 14 2009

Atomic number of first transition metal of period 2n (n>3) or of the element after n-th alkaline earth metal. This can be calculated by finding the sum of the first n even squares plus 1. - Natan Arie Consigli, Jul 03 2016

JANET,Charles, La structure du Noyau de l'atome,consideree dans la Classification periodique,des elements chimiques,1927 (Novembre),N. 2,BEAUVAIS,67 pages,3 leaflets.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002492, A010731, A016742, A017113, A018227.

[(3+2*n+6*n^2+4*n^3)/3: n in [0..60]]; // G. C. Greubel, Jul 27 2024

Table[(3+2*n+6*n^2+4*n^3)/3, {n,0,60}] (* G. C. Greubel, May 15 2016 *)

a(n)=(3+2*n+6*n^2+4*n^3)/3 \\ Charles R Greathouse IV, Oct 07 2015

[(3+2*n+6*n^2+4*n^3)//3 for n in range(61)] # G. C. Greubel, Jul 27 2024

a(n) - a(n-1) = 4*(n+1)^2 = A016742(n+1).
a(n) - 2*a(n-1) + a(n-2) = -4 + 8*n = A017113(n+1).
a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 8 = A010731(n).
a(n) - 4*a(n-1) + 6*a(n-2) - 4*a(n-3) + a(n-4) = 0.
Binomial transform of quasi-finite sequence 1,4,12,8,0,(0 continued).
G.f.: (1+x+7*x^2-x^3)/(1-x)^4. - R. J. Mathar, Feb 15 2010
From Natan Arie Consigli, Jul 03 2016: (Start)
a(n) = A018227(2*n) + 3.
a(n) = A002492(n) + 1. (End)
E.g.f.: (1/3)*(3 + 12*x + 18*x^2 + 4*x^3)*exp(x). - G. C. Greubel, Jul 27 2024

Edited by N. J. A. Sloane, Oct 17 2009

More terms a(11)-a(35) from Vincenzo Librandi, Oct 17 2009

A035005 Number of possible queen moves on an n X n chessboard.

Original entry on oeis.org

0, 12, 56, 152, 320, 580, 952, 1456, 2112, 2940, 3960, 5192, 6656, 8372, 10360, 12640, 15232, 18156, 21432, 25080, 29120, 33572, 38456, 43792, 49600, 55900, 62712, 70056, 77952, 86420, 95480, 105152, 115456, 126412, 138040, 150360

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

The number of (2 to n) digit sequences that can be found reading in any orientation, including diagonals, in an (n X n) grid. - Paul Cleary, Aug 12 2005

			3 X 3 board: queen has 8*6 moves and 1*8 moves, so a(3)=56.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A033586 (King), A035006 (Rook), A035008 (Knight), A002492 (Bishop) and A049450 (Pawn).

Cf. A162147.

[(n-1)*2*n^2 + (4*n^3-6*n^2+2*n)/3: n in [1..40]]; // Vincenzo Librandi, Jun 16 2011

Table[(n-1)2n^2+(4n^3-6n^2+2n)/3,{n,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,12,56,152},40] (* Harvey P. Dale, Aug 24 2011 *)

a(n) = (n-1)*2*n^2 + (4*n^3-6*n^2+2*n)/3.
From Johannes W. Meijer, Feb 04 2010: (Start)
a(n) = A002492(n-1) + A035006(n) since Queen = Bishop + Rook.
a(n) = 4 * A162147(n-1). (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=12, a(2)=56, a(3)=152. - Harvey P. Dale, Aug 24 2011
From Colin Barker, Mar 11 2012: (Start)
a(n) = 2*n*(1-6*n+5*n^2)/3.
G.f.: 4*x^2*(3+2*x)/(1-x)^4. (End)
E.g.f.: 2*exp(x)*x^2*(9 + 5*x)/3. - Stefano Spezia, Jul 31 2022

More terms from Erich Friedman

A112742 a(n) = n^2*(n^2 - 1)/3.

Original entry on oeis.org

0, 0, 4, 24, 80, 200, 420, 784, 1344, 2160, 3300, 4840, 6864, 9464, 12740, 16800, 21760, 27744, 34884, 43320, 53200, 64680, 77924, 93104, 110400, 130000, 152100, 176904, 204624, 235480, 269700, 307520, 349184, 394944, 445060, 499800, 559440

Offset: 0

Matthew T. Cornick (maruth(AT)gmail.com), Sep 16 2005

Second derivative of the n-th Chebyshev polynomial (of the first kind) evaluated at x=1.
The second derivative at x=-1 is just (-1)^n * a(n).
The difference between two consecutive terms generates the sequence a(n+1) - a(n) = A002492(n).
Consider the partitions of 2n into two parts (p,q) where p <= q. Then a(n) is the total volume of the family of rectangular prisms with dimensions p, |q-p| and |q-p|. - Wesley Ivan Hurt, Apr 15 2018

			a(4)=80 because
C_4(x) = 1 - 8x^2 + 8x^4,
C'_4(x) = -16x + 32x^3,
C''_4(x) = -16 + 96x^2,
C''_4(1) = -16 + 96 = 80.

Eric Weisstein's World of Mathematics, Chebyshev polynomials of the first kind
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000914, A002415, A002492.

Table[D[ChebyshevT[n, x], {x, 2}], {n, 0, 100}] /. x -> 1

a(n)=n^2*(n^2-1)/3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (n-1)*n^2*(n+1)/3 = 4*A002415(n).
a(n) = 2*( A000914(n-1) + C(n+1,4) ). - David Scambler, Nov 27 2006
From Colin Barker, Jan 26 2012: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: 4*x^2*(1+x)/(1-x)^5. (End)
E.g.f.: exp(x)*x^2*(6 + 6*x + x^2)/3. - Stefano Spezia, Dec 11 2021
a(n) = A053126(n+2) - A006324(n-1). - Yasser Arath Chavez Reyes, Feb 22 2024

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

A062344 Triangle of binomial(2*n, k) with n >= k.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 15, 20, 1, 8, 28, 56, 70, 1, 10, 45, 120, 210, 252, 1, 12, 66, 220, 495, 792, 924, 1, 14, 91, 364, 1001, 2002, 3003, 3432, 1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 1, 18, 153, 816, 3060, 8568, 18564, 31824, 43758, 48620

Offset: 0

Henry Bottomley, Jul 06 2001

From Wolfdieter Lang, Sep 19 2012: (Start)
The triangle a(n,k) appears in the formula F(2*l+1)^(2*n) = (sum(a(n,k)*L(2*(n-k)*(2*l+1)),k=0..n-1) + a(n,n))/5^n, n>=0, l>=0, with F=A000045 (Fibonacci) and L=A000032 (Lucas).
The signed triangle as(n,k):=a(n,k)*(-1)^k appears in the formula F(2*l)^(2*n) = (sum(as(n,k)*L(4*(n-k)*l),k=0..n-1) + as(n,n))/5^n, n>=0, l>=0. Proof with the Binet-de Moivre formula for F and L and the binomial formula. (End)

			Rows start
  (1),
  (1,2),
  (1,4,6),
  (1,6,15,20)
  etc.
Row n=2, (1,4,6):
F(2*l+1)^4 = (1*L(4*(2*l+1)) + 4*L(2*(2*l+1)) + 6)/25,
F(2*l)^4 = (1*L(8*l) - 4*L(4*l) + 6)/25, l>=0, F=A000045, L=A000032. See a comment above. - _Wolfdieter Lang_, Sep 19 2012

G. C. Greubel, Rows n = 0..100 of triangle, flattened
E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
Index entries for triangles and arrays related to Pascal's triangle

Columns include (sometimes truncated) A000012, A005843, A000384, A002492, A053134 etc. Right hand side includes A000984, A001791, A002694, A002696 etc. Row sums are A032443. Row alternate differences (e.g., 6-4+1=3 or 20-15+6-1=10) are A001700.
Cf. A122366.
a(2*n,n) gives A005810.

[[Binomial(2*n, k): k in [0..n]]: n in [0..20]]; // G. C. Greubel, Jun 28 2018

Flatten[Table[Binomial[2 n, k], {n, 0, 20}, {k, 0, n}]] (* G. C. Greubel, Jun 28 2018 *)

create_list(binomial(2*n,k),n,0,9,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

for(n=0, 20, for(k=0, n, print1(binomial(2*n, k), ", "))) \\ G. C. Greubel, Jun 28 2018

a(n,k) = a(n,k-1)*((2*n+1)/k-1) with a(n,0)=1.

G.f.: 1/((1-sqrt(1-4*x*y))^4/(16*x*y^2) + sqrt(1-4*x*y) - x). - Vladimir Kruchinin, Jan 26 2021

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A053123 Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order).

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A053126 Binomial coefficients binomial(2*n-3,4).

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A035006 Number of possible rook moves on an n X n chessboard.

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A202076 T(n,k)=Number of arrays of n+2 integers in -k..k with sum zero and the sum of every adjacent pair being odd.

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A128908 Riordan array (1, x/(1-x)^2).

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A166464 a(n) = (3 + 2*n + 6*n^2 + 4*n^3)/3.

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A166464 a(n) = (3 + 2n + 6n^2 + 4*n^3)/3.