A053122 -id:A053122 - cp's OEIS frontend

A159764 Riordan array (1/(1+4x+x^2), x/(1+4x+x^2)).

1, -4, 1, 15, -8, 1, -56, 46, -12, 1, 209, -232, 93, -16, 1, -780, 1091, -592, 156, -20, 1, 2911, -4912, 3366, -1200, 235, -24, 1, -10864, 21468, -17784, 8010, -2120, 330, -28, 1, 40545, -91824, 89238, -48624, 16255, -3416, 441, -32, 1, -151316, 386373

Offset: 0

Paul Barry, Apr 21 2009

Row sums are (-1)^n*F(2n+2). Diagonal sums are (-1)^n*4^n. Inverse is A052179.
The positive matrix is (1/(1-4x+x^2), x/(1-4x+x^2)) with general term T(n,k) = if(k<=n, Gegenbauer_C(n-k,k+1,2),0).
For another version, see A124029.
Triangle of coefficients of Chebyshev's S(n,x-4) polynomials (exponents of x in increasing order). - Philippe Deléham, Feb 22 2012
Subtriangle of triangle given by (0, -4, 1/4, -1/4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 22 2012

			Triangle begins
     1;
    -4,     1;
    15,    -8,     1;
   -56,    46,   -12,     1;
   209,  -232,    93,   -16,     1;
  -780,  1091,  -592,   156,   -20,     1;
  2911, -4912,  3366, -1200,   235,   -24,     1;
Triangle (0, -4, 1/4, -1/4, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1;
  0,    1;
  0,   -4,    1;
  0,   15,   -8,    1;
  0,  -56,   46,  -12,    1;
  0,  209, -232,   93,  -16,    1;

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. Triangle of coefficients of Chebyshev's S(n,x+k) polynomials : A207824, A207823, A125662, A078812, A101950, A049310, A104562, A053122, A207815, A159764, A123967 for k = 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5 respectively.

CoefficientList[CoefficientList[Series[1/(1 + 4*x + x^2 - y*x), {x, 0, 10}, {y, 0, 10}], x], y]//Flatten (* G. C. Greubel, May 21 2018 *)

@CachedFunction
def A159764(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    return A159764(n-1,k-1)-A159764(n-2,k)-4*A159764(n-1,k)
for n in (0..9): [A159764(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Number triangle T(n,k) = if(k<=n, Gegenbauer_C(n-k,k+1,-2),0).
G.f.: 1/(1+4*x+x^2-y*x). - Philippe Deléham, Feb 22 2012
T(n,k) = (-4)*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Feb 22 2012

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

A103328 Triangle T(n, k) read by rows: binomial(2n, 2k+1).

Original entry on oeis.org

0, 2, 0, 4, 4, 0, 6, 20, 6, 0, 8, 56, 56, 8, 0, 10, 120, 252, 120, 10, 0, 12, 220, 792, 792, 220, 12, 0, 14, 364, 2002, 3432, 2002, 364, 14, 0, 16, 560, 4368, 11440, 11440, 4368, 560, 16, 0, 18, 816, 8568, 31824, 48620, 31824, 8568, 816, 18, 0, 20, 1140, 15504

Offset: 0

Ralf Stephan, Feb 06 2005

A subset of Pascal's triangle A007318 with only even elements.

			Triangle begins
   0;
   2,   0;
   4,   4,    0;
   6,  20,    6,     0;
   8,  56,   56,     8,     0;
  10, 120,  252,   120,    10,    0;
  12, 220,  792,   792,   220,   12,   0;
  14, 364, 2002,  3432,  2002,  364,  14,  0;
  16, 560, 4368, 11440, 11440, 4368, 560, 16, 0;
  ...
From _Peter Bala_, Jan 30 2022: (Start)
(1/2)*(N^2 + N)^2 = 2*Sum_{j = 1..N} j^3.
(1/2)*(N^2 + N)^4 = 4*Sum_{j = 1..N} j^5 + 4*Sum_{j = 1..N} j^7.
(1/2)*(N^2 + N)^6 = 6*Sum_{j = 1..N} j^7 + 20*Sum_{j = 1..N} j^9 + 6*Sum_{j = 1..N} j^11.
(1/2)*(N^2 + N)^8 = 8*Sum_{j = 1..N} j^9 + 56*Sum_{j = 1..N} j^11 + 56*Sum_{j = 1..N} j^13 + 8*Sum_{j = 1..N} j^15. (End)

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.

Cf. A091042, A086645, A103327.

From Peter Bala, Jan 31 2022: (Start)
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) + 2*T(n-2,k-1) - T(n-2,k-2), with T(0,0) = 0, T(1,0) = 2 and T(n,k) = 0 if k < 0 or if k > n-1.
n-th row polynomial R(n,x) = (1/(2*sqrt(x)))*( (1 + sqrt(x))^(2*n) - (1 - sqrt(x))^(2*n) ).
O.g.f.: A(x,t) = 2*t/(1 - 2*(x + 1)*t + (x - 1)^2*t^2) = 2*t + (4 + 4*x)*t^2 + (6 + 20*x + 6*x^2)*t^3 + ....
G.f.: (1/sqrt(x))*sinh(t)*sinh(sqrt(x)*t) = 2*t^2/2! + (4 + 4*x)*t^4/4! + (6 + 20*x^2 + 6*x^3)*t^6/6! + ....
O.g.f. for n-th diagonal: ( Sum_{k = 0..n} binomial(2*n,2*k+1)*x^k )/(1 - x)^(2*n) = 1/(2*sqrt(x))*((1 - sqrt(x))^(-2*n) - (1 + sqrt(x))^(-2*n)).
With a different offset, 2/(x-4)*A(x/(x-4), t*(x-4)/4) = t/(1 + t*(2 - x) + t^2) is a g.f. of A053122.
Define S(r,N) = Sum_{j = 1..N} j^r. Then the following identity holds for n >= 1:
(1/2)*(N^2 + N)^(2*n) = T(n,0)*S(2*n+1,N) + T(n,1)*S(2*n+3,N) + ... + T(n,n-1)*S(4*n-1,N). Some examples are given below. (End)

A123967 Triangle read by rows: T(0,0)=1; for n >= 1 T(n,k) is the coefficient of x^k in the monic characteristic polynomial of the tridiagonal n X n matrix with main diagonal 5,5,5,... and sub- and superdiagonals 1,1,1,... (0 <= k <= n).

Original entry on oeis.org

1, -5, 1, 24, -10, 1, -115, 73, -15, 1, 551, -470, 147, -20, 1, -2640, 2828, -1190, 246, -25, 1, 12649, -16310, 8631, -2400, 370, -30, 1, -60605, 91371, -58275, 20385, -4225, 519, -35, 1, 290376, -501150, 374115, -157800, 41140, -6790, 693, -40, 1, -1391275, 2704755, -2313450, 1142730, -359275, 74571, -10220, 892, -45, 1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Oct 28 2006

Riordan array (1/(1+5*x+x^2), x/(1+5*x+x^2)). - Philippe Deléham, Feb 03 2007
Chebyshev's S(n,x-5) polynomials (exponents of x in increasing order). - Philippe Deléham, Feb 22 2012
Row sums are A125905(n). - Philippe Deléham, Feb 22 2012
Diagonal sums are (-5)^n. - Philippe Deléham, Feb 22 2012
Subtriangle of triangle given by (0, -5, 1/5, -1/5, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 22 2012
Inverse of triangle in A125906. - Philippe Deléham, Feb 22 2012

			Triangle starts:
      1;
     -5,      1;
     24,    -10,     1;
   -115,     73,   -15,     1;
    551,   -470,   147,   -20,   1;
  -2640,   2828, -1190,   246, -25,   1;
  12649, -16310,  8631, -2400, 370, -30, 1;
  ...
Triangle (0, -5, 1/5, -1/5, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1;
  0,     1;
  0,    -5,    1;
  0,    24,  -10,     1:
  0,  -115,   73,   -15,   1;
  0,   551, -470,   147, -20,   1;
  0, -2640, 2828, -1190, 246, -25, 1;
  ...

Eric Weisstein's World of Mathematics, Tridiagonal Matrix

Cf. A123343, A004254.

Cf. Chebyshev's S(n,x+k) polynomials : A207824 (k = 5), A207823 (k = 4), A125662 (k = 3), A078812 (k=2), A101950 (k = 1), A049310 (k = 0), A104562 (k = -1), A053122 (k = -2), A207815 (k = -3), A159764 (k = -4), A123967 (k = -5).

with(linalg): m:=proc(i,j) if i=j then 5 elif abs(i-j)=1 then 1 else 0 fi end: T:=(n,k)->coeff(charpoly(matrix(n,n,m),x),x,k): 1; for n from 1 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] - 5 T[n-1, k] - T[n-2, k]; T[0, 0] = 1; T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018, after Philippe Deléham *)

@CachedFunction
def A123967(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    return A123967(n-1,k-1)-A123967(n-2,k)-5*A123967(n-1,k)
for n in (0..9): [A123967(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

T(n,0) = (-1)^n*A004254(n+1).
G.f.: 1/(1+5*x+x^2 - y*x). - Philippe Deléham, Feb 22 2012
T(n,k) = T(n-1,k-1) - 5*T(n-1,k) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 22 2014

Edited by N. J. A. Sloane, Dec 03 2006

A125662 A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).

Original entry on oeis.org

1, 3, 1, 8, 6, 1, 21, 25, 9, 1, 55, 90, 51, 12, 1, 144, 300, 234, 86, 15, 1, 377, 954, 951, 480, 130, 18, 1, 987, 2939, 3573, 2305, 855, 183, 21, 1, 2584, 8850, 12707, 10008, 4740, 1386, 245, 24, 1, 6765, 26195, 43398, 40426, 23373, 8715, 2100, 316, 27, 1

Offset: 0

Philippe Deléham, Jan 28 2007

Subtriangle of the triangle given by [0,3,-1/3,1/3,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Unsigned version of A123965.
From Philippe Deléham, Feb 19 2012: (Start)
Riordan array (1/(1-3*x+x^2), x/(1-3*x+x^2)).
Equals A078812*A007318 as infinite lower triangular matrices.
Triangle of coefficients of Chebyshev's S(n,x+3) polynomials (exponents of x in increasing order). (End)
For 1 <= k <= n, T(n,k) equals the number of (n-1)-length  words over {0,1,2,3} containing k-1 letters equal 3 and avoiding 01. - Milan Janjic, Dec 20 2016

			Triangle begins:
   1;
   3,  1;
   8,  6,  1;
  21, 25,  9,  1;
  55, 90, 51, 12,  1;
  ...
Triangle [0,3,-1/3,1/3,0,0,0,...] DELTA [1,0,0,0,0,0,...] begins:
  1;
  0,  1;
  0,  3,  1;
  0,  8,  6,  1;
  0, 21, 25,  9,  1;
  0, 55, 90, 51, 12,  1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Diagonal sums: A000244(powers of 3).
Row sums: A001353 (n+1).
Diagonals: A001906(n+1), A001871.
Cf. A000012, A008585, A062728.
Cf. Triangle of coefficients of Chebyshev's S(n,x+k) polynomials: A207824, A207823, A125662, A078812, A101950, A049310, A104562, A053122, A207815, A159764, A123967 for k = 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5 respectively.

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
A125662:= func< n,k | Abs( Coefficient(R!( Evaluate(ChebyshevU(n+1), (3-x)/2) ), k) ) >;
[A125662(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Aug 20 2023

With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 3 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)
Table[Abs[CoefficientList[ChebyshevU[n,(x-3)/2], x]], {n,0,12}]//Flatten (* G. C. Greubel, Aug 20 2023 *)

def A125662(n,k): return abs( ( chebyshev_U(n, (3-x)/2) ).series(x, n+2).list()[k] )
flatten([[A125662(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 20 2023

T(n,k) = T(n-1,k-1) + 3*T(n-1,k) - T(n-2,k); T(0,0)=1; T(n,k)=0 if k < 0 or k > n.
Sum_{k=0..n} T(n, k) = A001353(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000244(n+1).
G.f.: 1/(1-3*x+x^2-y*x). - Philippe Deléham, Feb 19 2012
From G. C. Greubel, Aug 20 2023: (Start)
T(n, k) = abs( [x^k]( ChebyshevU(n, (3-x)/2) ) ).
Sum_{k=0..n} (-1)^k*T(n, k) = A000027(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000225(n). (End)

a(45) corrected and a(51) added by Philippe Deléham, Feb 19 2012

A207823 Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).

Original entry on oeis.org

1, 4, 1, 15, 8, 1, 56, 46, 12, 1, 209, 232, 93, 16, 1, 780, 1091, 592, 156, 20, 1, 2911, 4912, 3366, 1200, 235, 24, 1, 10864, 21468, 17784, 8010, 2120, 330, 28, 1, 40545, 91824, 89238, 48624, 16255, 3416, 441, 32, 1, 151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1

Offset: 0

Philippe Deléham, Feb 20 2012

Riordan array (1/(1-4*x+x^2), x/(1-4*x+x^2)).
Subtriangle of the triangle given by (0, 4, -1/4, 1/4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Unsigned version of triangles in A124029 and in A159764.
For 1<=k<=n, T(n,k) equals the number of (n-1)-length  words over {0,1,2,3,4} containing k-1 letters equal 4 and avoiding 01. - Milan Janjic, Dec 20 2016

			Triangle begins:
  1
  4, 1
  15, 8, 1
  56, 46, 12, 1
  209, 232, 93, 16, 1
  780, 1091, 592, 156, 20, 1
  2911, 4912, 3366, 1200, 235, 24, 1
  10864, 21468, 17784, 8010, 2120, 330, 28, 1
  40545, 91824, 89238, 48624, 16255, 3416, 441, 32, 1
  151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1
  ...
Triangle (0, 4, -1/4, 1/4, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1
  0, 1
  0, 4, 1
  0, 15, 8, 1
  0, 56, 46, 12, 1
  0, 209, 232, 93, 16, 1
  ...

Rigoberto Flórez, Leandro Junes, José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, J. Int. Seq. 21 (2018), #18.1.2.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. Triangle of coefficients of Chebyshev's S(n,x+k) polynomials: A207824 (k = 5), A207823 (k = 4), A125662 (k = 3), A078812 (k = 2), A101950 (k = 1), A049310 (k = 0), A104562 (k = -1), A053122 (k = -2), A207815 (k = -3), A159764 (k = -4), A123967 (k = -5).

With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 4 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Recurrence: T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - T(n-2,k).
Diagonal sums are 4^n = A000302(n).
Row sums are A004254(n+1).
G.f.: 1/(1-4*x+x^2-y*x)
T(n,n) = 1, T(n+1,n) = 4*n+4 = A008586(n+1), T(n+2,n) = (n+1)*(8n+15) = A139278(n+1).
T(n,0) = A001353(n+1).

Offset changed to 0 by Georg Fischer, Feb 18 2020

A207815 Triangle of coefficients of Chebyshev's S(n,x-3) polynomials (exponents of x in increasing order).

Original entry on oeis.org

1, -3, 1, 8, -6, 1, -21, 25, -9, 1, 55, -90, 51, -12, 1, -144, 300, -234, 86, -15, 1, 377, -954, 951, -480, 130, -18, 1, -987, 2939, -3573, 2305, -855, 183, -21, 1, 2584, -8850, 12707, -10008, 4740, -1386, 245, -24, 1, -6765, 26195, -43398, 40426, -23373, 8715

Offset: 0

Philippe Deléham, Feb 20 2012

Riordan array (1/(1+3*x+x^2), x/(1+3*x+x^2)).
Subtriangle of the triangle given by (0, -3, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Diagonal sums are (-3)^n.
Inverse array is A091965.

			Triangle begins:
      1;
     -3,     1;
      8,    -6,      1;
    -21,    25,     -9,      1;
     55,   -90,     51,    -12,      1;
   -144,   300,   -234,     86,    -15,     1;
    377,  -954,    951,   -480,    130,   -18,     1;
   -987,  2939,  -3573,   2305,   -855,   183,   -21,   1;
   2584, -8850,  12707, -10008,   4740, -1386,   245, -24,   1;
  -6765, 26195, -43398,  40426, -23373,  8715, -2100, 316, -27, 1;
Triangle (0, -3, 1/3, -1/3, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1;
  0,    1;
  0,   -3,   1;
  0,    8,  -6,    1;
  0,  -21,  25,   -9,   1;
  0,   55, -90,   51, -12,   1;
  0, -144, 300, -234,  86, -15, 1;
  ...

Cf. Chebyshev's S(n,x+k) polynomials: A207824 (k = 5), A207823 (k = 4), A125662 (k = 3), A078812 (k = 2), A101950 (k = 1), A049310 (k = 0), A104562 (k = -1), A053122 (k = -2), A207815 (k = -3), A159764 (k = -4), A123967 (k = -5).

T[?Negative, ] = 0; T[0, 0] = 1; T[0, ] = 0; T[n, n_] = 1; T[n_, k_] := T[n, k] = T[n - 1, k - 1] - T[n - 2, k] - 3 T[n - 1, k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 22 2018 *)

row(n) = Vecrev(subst(polchebyshev(n,2,x/2), x, x-3))
tabf(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Jun 22 2018

@CachedFunction
def A207815(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    return A207815(n-1,k-1)-A207815(n-2,k)-3*A207815(n-1,k)
for n in (0..9): [A207815(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

T(n,k) = (-1)^(n-k)*A125662(n,k).
Recurrence: T(n,k) = (-3)*T(n-1,k) + T(n-1,k-1) - T(n-2,k).
G.f.: 1/(1+3*x+x^2-y*x).

T(8,0) corrected by Jean-François Alcover, Jun 22 2018

A207824 Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).

Original entry on oeis.org

1, 5, 1, 24, 10, 1, 115, 73, 15, 1, 551, 470, 147, 20, 1, 2640, 2828, 1190, 246, 25, 1, 12649, 16310, 8631, 2400, 370, 30, 1, 60605, 91371, 58275, 20385, 4225, 519, 35, 1, 290376, 501150, 374115, 157800, 41140, 6790, 693, 40, 1

Offset: 0

Philippe Deléham, Feb 20 2012

Riordan array (1/(1-5*x+x^2), x/(1-5*x+x^2)).
Subtriangle of triangle given by (0, 5, -1/5, 1/5, 0, 0, 0, 0, 0, 0, 0, 0...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Unsigned version of A123967 and A179900.
For 1<=k<=n, T(n,k) equals the number of (n-1)-length  words over {0,1,2,3,4,5} containing k-1 letters equal 5 and avoiding 01. -  Milan Janjic, Dec 20 2016

			Triangle begins :
  1
  5, 1
  24, 10, 1
  115, 73, 15, 1
  551, 470, 147, 20, 1
  2640, 2828, 1190, 246, 25, 1
  12649, 16310, 8631, 2400, 370, 30, 1
  ...
Triangle (0, 5, -1/5, 1/5, 0, 0, 0,...) DELTA (1, 0, 0, 0, ...) begins :
  1
  0, 1
  0, 5, 1
  0, 24, 10, 1
  0, 115, 73, 15, 1
  0, 551, 470, 147, 20, 1
  0, 2640, 2828, 1190, 246, 25, 1
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. Triangles of coefficients of Chebyshev's S(n,x+k) polynomials : A207824 (k = 5), A207823 (k = 4), A125662 (k = 3), A078812 (k = 2), A101950 (k = 1), A049310 (k = 0), A104562 (k = -1), A053122 (k = -2), A207815 (k = -3), A159764 (k = -4), A123967 (k = -5).

With[{n = 8}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 5 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

row(n) = Vecrev(polchebyshev(n, 2, (x+5)/2)); \\ Michel Marcus, Apr 26 2018

Recurrence : T(n,k) = 5*T(n-1,k) + T(n-1,k-1) - T(n-2,k).
G.f.: 1/(1-5*x+x^2-y*x).
Diagonal sums are 5^n = A000351(n).
Row sums are A001109(n+1).
T(n,0) = A004254(n+1), T(n+1,n) = 5n+5 = A008587(n+1).

A053124 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in increasing order).

Original entry on oeis.org

1, -2, 4, 3, -16, 16, -4, 40, -96, 64, 5, -80, 336, -512, 256, -6, 140, -896, 2304, -2560, 1024, 7, -224, 2016, -7680, 14080, -12288, 4096, -8, 336, -4032, 21120, -56320, 79872, -57344, 16384, 9, -480, 7392, -50688, 183040, -372736, 430080, -262144, 65536, -10, 660, -12672, 109824, -512512, 1397760, -2293760, 2228224

Offset: 0

Wolfdieter Lang

a(n,m) = (4^m)*A053122(n,m).
G.f. for row polynomials U^{*}(n,x) = U(n,2*x-1) (signed triangle): 1/(1+2*z*(1-2*x) + z^2). Unsigned triangle |a(n,m)| has g.f. 1/(1-2*z*(1+2*x)+z^2) for the row polynomials.
Row sums (signed triangle) A000027(n+1) (natural numbers). Row sums (unsigned triangle) A001109(n+1).
In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to a Riordan group.

			{1}; {-2,4}; {3,-16,16}; {-4,40,-96,64}; {5,-80,336,-512,256};... E.g., fourth row (n=3) {-4,40,-96,64} corresponds to polynomial U(3,2*x-1)= -4+40*x-96*x^2+64*x^3.

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 1, problem 39, page 7.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

T. D. Noe, Rows n = 0..50 of triangle, flattened
C. Lanczos, Applied Analysis (Annotated scans of selected pages). See page 518.
Index entries for sequences related to Chebyshev polynomials.

Cf. A053122, A053125.

Table[ CoefficientList[ ChebyshevU[n, 2x - 1], x], {n, 0, 9}] // Flatten (* Jean-François Alcover, Dec 05 2012 *)

a(n, m) := 0 if n < m, otherwise (4^m)*((-1)^(n-m))*binomial(n+m+1, 2*m+1);
a(n, m) = -2*a(n-1, m) + 4*a(n-1, m-1) - a(n-2, m), a(n, m) := 0 if n=-1 or m=-1 or n < m, a(0, 0)=1;
g.f. for m-th column (signed triangle): ((4*x/(1+x)^2)^m)/(1+x)^2.
In other words, Riordan array (1/(1+x)^2, 4x/(1+x)^2). - Ralf Stephan, Jan 21 2014

A073135 Table by antidiagonals of T(n,k) = 2nT(n,k-1) - n^2*T(n,k-2) + T(n,k-4) starting with T(n,1) = 1.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 12, 6, 1, 6, 32, 27, 8, 1, 10, 81, 108, 48, 10, 1, 17, 200, 406, 256, 75, 12, 1, 28, 488, 1470, 1281, 500, 108, 14, 1, 45, 1184, 5193, 6160, 3126, 864, 147, 16, 1, 72, 2865, 18036, 28832, 18770, 6481, 1372, 192, 18, 1, 116, 6924, 61885, 132352

Offset: 0

Henry Bottomley, Jul 16 2002

			Rows start:
  1, 2,  3,   4,    6,   10,    17, ...;
  1, 4, 12,  32,   81,  200,   488, ...;
  1, 6, 27, 108,  406, 1470,  5193, ...;
  1, 8, 48, 256, 1281, 6160, 28832, ...;
  ...

Cf. A053122, A073133, A073134.
Rows include A024490, A048776.
Columns include A000012, A005843, A033428, A033430.

T(n, k) = (A073133(n, k+2) - A073134(n, k+2))/2.

T(n, k) = Sum_{j=0..floor((k-1)/4)} abs(A053122(k-3*j-1, j)*n^(k-4*j-1)).

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A159764 Riordan array (1/(1+4x+x^2), x/(1+4x+x^2)).

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

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A103328 Triangle T(n, k) read by rows: binomial(2n, 2k+1).

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A123967 Triangle read by rows: T(0,0)=1; for n >= 1 T(n,k) is the coefficient of x^k in the monic characteristic polynomial of the tridiagonal n X n matrix with main diagonal 5,5,5,... and sub- and superdiagonals 1,1,1,... (0 <= k <= n).

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A125662 A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).

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A207823 Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).

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A207815 Triangle of coefficients of Chebyshev's S(n,x-3) polynomials (exponents of x in increasing order).

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A073135 Table by antidiagonals of T(n,k) = 2nT(n,k-1) - n^2*T(n,k-2) + T(n,k-4) starting with T(n,1) = 1.