A078812 -id:A078812 - cp's OEIS frontend

A030523 A convolution triangle of numbers obtained from A001792.

1, 3, 1, 8, 6, 1, 20, 25, 9, 1, 48, 88, 51, 12, 1, 112, 280, 231, 86, 15, 1, 256, 832, 912, 476, 130, 18, 1, 576, 2352, 3276, 2241, 850, 183, 21, 1, 1280, 6400, 10976, 9424, 4645, 1380, 245, 24, 1, 2816, 16896, 34848, 36432, 22363, 8583, 2093, 316, 27, 1

Offset: 1

Wolfdieter Lang

a(n,m) := s1p(3; n,m), a member of a sequence of unsigned triangles including s1p(2; n,m)= A007318(n-1,m-1) (Pascal's triangle). Signed version: (-1)^(n-m)*a(n,m) := s1(3; n,m).
With offset 0, this is T(n,k) = Sum_{i=0..n} C(n,i)*C(i+k+1,2k+1). Binomial transform of A078812 (product of lower triangular matrices). - Paul Barry, Jun 22 2004
Subtriangle of the triangle T(n,k) given by (0, 3, -1/3, 4/3, 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 20 2013

			{1}; {3,1}; {8,6,1}; {20,25,9,1}; {48,88,51,12,1}; ...
(0, 3, -1/3, 4/3, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
1
0   1
0   3   1
0   8   6   1
0  20  25   9   1
0  48  88  51  12   1
...
- _Philippe Deléham_, Feb 20 2013

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First ten rows.

Cf. A057682 (alternating row sums).

a[n_, m_] := SeriesCoefficient[(1-2*x)^2/((x^2-x)*y + (1-2*x)^2) - 1, {x, 0, n}, {y, 0, m}]; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Apr 28 2015, after Vladimir Kruchinin *)

a(n, 1) = A001792(n-1).
Row sums = A039717(n).
a(n, m) = 2*(2*m+n-1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, nT(n,k) = 4*T(n-1,k) - 4*T(n-2,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Feb 20 2013
Sum_{k=1..n} T(n,k)*2^(k-1) = A140766(n). -Philippe Deléham, Feb 20 2013
G.f.: (1-2*x)^2/((x^2-x)*y+(1-2*x)^2)-1. - Vladimir Kruchinin, Apr 28 2015

A092276 Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k.

Original entry on oeis.org

1, 2, 1, 7, 4, 1, 30, 18, 6, 1, 143, 88, 33, 8, 1, 728, 455, 182, 52, 10, 1, 3876, 2448, 1020, 320, 75, 12, 1, 21318, 13566, 5814, 1938, 510, 102, 14, 1, 120175, 76912, 33649, 11704, 3325, 760, 133, 16, 1, 690690, 444015, 197340, 70840, 21252, 5313, 1078, 168, 18, 1

Offset: 1

Emeric Deutsch, Feb 24 2004

With offset 0, Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A006013. - Philippe Deléham, Jan 23 2010

			Triangle begins:
     1;
     2,    1;
     7,    4,    1;
    30,   18,    6,   1;
   143,   88,   33,   8,  1;
   728,  455,  182,  52, 10,  1;
  3876, 2448, 1020, 320, 75, 12, 1;
  ...
Top row of M^3 = (30, 18, 6, 1)
From _Peter Bala_, Nov 25 2024: (Start)
The transposed array as an infinite product of upper triangular arrays:
  /1 2 3 4 5 ... \/1            \/1              \       /1 2 7 30 143 ...\
  |  1 2 3 4 ... ||  1 2 3 4 ...||  1            |       |  1 4 18  88 ...|
  |    1 2 3 ... ||    1 2 3 ...||    1 2 3 4 ...| ... = |    1  6  33 ...|
  |      1 2 ... ||      1 2 ...||      1 2 3 ...|       |       1   8 ...|
  |        1 ... ||        1 ...||        1 2 ...|       |           1 ...|
  |          ... ||          ...||            ...|       |             ...|
Cf. A078812. (End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Peter Bala, Factorisations of some Riordan arrays as infinite products
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Paul Barry, The second production matrix of a Riordan array, arXiv:2011.13985 [math.CO], 2020.
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Row sums give sequence A001764.

Columns 1..5 are A006013, A006629, A006630, A006631, A233657.

T := proc(n,k) if k=n then 1 else 2*k*binomial(3*n-k,n-k)/(3*n-k) fi end: seq(seq(T(n,k),k=1..n),n=1..11);

t[n_, n_] = 1; t[n_, k_] := 2*k*Binomial[3*n-k, n-k]/(3*n-k); Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 22 2012, after Maple *)

T(n, k) = 2*k*binomial(3*n-k, n-k)/(3*n-k); \\ Andrew Howroyd, Nov 06 2017

T(n, k) = 2*k*binomial(3n-k, n-k)/(3n-k).
G.f.: 1/(1-t*z*g^2), where g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z) is the g.f. of the sequence A001764.
T(n, k) = Sum_{j>=1} j*T(n-1, k-2+j). - Philippe Deléham, Sep 14 2005
With offset 0, T(n,k) = ((n+1)/(k+1))*binomial(3n-k+1, n-k). - Philippe Deléham, Jan 23 2010
From Gary W. Adamson, Jul 07 2011: (Start)
Let M = the production matrix
  2, 1;
  3, 2, 1;
  4, 3, 2, 1;
  5, 4, 3, 2, 1;
  ...
Top row of M^(n-1) generates n-th row terms of triangle A092276. Leftmost terms of each row = A006013 starting (1, 2, 7, 30, 143, ...). (End)
Working with an offset of 0, the inverse array is the Riordan array ((1 - x)^2, x*(1 - x)^2). - Peter Bala, Apr 30 2024

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

A208513 Triangle of coefficients of polynomials u(n,x) jointly generated with A111125; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 6, 1, 1, 16, 20, 8, 1, 1, 25, 50, 35, 10, 1, 1, 36, 105, 112, 54, 12, 1, 1, 49, 196, 294, 210, 77, 14, 1, 1, 64, 336, 672, 660, 352, 104, 16, 1, 1, 81, 540, 1386, 1782, 1287, 546, 135, 18, 1, 1, 100, 825, 2640, 4290, 4004, 2275, 800, 170, 20, 1

Offset: 1

Clark Kimberling, Feb 28 2012

The columns of A208513 are identical to those of A208509.  Here, however, the alternating row sums are periodic (with period 1,0,-2,-3,-2,0).
From Tom Copeland, Nov 07 2015: (Start)
These polynomials may be expressed in terms of the Faber polynomials of A263916, similar to A127677.
Rephrasing notes in A111125: Append an initial column of zeros except for a 1 at the top to A111125. Then the rows of this entry contain the partial sums of the column sequences of modified A111125; therefore, the difference of consecutive pairs of rows of this entry, modified by appending an initial row of zeros to it, generates the modified A111125. (End)

			First five rows:
  1;
  1,  1;
  1,  4,  1;
  1,  9,  6, 1;
  1, 16, 20, 8, 1;
First five polynomials u(n,x):
  u(1,x) = 1;
  u(2,x) = 1 +    x;
  u(3,x) = 1 +  4*x +    x^2;
  u(4,x) = 1 +  9*x +  6*x^2 +   x^3;
  u(5,x) = 1 + 16*x + 20*x^2 + 8*x^3 + x^4;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials

Cf. A085478, A078812, A111125, A156308, A208509, A211956.

Cf. A127677, A263916.

A208513:= func< n,k | k eq 1 select 1 else (2*(n-1)/(n+k-2))*Binomial(n+k-2, 2*k-2) >;
[A208513(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 02 2022

(* First program *)
u[1, x_]:=1; v[1, x_]:=1; z=16;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n,z}];
TableForm[cu]
Flatten[%]  (* A208513 *)
Table[Expand[v[n, x]], {n,z}]
cv = Table[CoefficientList[v[n, x], x], {n,z}];
TableForm[cv]
Flatten[%]  (* A111125 *)
(* Second program *)
T[n_, k_]:= If[k==1, 1, ((n-1)/(k-1))*Binomial[n+k-3, 2*k-3]];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Feb 02 2022 *)

def A208513(n,k): return 1 if (k==1) else ((n-1)/(k-1))*binomial(n+k-3, 2*k-3)
flatten([[A208513(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 02 2022

Coefficients of u(n, x) from the mixed recurrence relations:
  u(n,x) = u(n-1,x) + x*v(n-1,x),
  v(n,x) = u(n-1,x) + (x+1)*v(n-1,x) + 1,
where u(1,x) = 1, u(2,x) = 1+x, v(1,x) = 1, v(2,x) = 3+x.
From Peter Bala, May 01 2012: (Start)
Working with an offset of 0: T(n,0) = 1; T(n,k) = (n/k)*binomial(n+k-1,2*k-1) = (n/k)*A078812(n,k) for k > 0. Cf. A156308.
O.g.f.: ((1-t)^2 + t^2*x)/((1-t)*((1-t)^2-t*x)) = 1 + (1+x)*t + (1+4*x+x^2)*t^2 + ....
u(n+1,x) = -1 + (b(2*n,x) + 1)/b(n,x), where b(n,x) = Sum_{k = 0..n} binomial(n+k, 2*k)*x^k are the Morgan-Voyce polynomials of A085478.
This triangle is formed from the even numbered rows of A211956 with a factor of 2^(k-1) removed from the k-th column entries.
(End)
T(n, k) = (2*(n-1)/(n+k-2))*binomial(n+k-2, 2*k-2). - G. C. Greubel, Feb 02 2022

A172431 Even row Pascal-square read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Mark Dols, Feb 02 2010

Apart from signs identical to A053123. Mirror of A078812.
As a triangle, row n consists of the coefficients of Morgan-Voyce polynomial B(n,x); e.g., B(3,x)=x^3+6x^2+10x+4. As a triangle, rows 0 to 4 are as follows: 1 1...2 1...4...3 1...6...10...4 1...8...21...20...5 See A054142 for coefficients of Morgan-Voyce polynomial b(n,x).
Scaled version of A119900. - Philippe Deléham, Feb 24 2012
A172431 is jointly generated with A054142 as an array of coefficients of polynomials v(n,x):  initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=x*u(n-1,x)+v(n-1,x) and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 09 2012
Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 22 2012

			Array begins:
  1,  2,  3,  4,  5,  6, ...
  1,  4, 10, 20, 35, ...
  1,  6, 21, 56, ...
  1,  8, 36, ...
  1, 10, ...
  1, ...
  ...
Example:
Starting with 1, every entry is twice the one to the left minus the second one to the left, plus the one above.
For n = 9 the a(9) = 10 solution is 2*4 - 1 + 3.
From _Philippe Deléham_, Feb 24 2012: (Start)
Triangle T(n,k) 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; (End)
From _Philippe Deléham_, Mar 22 2012: (Start)
(1, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, 1/2, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   4,   3,   0;
  1,   6,  10,   4,   0;
  1,   8,  21,  20,   5,   0;
  1,  10,  36,  56,  35,   6,   0;
  1,  12,  55, 120, 126,  56,   7,   0; (End)

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A078812, A053123, A007318, A001906 (antidiagonals sums), A007685.

Cf. also A054142, A082985.

F:=Factorial;; Flat(List([1..15], n-> List([1..n], k-> Sum([0..Int((k-1)/2)], j-> (-1)^j*F(n-j-1)*2^(k-2*j-1)/(F(j)*F(n-k)*F(k-2*j-1)) )))); # G. C. Greubel, Dec 15 2019

F:=Factorial; [ &+[(-1)^j*F(n-j-1)*2^(k-2*j-1)/(F(j)*F(n-k)*F(k-2*j-1)): j in [0..Floor((k-1)/2)]]: k in [1..n], n in [1..15]]; // G. C. Greubel, Dec 15 2019

T := (n, k) -> simplify(GegenbauerC(k, n-k, 1)):
for n from 0 to 10 do seq(T(n,k), k=0..n-1) od; # Peter Luschny, May 10 2016

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A054142 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A172431 *)
(* Clark Kimberling, Mar 09 2012 *)
Table[GegenbauerC[k-1, n-k+1, 1], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Dec 15 2019 *)

T(n,k) = sum(j=0, (k-1)\2, (-1)^j*(n-j-1)!*2^(k-2*j-1)/(j!*(n-k)!*(k-2*j-1)!) );
for(n=1, 10, for(k=1, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 15 2019

[[gegenbauer(k-1, n-k+1, 1) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Dec 15 2019

As a decimal sequence: a(n)= 12*a(n-1)- a(n-2) with a(1)=1. [I interpret this remark as: 1, 12=1,2, 143=1,4,3, 1704=1,6,10,4,... taken from A004191 are decimals on the diagonal. - R. J. Mathar, Sep 08 2013]
As triangle T(n,k): T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2). - Philippe Deléham, Feb 24 2012
As DELTA-triangle T(n,k) with 0<=k<=n: G.f.: (1-y*x)^2/((1-y*x)^2-x). - Philippe Deléham, Mar 22 2012
T(n, k) = GegenbauerC(k, n-k, 1). - Peter Luschny, May 10 2016
As triangle T(n,k): Product_{k=1..n} T(n,k) = Product_{k=0..n-1} binomial(2*k,k) = A007685(n-1)  for n >= 1. - Werner Schulte, Apr 26 2017
As triangle T(n,k) with 1 <= k <= n: T(n,k) = binomial(2*n-k, k-1). - Paul Weisenhorn, Nov 25 2019

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

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

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A156308 Inverse of triangle S(n,m) defined by sequence A156290, n >= 1, 1 <= m <= n.

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A030523 A convolution triangle of numbers obtained from A001792.

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A092276 Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k.

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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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