A062110 -id:A062110 - cp's OEIS frontend

A147703 Triangle [1,1,1,0,0,0,...] DELTA [1,0,0,0,...] with Deléham DELTA defined in A084938.

1, 1, 1, 2, 3, 1, 5, 9, 5, 1, 13, 27, 20, 7, 1, 34, 80, 73, 35, 9, 1, 89, 234, 252, 151, 54, 11, 1, 233, 677, 837, 597, 269, 77, 13, 1, 610, 1941, 2702, 2225, 1199, 435, 104, 15, 1, 1597, 5523, 8533, 7943, 4956, 2158, 657, 135, 17, 1

Offset: 0

Paul Barry, Nov 10 2008

Equal to A062110*A007318 when A062110 is regarded as a triangle read by rows.

			Triangle begins
   1;
   1,   1;
   2,   3,   1;
   5,   9,   5,   1;
  13,  27,  20,   7,  1;
  34,  80,  73,  35,  9,  1;
  89, 234, 252, 151, 54, 11, 1;

Indranil Ghosh, Rows 0..100, flattened

Row sums are A006012. Diagonal sums are A147704.

Cf. A062110, A007318, A206800, A001519, A321620.

# The function RiordanSquare is defined in A321620:
RiordanSquare(1 / (1 - x / (1 - x / (1 - x))), 10); # Peter Luschny, Jan 26 2020

nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[(1 - 2*x)/(1 - (3 + y)*x + (1 + y)*x^2), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 11 2017 *)

Riordan array ((1-2x)/(1-3x+x^2), x(1-x)/(1-3x+x^2)).
Sum_{k=0..n} T(n,k)*x^k = A152239(n), A152223(n), A152185(n), A152174(n), A152167(n), A152166(n), A152163(n), A000007(n), A001519(n), A006012(n), A081704(n), A082761(n), A147837(n), A147838(n), A147839(n), A147840(n), A147841(n), for x = -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Dec 01 2008
G.f.: (1-2*x)/(1-(3+y)*x+(1+y)*x^2). - Philippe Deléham, Nov 26 2011
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), for n > 1. - Philippe Deléham, Feb 12 2012
The Riordan square of the odd indexed Fibonacci numbers A001519. - Peter Luschny, Jan 26 2020

A106344 Triangle read by rows: T(n,k) = binomial(k,n-k) mod 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1

Offset: 0

Paul Barry, Apr 29 2005

A skew version of Sierpinski’s triangle A047999. - Johannes W. Meijer, Jun 05 2011
Row sums are A002487(n+1). Diagonal sums are A106345. Inverse is A106346.
Triangle formed by reading T triangle mod 2 with T := A026729, A062110, A084938, A099093, A106344, A109466, A110517, A112883, A130167. - Philippe Deléham, Dec 18 2008

			Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 0, 0, 1;
  0, 0, 1, 1, 1;
  0, 0, 0, 1, 0, 1;

G. C. Greubel, Rows n = 0..50, flattened
Thomas Baruchel, Flattening Karatsuba's Recursion Tree into a Single Summation, SN Computer Science (2020) Vol. 1, Article No. 48.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (1.6) p. 2.

Cf. A047999, A002487.
Cf. A106345 (diagonal sums), A106346 (inverse).
Cf. A026729, A062110, A084938, A099093, A106344, A109466, A110517, A112883, A130167.

Flat(List([0..15], n-> List([0..n], k-> (Binomial(k,n-k) mod 2) ))); # G. C. Greubel, Feb 07 2020

[ Binomial(k,n-k) mod 2: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 07 2020

seq(seq(`mod`(binomial(k, n-k), 2), k = 0..n), n = 0..15); # G. C. Greubel, Feb 07 2020

Table[Mod[Binomial[k, n-k], 2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 18 2017 *)

T(n,k) = binomial(k,n-k)%2;
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 07 2020

[[ mod(binomial(k,n-k), 2) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Feb 07 2020

A062109 Expansion of ((1-x)/(1-2*x))^4.

Original entry on oeis.org

1, 4, 14, 44, 129, 360, 968, 2528, 6448, 16128, 39680, 96256, 230656, 546816, 1284096, 2990080, 6909952, 15859712, 36175872, 82051072, 185139200, 415760384, 929562624, 2069889024, 4591714304, 10150215680, 22364028928, 49123688448, 107592286208, 235015241728, 512040632320

Offset: 0

Henry Bottomley, May 30 2001

If X_1,X_2,...,X_n are 2-blocks of a (2n+4)-set X then, for n >= 1, a(n+1) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 23 2007
If the offset here is set to zero, the binomial transform of A006918. - R. J. Mathar, Jun 29 2009
a(n) is the number of weak compositions of n with exactly 3 parts equal to 0. - Milan Janjic, Jun 27 2010
Binomial transform of A002623. - Carl Najafi, Jan 22 2013
Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^4; see A291000. - Clark Kimberling, Aug 24 2017

Harry J. Smith, Table of n, a(n) for n = 0..200
Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Milan Janjic, Two Enumerative Functions
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^4)); // G. C. Greubel, Oct 16 2018

seq(coeff(series(((1-x)/(1-2*x))^4, x,n+1),x,n),n=0..30); # Muniru A Asiru, Jul 01 2018

CoefficientList[Series[(1 - x)^4/(1 - 2 x)^4, {x, 0, 26}], x] (* Michael De Vlieger, Jul 01 2018 *)
LinearRecurrence[{8,-24,32,-16},{1,4,14,44,129},30] (* Harvey P. Dale, Sep 02 2022 *)

a(n)=if(n<1,n==0,(n+5)*(n^2+13*n+18)*2^n/96)

a(n) = (n+5)*(n^2 + 13*n + 18)*2^(n-5)/3, with a(0)=1.
a(n) = A055809(n-5)*2^(n-4).
a(n) = 2*a(n-1) + A058396(n) - A058396(n-1).
a(n) = Sum_{kA058396(n).
a(n) = A062110(4, n).
G.f.: (1-x)^4/(1-2*x)^4.

A160232 Array read by antidiagonals: row n has g.f. ((1-x)/(1-2x))^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 12, 8, 1, 5, 14, 25, 28, 16, 1, 6, 20, 44, 66, 64, 32, 1, 7, 27, 70, 129, 168, 144, 64, 1, 8, 35, 104, 225, 360, 416, 320, 128, 1, 9, 44, 147, 363, 681, 968, 1008, 704, 256, 1, 10, 54, 200, 553, 1182, 1970, 2528, 2400, 1536, 512, 1, 11, 65

Offset: 1

N. J. A. Sloane, May 15 2010

Suggested by a question from Phyllis Chinn (Humboldt State University).
As triangle, mirror image of A105306. - Philippe Deléham, Nov 01 2011
A160232 is jointly generated with A208341 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n > 1, u(n,x) = u(n-1,x) + x*v(n-1)x and v(n,x) = u(n-1,x) + 2x*v(n-1,x).  See the Mathematica section. - Clark Kimberling, Feb 25 2012
Subtriangle of the triangle T(n,k) given by (1, 0, 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 08 2012

			Array begins:
  1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, ...
  1, 2, 5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, 32768, 69632, 147456, 311296, 655360, 1376256, ...
  1, 3, 9, 25, 66, 168, 416, 1008, 2400, 5632, 13056, 29952, 68096, 153600, 344064, 765952, 1695744, 3735552, ...
  1, 4, 14, 44, 129, 360, 968, 2528, 6448, 16128, 39680, 96256, 230656, 546816, 1284096, 2990080, 6909952, ...
  1, 5, 20, 70, 225, 681, 1970, 5500, 14920, 39520, 102592, 261760, 657920, 1632000, 4001280, 9708544, ...
  1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, 236640, 632448, 1661056, 4296192, 10961664, 27630592, ...
From _Clark Kimberling_, Feb 25 2012: (Start)
As a triangle (see Comments):
  1;
  1,  1;
  1,  2,  2;
  1,  3,  5,  4;
  1,  4,  9, 12,  8;  (End)
From _Philippe Deléham_, Mar 08 2012: (Start)
(1, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  2,  0;
  1,  3,  5,  4,  0;
  1,  4,  9, 12,  8,  0;
  1,  5, 14, 25, 28, 16,  0; (End)

Rows give A011782, A045623, A058396, A062109, A169792-A169797.

Cf. A062110, A105306, A208341.

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2*x*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[%]  (* A160232 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208341 *)
(* Clark Kimberling, Feb 25 2012 *)

From Philippe Deléham, Mar 08 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1-2*y*x)/(1-2*y*x-x+y*x^2).
Sum_{k=0..n, n>0} T(n,k)*x^k = A000012(n), A001519(n), A052984(n-1) for x = 0, 1, 2 respectively. (End)

A105306 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n, having the top of the rightmost column at height k.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 8, 12, 9, 4, 1, 16, 28, 25, 14, 5, 1, 32, 64, 66, 44, 20, 6, 1, 64, 144, 168, 129, 70, 27, 7, 1, 128, 320, 416, 360, 225, 104, 35, 8, 1, 256, 704, 1008, 968, 681, 363, 147, 44, 9, 1, 512, 1536, 2400, 2528, 1970, 1182, 553, 200, 54, 10, 1, 1024

Offset: 1

Emeric Deutsch, Apr 25 2005

From Gary W. Adamson, Apr 24 2005: (Start)
Let A be the array
  1, 0, 0, 0, 0, 0, ...
  0, 1, 0, 0, 0, 0, ...
  1, 0, 1, 0, 0, 0, ...
  0, 2, 0, 1, 0, 0, ...
  1, 0, 3, 0, 1, 0, ...
  0, 3, 0, 4, 0, 1, ...
  ...
where columns are bin(n,k) with alternating zeros. (Row sums = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...(Fibonacci numbers).) Let P = infinite lower triangular Pascal triangle matrix (A007318). Form P * A: this gives the rows of the present sequence. [Comment corrected by Philippe Deléham, Dec 09 2008] (End)
T(n,k) is the number of nondecreasing Dyck paths of semilength n, having height of rightmost peak equal to k. Example: T(4,1)=4 because we have UDUDUDUD, UDUUDDUD, UUDDUDUD and UUUDDDUD, where U=(1,1) and D=(1,-1). Sum of row n = Fibonacci(2n-1) (A001519). Basically the same as A062110.
T(n,k) is the number of permutations of [n] with length n-k that avoid the patterns 321 and 3412. - Bridget Tenner, Sep 28 2005
T(2*n-1,n)/n = A001003(n-1) (little Schroeder numbers). Proof with Lagrange inversion of inverse of g.f. of A001003.
Row sums = odd-indexed Fibonacci numbers.
Diagonal sums: A077998. - Philippe Deléham, Nov 16 2008
Central coefficients are A176479. Inverse is A125692. - Paul Barry, Apr 18 2010
Riordan matrix ((1-x)/(1-2x),(x-x^2)/(1-2x)). - Emanuele Munarini, Mar 22 2011
T(n,k) is the number of ideals in the fence Z(2n-1) with  n-k elements of rank 0. - Emanuele Munarini, Mar 22 2011
Triangle T(n,k), 1 <= k <= n, read by rows, given by (0,1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 30 2011
T(n,k) is the number of permutations of [n] for which k is equal to both the length and reflection length. - Bridget Tenner, Feb 22 2012

			Triangle begins:
   1;
   1,   1;
   2,   2,   1;
   4,   5,   3,   1;
   8,  12,   9,   4,   1;
  16,  28   25,  14,   5,   1;
  32,  64,  66,  44,  20,   6,   1;
  64, 144, 168, 129,  70,  27,   7,   1;
  ...
From _Paul Barry_, Apr 18 2010: (Start)
Production matrix is
   1,  1;
   1,  1,  1;
   0,  1,  1,  1;
  -1,  0,  1,  1,  1;
   0, -1,  0,  1,  1,  1;
   2,  0, -1,  0,  1,  1,  1;
   0,  2,  0, -1,  0,  1,  1,  1;
  -5,  0,  2,  0, -1,  0,  1,  1,  1;
   0, -5,  0,  2,  0, -1,  0,  1,  1,  1;
  14,  0, -5,  0,  2,  0, -1,  0,  1,  1,  1; (End)

V. E. Hoggatt, Jr. and Marjorie Bicknell, editors: "A Primer for the Fibonacci Numbers", 1970, p. 87.

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Jean-Luc Baril, Javier F. González, and José L. Ramírez, Last symbol distribution in pattern avoiding Catalan words, Univ. Bourgogne (France, 2022).
Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega, J. Rafael Sendra, Juana Sendra, and Steven E. Thornton, Bohemian Upper Hessenberg Toeplitz Matrices, arXiv:1809.10664 [cs.SC], 2018.
Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega, J. Rafael Sendra, and Juana Sendra, Upper Hessenberg and Toeplitz Bohemians, arXiv:1907.10677 [cs.SC], 2019.
Emeric Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021).
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 9-10, 19.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
V. V. Kruchinin and D. V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, arXiv preprint arXiv:1206.0877 [math.CO], 2012, and J. Int. Seq. 15 (2012) #12.9.3.
E. Munarini and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
T. K. Petersen and B. E. Tenner, The depth of a permutation, arXiv:1202.4765 [math.CO], 2012-2014..
M. Pétréolle, Characterization of Cyclically Fully commutative elements in finite and affine Coxeter Groups, arXiv preprint arXiv:1403.1130 [math.GR], 2014.
Bridget Eileen Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
S.-n. Zheng and S.-l. Yang, On the-Shifted Central Coefficients of Riordan Matrices, Journal of Applied Mathematics, Volume 2014, Article ID 848374, 8 pages.

Cf. A001519. Essentially the same array as A062110.

Row sums = A001519(n-1), n >= 1.

Maple
```
T:=proc(n,k) if k
				
```

t[n_, k_] := 2^(n-2*k-1)*Binomial[n, k]*Hypergeometric2F1[-k-1, -k, -n, -1]; t[n_, n_] = 1; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 28 2014 *)

create_list(sum(binomial(k+1,i)*binomial(n-i,k)*(-1)^i*2^(n-k-i),i,0,n-k),n,0,8,k,0,n); /* Emanuele Munarini, Mar 22 2011 */

T(n,k) = sum(binomial(k+j, k-1)*binomial(n-k-1, j), j=0..n-k-1) (0<=k<=n). (This appears to be incorrect. - Emanuele Munarini, Mar 22 2011)
G.f.: t*z*(1-z)/(1 - 2*z - t*z*(1-z)).
From Emanuele Munarini, Mar 22 2011: (Start)
T(n,k) = Sum_{i=0..n-k} binomial(k+1,i)*binomial(n-i,k)*(-1)^i*2^(n-k-i).
T(n,k) = Sum_{i=0..n-k} binomial(k+1,i)*M(i,n-k-i)*2^(n-k-i), where M(n,k) = n*(n+1)*(n+2)*...*(n+k-1)/k!. (End)
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Oct 30 2013

Entry revised by N. J. A. Sloane, Apr 27 2007

A121462 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having pyramid weight k (1 <= k <= n).

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 1, 4, 8, 0, 1, 5, 12, 16, 0, 1, 6, 18, 32, 32, 0, 1, 7, 25, 56, 80, 64, 0, 1, 8, 33, 88, 160, 192, 128, 0, 1, 9, 42, 129, 280, 432, 448, 256, 0, 1, 10, 52, 180, 450, 832, 1120, 1024, 512, 0, 1, 11, 63, 242, 681, 1452, 2352, 2816, 2304, 1024, 0, 1, 12, 75, 316

Offset: 1

Emeric Deutsch, Jul 31 2006

A pyramid in a Dyck word (path) is a factor of the form U^h D^h, where U=(1,1), D=(1,-1) and h is the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a u and immediately followed by a d. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids.
Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,n)=2^(n-1). Sum_{k=1..n} k*T(n,k) = A030267(n).
Mirror image of triangle in A153342. - Philippe Deléham, Dec 31 2008
Essentially triangle given by (0,1/2,1/2,0,0,0,0,0,0,0,...) DELTA (2,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 30 2011
A121462 is jointly generated with A208341 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1; for n > 1, u(n,x) = x*u(n-1,x) + x*v(n-1) and v(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x).  See the Mathematica section. - Clark Kimberling, Mar 11 2012

			T(4,3)=4 because we have (UD)U(UD)(UD)D, U(UD)(UD)(UD)D, U(UD)(UUDD)D and U(UUDD)(UD)D, where U=(1,1) and D=(1,-1) (the maximal pyramids are shown between parentheses).
Triangle starts:
  1;
  0,  2;
  0,  1,  4;
  0,  1,  4,  8;
  0,  1,  5, 12, 16;
  0,  1,  6, 18, 32, 32;

Elena Barcucci, Alberto del Lungo, S. Fezzi and Renzo Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
Alain Denise and Rodica Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
Emeric Deutsch and Helmut Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.

Cf. A001519, A030267, A091866.

T:=proc(n,k) if n=1 and k=1 then 1 elif k=1 then 0 elif k<=n then sum(binomial(k-1,j)*binomial(n-k-1+j,j-1),j=0..k-1) else 0 fi end: for n from 1 to 13 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + 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[%]    (* A121462 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208341 *)
(* Clark Kimberling, Mar 11 2012 *)

T(n,k) = Sum_{j=0..k-1} binomial(k-1,j)*binomial(n-k-1+j,j-1) for 2 <= k <= n; T(1,1)=1; T(n,1)=0 for n >= 2.
G.f.: G = G(t,z) = tz(1-z)/(1-2tz-z+tz^2).
T(n+1,k+1) = A062110(n,k)*2^(2*k-n). - Philippe Deléham, Aug 01 2006

cp's OEIS Frontend

A147703 Triangle [1,1,1,0,0,0,...] DELTA [1,0,0,0,...] with Deléham DELTA defined in A084938.

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A106344 Triangle read by rows: T(n,k) = binomial(k,n-k) mod 2.

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A062109 Expansion of ((1-x)/(1-2*x))^4.

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A160232 Array read by antidiagonals: row n has g.f. ((1-x)/(1-2x))^n.

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A105306 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n, having the top of the rightmost column at height k.

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A121462 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having pyramid weight k (1 <= k <= n).

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