A080246 -id:A080246 - cp's OEIS frontend

A080247 Formal inverse of triangle A080246. Unsigned version of A080245.

1, 2, 1, 6, 4, 1, 22, 16, 6, 1, 90, 68, 30, 8, 1, 394, 304, 146, 48, 10, 1, 1806, 1412, 714, 264, 70, 12, 1, 8558, 6752, 3534, 1408, 430, 96, 14, 1, 41586, 33028, 17718, 7432, 2490, 652, 126, 16, 1, 206098

Offset: 0

Paul Barry, Feb 15 2003

Row sums are little Schroeder numbers A001003. Diagonal sums are generalized Fibonacci numbers A006603. Columns include A006318, A006319, A006320, A006321.
T(n,k) is the number of dissections of a convex (n+3)-gon by nonintersecting diagonals with exactly k diagonals emanating from a fixed vertex. Example: T(2,1)=4 because the dissections of the convex pentagon ABCDE having exactly one diagonal emanating from the vertex A are: {AC}, {AD}, {AC,EC} and {AD,BD}. - Emeric Deutsch, May 31 2004
For more triangle sums, see A180662, see the Schroeder triangle A033877 which is the mirror of this triangle. - Johannes W. Meijer, Jul 15 2013

			Triangle starts:
[0]    1
[1]    2,    1
[2]    6,    4,   1
[3]   22,   16,   6,   1
[4]   90,   68,  30,   8,  1
[5]  394,  304, 146,  48, 10,  1
[6] 1806, 1412, 714, 264, 70, 12, 1
...
From _Gary W. Adamson_, Jul 25 2011: (Start)
n-th row = top row of M^n, M = the following infinite square production matrix:
  2, 1, 0, 0, 0, ...
  2, 2, 1, 0, 0, ...
  2, 2, 2, 1, 0, ...
  2, 2, 2, 2, 1, ...
  ... (End)

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Paul Barry, Laurent Biorthogonal Polynomials and Riordan Arrays, arXiv preprint arXiv:1311.2292 [math.CA], 2013.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 21, 29.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
W.-j. Woan, The Lagrange inversion formula and divisibility properties, JIS 10 (2007) 07.7.8, example 5.

Cf. A000007, A033877 (mirror), A084938.

A080247:=(n,k)->(k+1)*add(binomial(n+1,k+j+1)*binomial(n+j,j),j=0..n-k)/(n+1):
seq(seq(A080247(n,k),k=0..n),n=0..9);

Clear[w] w[n_, k_] /; k < 0 || k > n := 0 w[0,0]=1 ; w[n_, k_] /; 0 <= k <= n && !n == k == 0 := w[n, k] = w[n-1,k-1] + w[n-1,k] + w[n,k+1] Table[w[n,k],{n,0,10},{k,0,n}] (* David Callan, Jul 03 2006 *)
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Peter Luschny, Jan 08 2018 *)

T(n,k):=((k+1)*sum(2^m*binomial(n+1,m)*binomial(n-k-1,n-k-m),m,0,n-k))/(n+1); /* Vladimir Kruchinin, Jan 10 2022 */

def A080247_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^(n-k)*prec(n, k) for k in (1..n)]
for n in (1..10): print(A080247_row(n)) # Peter Luschny, Mar 16 2016

G.f.: 2/(2+y*x-y+y*(x^2-6*x+1)^(1/2))/y/x. - Vladeta Jovovic, Feb 16 2003
Essentially same triangle as triangle T(n,k), n > 0 and k > 0, read by rows; given by [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA A000007 where DELTA is Deléham's operator defined in A084938.
T(n, k) = T(n-1, k-1) + 2*Sum_{j>=0} T(n-1, k+j) with T(0, 0) = 1 and T(n, k)=0 if k < 0. - Philippe Deléham, Jan 19 2004
T(n, k) = (k+1)*Sum_{j=0..n-k} (binomial(n+1, k+j+1)*binomial(n+j, j))/(n+1). - Emeric Deutsch, May 31 2004
Recurrence: T(0,0)=1; T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n,k+1). - David Callan, Jul 03 2006
T(n, k) = binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], -1). - Peter Luschny, Jan 08 2018
T(n,k) = (k+1)/(n+1)*Sum_{m=0..n-k} 2^m*binomial(n+1,m)*binomial(n-k-1,n-k-m). - Vladimir Kruchinin, Jan 10 2022
From Peter Bala, Sep 16 2024: (Start)
Riordan array (S(x), x*S(x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the g.f. of the large Schröder numbers A006318.
For integer m and n >= 1, (m + 2)*[x^n] S(x)^(m*n) = m*[x^n] (1/S(-x))^((m+2)*n). For cases of this identity see A103885 (m = 1), A333481 (m = 2) and A370102 (m = 3). (End)

A002464 Hertzsprung's problem: ways to arrange n non-attacking kings on an n X n board, with 1 in each row and column. Also number of permutations of length n without rising or falling successions.

Original entry on oeis.org

1, 1, 0, 0, 2, 14, 90, 646, 5242, 47622, 479306, 5296790, 63779034, 831283558, 11661506218, 175203184374, 2806878055610, 47767457130566, 860568917787402, 16362838542699862, 327460573946510746, 6880329406055690790, 151436547414562736234, 3484423186862152966838

Offset: 0

N. J. A. Sloane

Permutations of 12...n such that none of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).
This sequence is also the solution to the 'toast problem' devised by my house-mates and me as math undergraduates some 27 years ago: Given a toast rack with n slots, how many ways can the slices be removed so that no two consecutive slices are removed from adjacent slots? - David Jones (david.jones(AT)zetnet.co.uk), Oct 24 2003
This sequence was also derived by the late D. P. Robbins. - David Callan, Nov 04 2003
Another interpretation: number of permutations of n containing exactly n different patterns of size n-1. - Olivier Gérard, Nov 05 2007
Number of directed Hamiltonian paths in the complement of the n-path graph P_n. - Andrew Howroyd, Mar 16 2016
There is an obvious connection between the two descriptions of the sequence: Replace the chessboard with a n X n zero-matrix and each king with "1". This matrix will transform the vector (1,2,..,n) into a permutation such that adjacent components do not differ by 1. The reverse is also true: Any such transformation is a solution of the king problem. - Gerhard Kirchner, Feb 10 2017
A formula of Poulet (1919) relates this to A326411: a(n) = T(n+2,1)/(n+2) + 2*T(n+1,1)/(n+1) + T(n,1)/n, where T(i,j) = A326411(i,j). - N. J. A. Sloane, Mar 08 2022
For the number of these permutations without fixed points see A288208. - Wolfdieter Lang, May 22 2025

			a(4) = 2: 2413, 3142.
a(5) = 14 corresponds to these 14 permutations of length 5: 13524, 14253, 24135, 24153, 25314, 31425, 31524, 35142, 35241, 41352, 42513, 42531, 52413, 53142.

W. Ahrens, Mathematische Unterhaltungen und Spiele. Teubner, Leipzig, Vol. 1, 3rd ed., 1921; Vol. 2, 2nd ed., 1918. See Vol. 1, p. 271.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.40.

N. J. A. Sloane, Table of n, a(n) for n = 0..500
M. Abramson and W. O. J. Moser, Permutations without rising or falling w-sequences, Ann. Math. Stat., 38 (1967), 1245-1254.
W. Ahrens, Mathematische Unterhaltungen und Spiele, Leipzig: B. G. Teubner, 1901.
Another Roof, A Lifelong Mathematical Obsession, YouTube video, 2023.
Art of Problem Solving Forum, Number of permutative
Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, On the poset of King-Non-Attacking permutations, arXiv:1905.02387 [math.CO], 2019.
Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Counting King Permutations on the Cylinder, arXiv:2001.02948 [math.CO], 2020.
Christoph Bärligea, On the dimension of the Fomin-Kirillov algebra and related algebras, arXiv:2001.04597 [math.QA], 2020.
Robert Brignall, Vít Jelínek, Jan Kynčl, and David Marchant, Zeros of the Möbius function of permutations, arXiv:1810.05449 [math.CO], 2018.
Doug Chatham, Independence and domination on shogiboard graphs, Recreational Mathematics Magazine 4.8 (2017), pp. 25-37.
Doug Chatham, Reflections on the n+k dragon kings problem, Games and Puzzles, Recreational Mathematics Magazine (2018) 5.10, 39-55.
Anders Claesson, From Hertzsprung's problem to pattern-rewriting systems, University of Iceland (2020).
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 373.
Sela Fried, Further results on staircase (cyclic) words, arXiv:2501.11991 [math.CO], 2025. See p. 12.
Sela Fried, Toufik Mansour, and Mark Shattuck, Counting k-ary words by number of adjacency differences of a prescribed size, arXiv:2504.03013 [math.CO], 2025. See p. 2.
Michael Han, Tanya Khovanova, Ella Kim, Evin Liang, Miriam Lubashev, Oleg Polin, Vaibhav Rastogi, Benjamin Taycher, Ada Tsui and Cindy Wei, Fun with Latin Squares, arXiv:2109.01530 [math.HO], 2021.
C. Homberger, Counting Fixed Length Permutation Patterns, Online Journal of Analytic Combinatorics, 7 (2012).
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
Mark Huibregtse, Cristobal Lemus-Vidales, and David Vella, Bootstrap Percolation, Indecomposable Permutations, and the n-Kings problem, arXiv:2508.02030 [math.CO], 2025. See p. 2.
Irving Kaplansky, The asymptotic distribution of runs of consecutive elements, Ann. Math. Statistics, 16 (1945), 200-203
Sergey Kitaev and Jeffrey Remmel, (a,b)-rectangle patterns in permutations and words, arXiv:1304.4286 [math.CO], (2013).
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 625-635.
Igor Minevich, Gabe Cunningham, Aditya Karan, and Joshua V. Gyllinsky, Parks: A Doubly Infinite Family of NP-Complete Puzzles and Generalizations of A002464, arXiv:2411.02251 [cs.CC], 2024.
O. Patashnik and P. Flajolet, Email to N. J. A. Sloane, Nov 26 1989
P. Poulet, Query 4750: Permutations, L'Intermédiaire des Mathématiciens, 26 (1919), 117-121. (Page 117)
P. Poulet, Query 4750: Permutations, L'Intermédiaire des Mathématiciens, 26 (1919), 117-121. (Pages 118, 119)
P. Poulet, Query 4750: Permutations, L'Intermédiaire des Mathématiciens, 26 (1919), 117-121. (Pages 120, 121)
J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710.
D. P. Robbins, The probability that neighbors remain neighbors after random rearrangements, Amer. Math. Monthly 87 (1980), 122-124.
Josef Rukavicka, Secretary problem and two almost the same consecutive applicants, arXiv:2106.11244 [math.PR], 2021.
A. J. Schwenk, Letter to N. J. A. Sloane, Mar 24 1980 (with enclosure and notes)
L. Shapiro and A. B. Stephens, Bootstrap percolation, the Schroeder problems and the n-kings problem, SIAM J. Discrete Math., 4 (1991), 275-280.
Jason P. Smith, A Formula for the Mobius function of the Permutation Poset Based on a Topological Decomposition, arXiv preprint arXiv:1506.04406 [math.CO], 2015.
Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 6 (2006), #A11. See Column 3 in the table on page 3.
B. E. Tenner, Mesh patterns with superfluous mesh, arXiv preprint arXiv:1302.1883 [math.CO], 2013.
Eric Weisstein's World of Mathematics, Hamiltonian Path
Eric Weisstein's World of Mathematics, Path Complement Graph
Eric Weisstein's World of Mathematics, Permutation

Equals 2*A001266(n) for n >= 2. A diagonal of A001100. Cf. A010028.
Cf. A086852, A086853, A086854, A086855, A000130, A000349, A001267, A001268, A010028, A002493.
Cf. A089222, A104698, A288208.
Column k=1 of A333706.

A002464 := proc(n) options remember; if n <= 1 then 1 elif n <= 3 then 0 else (n+1)*A002464(n-1)-(n-2)*A002464(n-2)-(n-5)*A002464(n-3)+(n-3)*A002464(n-4); fi; end;

(* computation from the permutation class *)
g[ L_ ] := Apply[ And, Map[ #>1&, L ] ]; f[ n_ ] := Length[ Select[ Permutations[ Range[ n ] ], g[ Rest[ Abs[ RotateRight[ # ]-# ] ] ]& ] ]; Table[ f[ n ], {n, 1, 8} ] (* Erich Friedman *)
(* or direct computation of terms *)
Table[n! + Sum[(-1)^r*(n-r)!*Sum[2^c *Binomial[r-1,c-1] *Binomial[n-r,c], {c,1,r}], {r,1,n-1}], {n,1,30}] (* Vaclav Kotesovec, Mar 28 2011 *)
(* or from g.f. *)
M = 30; CoefficientList[Sum[n!*x^n*(1-x)^n/(1+x)^n, {n, 0, M}] + O[x]^M, x] (* Jean-François Alcover, Jul 07 2015 *)
CoefficientList[Series[Exp[(1 + x)/((-1 + x) x)] (1 + x) Gamma[0, (1 + x)/((-1 + x) x)]/((-1 + x) x), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 11 2018 *)
RecurrenceTable[{a[n] == (n + 1) a[n - 1] - (n - 2) a[n - 2] - (n - 5) a[n - 3] + (n - 3) a[n - 4], a[0] == a[1] == 1, a[2] == a[3] == 0}, a, {n, 0, 20}] (* Eric W. Weisstein, Apr 11 2018 *)

N = 66;  x = 'x + O('x^N);
gf = sum(n=0,N, n!*(x*(1-x))^n/(1+x)^n );
v = Vec(gf) /* Joerg Arndt, Apr 17 2013 */

from math import factorial, comb
def A002464(n): return factorial(n)+sum((-1 if k&1 else 1)*factorial(n-k)*sum(comb(k-1,t-1)*comb(n-k,t)<Chai Wah Wu, Feb 19 2024

If n = 0 or 1 then a(n) = 1; if n = 2 or 3 then a(n) = 0; otherwise a(n) = (n+1)*a(n-1) - (n-2)*a(n-2) - (n-5)*a(n-3) + (n-3)*a(n-4).
G.f.: Sum_{n >= 0} n!*x^n*(1-x)^n/(1+x)^n. - Philippe Flajolet
G.f.: e^((1 + x)/((-1 + x) * x)) * (1 + x) * Gamma(0, (1 + x)/((-1 + x) * x))/((-1 + x) * x). - Eric W. Weisstein, May 16 2014
Let S_{n, k} = number of permutations of 12...n with exactly k rising or falling successions. Let S[n](t) = Sum_{k >= 0} S_{n, k}*t^k. Then S[0] = 1; S[1] = 1; S[2] = 2*t; S[3] = 4*t+2*t^2; for n >= 4, S[n] = (n+1-t)*S[n-1] - (1-t)*(n-2+3*t)*S[n-2] - (1-t)^2*(n-5+t)*S[n-3] + (1-t)^3*(n-3)*S[n-4].
a(n) = n! + Sum_{k=1..n} (-1)^k * Sum_{t=1..k} binomial(k-1,t-1) * binomial(n-k,t) * 2^t * (n-k)!. - Max Alekseyev, Jan 29 2006
a(n) = Sum_{k=0..n} (-1)^(n-k)*k!*b(n,k), where g.f. for b(n,k) is (1-x)/(1-(1+y)*x-y*x^2), cf. A035607. - Vladeta Jovovic, Nov 24 2007
Asymptotic (M. Abramson and W. Moser, 1966): a(n)/n! ~ (1 - 2/n^2 - 10/(3*n^3) - 6/n^4 - 154/(15*n^5) - 88/(9*n^6) + 5336/(105*n^7) + 1612/(3*n^8) + 2098234/(567*n^9) + 36500686/(1575*n^10) + ... )/e^2. - Vaclav Kotesovec, Apr 19 2011, extended Dec 27 2020
Conjecture: a(n) = Sum_{k=1..n} k!*A080246(n-1, k-1) for n > 0. - John Keith, Nov 02 2020
Proof: a(n) = Sum_{k=1..n} k!*A080246(n-1, k-1) for n > 0. Since a(n) = Sum_{k=0..n-1} (-1)^k*(n-k)!*Sum_{i=0..k} binomial(n-k,i)*binomial(n-1-i,k-i) (M. Abramson and W. Moser, 1966) which is Sum_{k=1..n} (-1)^(k-1)(n-k+1)!*Sum{i=0..k-1} binomial(n-k+1,i)*binomial(n-1-i,k-1-i) = Sum_{k=1..n} (-1)^(n-k)(k!)*Sum_{i=0..n-k} binomial(k,i)*binomial(n-1-i,n-k-i) = k!*A080246(n-1, k-1) as (-1)^(n-k) = (-1)^(n+k) and binomial(n-1-i,k-1) = binomial(n-1-i,n-k-i). - Alex McGaw, Apr 13 2023
a(n+2) = (n+2)! - Sum_{j=0..n} (-1)^j*(n+1-j)!*2*A104698(n, j), for n >= 0 (Abramson and Moser, p. 1250, (III), N_0(n+2), last line, rewritten). - Wolfdieter Lang, May 14 2025

Merged with the old A001100, Aug 19 2003
Kaplansky reference from David Callan, Oct 29 2003
Tauraso reference from Parthasarathy Nambi, Dec 21 2006
Edited by Jon E. Schoenfield, Jan 31 2015

A113413 A Riordan array of coordination sequences.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 2, 8, 6, 1, 2, 12, 18, 8, 1, 2, 16, 38, 32, 10, 1, 2, 20, 66, 88, 50, 12, 1, 2, 24, 102, 192, 170, 72, 14, 1, 2, 28, 146, 360, 450, 292, 98, 16, 1, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1, 2, 36, 258, 952, 1970, 2364, 1666, 688, 162, 20, 1, 2, 40, 326

Offset: 0

Paul Barry, Oct 29 2005

Columns include A040000, A008574, A005899, A008412, A008413, A008414. Row sums are A078057(n)=A001333(n+1). Diagonal sums are A001590(n+3). Reverse of A035607. Signed version is A080246. Inverse is A080245.
For another version see A122542. - Philippe Deléham, Oct 15 2006
T(n,k) is the number of length n words on alphabet {0,1,2} with no two consecutive 1's and no two consecutive 2's and having exactly k 0's. - Geoffrey Critzer, Jun 11 2015
From Eric W. Weisstein, Feb 17 2016: (Start)
Triangle of coefficients (from low to high degree) of x^-n * vertex cover polynomial of the n-ladder graph P_2 \square p_n:
  Psi_{L_1}: x*(2 + x) -> {2, 1}
  Psi_{L_2}: x^2*(2 + 4 x + x^2) -> {2, 4, 1}
  Psi_{L_3}: x^3*(2 + 8 x + 6 x^2 + x^3) -> {2, 8, 6, 1}
(End)
Let c(n, k), n > 0, be multiplicative sequences for some fixed integer k >= 0 with c(p^e, k) = T(e+k, k) for prime p and e >= 0. Then we have Dirichlet g.f.: Sum_{n>0} c(n, k) / n^s = zeta(s)^(2*k+2) / zeta(2*s)^(k+1). Examples: For k = 0 see A034444 and for k = 1 see A322328. Dirichlet convolution of c(n, k) and lambda(n) is Dirichlet inverse of c(n, k). - Werner Schulte, Oct 31 2022

			Triangle begins
  1;
  2,  1;
  2,  4,  1;
  2,  8,  6,  1;
  2, 12, 18,  8,  1;
  2, 16, 38, 32, 10,  1;
  2, 20, 66, 88, 50, 12,  1;

G. C. Greubel, Table of n, a(n) for the first 50 rows
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
P. Holub, M. Miller, H. Perez-Roses, and J. Ryan, Degree diameter problem on honeycomb networks, Disc. Appl. Math. 179 (2014) 139-151.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 2.
Mirka Miller, Hebert Perez-Roses, and Joe Ryan, The Maximum Degree-and-Diameter-Bounded Subgraph in the Mesh, arXiv preprint arXiv:1203.4069 [math.CO], 2012.

Other versions: A035607, A119800, A122542, A266213.

nn = 10; Map[Select[#, # > 0 &] &, CoefficientList[Series[1/(1 - 2 x/(1 + x) - y x), {x, 0, nn}], {x, y}]] // Grid (* Geoffrey Critzer, Jun 11 2015 *)
CoefficientList[CoefficientList[Series[1/(1 - 2 x/(1 + x) - y x), {x, 0, 10}, {y, 0, 10}], x], y] (* Eric W. Weisstein, Feb 17 2016 *)

T = lambda n,k : binomial(n, k)*hypergeometric([-k-1, k-n], [-n], -1).simplify_hypergeometric()
A113413 = lambda n,k : 1 if n==0 and k==0 else T(n, k)
for n in (0..12): print([A113413(n,k) for k in (0..n)]) # Peter Luschny, Sep 17 2014 and Mar 16 2016

# Alternatively:
def A113413_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)+2*sum(prec(n-i,k-1) for i in (2..n-k+1))
    return [prec(n, k) for k in (1..n)]
for n in (1..10): print(A113413_row(n)) # Peter Luschny, Mar 16 2016

From Paul Barry, Nov 13 2005: (Start)
Riordan array ((1+x)/(1-x), x(1+x)/(1-x)).
T(n, k) = Sum_{i=0..n-k} C(k+1, i)*C(n-i, k).
T(n, k) = Sum_{j=0..n-k} C(k+j, j)*C(k+1, n-k-j).
T(n, k) = D(n, k) + D(n-1, k) where D(n, k) = Sum_{j=0..n-k} C(n-k, j)*C(k, j)*2^j = A008288(n, k).
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k-1).
T(n, k) = Sum_{j=0..n} C(floor((n+j)/2), k)*C(k, floor((n-j)/2)). (End)
T(n, k) = C(n, k)*hypergeometric([-k-1, k-n], [-n], -1). - Peter Luschny, Sep 17 2014
T(n, k) = (Sum_{i=2..k+2} A137513(k+2, i) * (n-k)^(i-2)) / (k!) for 0 <= k < n (conjectured). - Werner Schulte, Oct 31 2022

cp's OEIS Frontend

A080247 Formal inverse of triangle A080246. Unsigned version of A080245.

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A002464 Hertzsprung's problem: ways to arrange n non-attacking kings on an n X n board, with 1 in each row and column. Also number of permutations of length n without rising or falling successions.

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A113413 A Riordan array of coordination sequences.

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A090727 a(n) = 16a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 16.

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A111806 Riordan array (1/(1+3x+2x^2),x/(1+3x+2x^2)).

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