A108666 -id:A108666 - cp's OEIS frontend

A001787 a(n) = n*2^(n-1).

0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120, 11264, 24576, 53248, 114688, 245760, 524288, 1114112, 2359296, 4980736, 10485760, 22020096, 46137344, 96468992, 201326592, 419430400, 872415232, 1811939328, 3758096384, 7784628224, 16106127360, 33285996544

Offset: 0

N. J. A. Sloane

Number of edges in an n-dimensional hypercube.
Number of 132-avoiding permutations of [n+2] containing exactly one 123 pattern. - Emeric Deutsch, Jul 13 2001
Number of ways to place n-1 nonattacking kings on a 2 X 2(n-1) chessboard for n >= 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 22 2001
Arithmetic derivative of 2^n: a(n) = A003415(A000079(n)). - Reinhard Zumkeller, Feb 26 2002
(-1) times the determinant of matrix A_{i,j} = -|i-j|, 0 <= i,j <= n.
a(n) is the number of ones in binary numbers 1 to 111...1 (n bits). a(n) = A000337(n) - A000337(n-1) for n = 2,3,... . - Emeric Deutsch, May 24 2003
The number of 2 X n 0-1 matrices containing n+1 1's and having no zero row or column. The number of spanning trees of the complete bipartite graph K(2,n). This is the case m = 2 of K(m,n). See A072590. - W. Edwin Clark, May 27 2003
Binomial transform of 0,1,2,3,4,5,... (A001477). Without the initial 0, binomial transform of odd numbers.
With an additional leading zero, [0,0,1,4,...] this is the binomial transform of the integers repeated A004526. Its formula is then (2^n*(n-1) + 0^n)/4. - Paul Barry, May 20 2003
Number of zeros in all different (n+1)-bit integers. - Ralf Stephan, Aug 02 2003
From Lekraj Beedassy, Jun 03 2004: (Start)
Final element of a summation table (as opposed to a difference table) whose first row consists of integers 0 through n (or first n+1 nonnegative integers A001477); illustrating the case n=5:
   0   1   2   3   4   5
     1   3   5   7   9
       4   8  12  16
        12  20  28
          32  48
            80
and the final element is a(5)=80. (End)
This sequence and A001871 arise in counting ordered trees of height at most k where only the rightmost branch at the root actually achieves this height and the count is by the number of edges, with k = 3 for this sequence and k = 4 for A001871.
Let R be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all elements x,y of P(A), xRy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. Then a(n) = |R|. - Ross La Haye, Sep 21 2004
Number of 2 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1) and (10;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004
Number of subsequences 00 in all binary words of length n+1. Example: a(2)=4 because in 000,001,010,011,100,101,110,111 the sequence 00 occurs 4 times. - Emeric Deutsch, Apr 04 2005
If you expand the n-factor expression (a+1)*(b+1)*(c+1)*...*(z+1), there are a(n) variables in the result. For example, the 3-factor expression (a+1)*(b+1)*(c+1) expands to abc+ab+ac+bc+a+b+c+1 with a(3) = 12 variables. - David W. Wilson, May 08 2005
An inverse Chebyshev transform of n^2, where g(x)->(1/sqrt(1-4*x^2))*g(x*c(x^2)), c(x) the g.f. of A000108. - Paul Barry, May 13 2005
Sequences A018215 and A058962 interleaved. - Graeme McRae, Jul 12 2006
The number of never-decreasing positive integer sequences of length n with a maximum value of 2*n. - Ben Paul Thurston, Nov 13 2006
Total size of all the subsets of an n-element set. For example, a 2-element set has 1 subset of size 0, 2 subsets of size 1 and 1 of size 2. - Ross La Haye, Dec 30 2006
Convolution of the natural numbers [A000027] and A045623 beginning [0,1,2,5,...]. - Ross La Haye, Feb 03 2007
If M is the matrix (given by rows) [2,1;0,2] then the sequence gives the (1,2) entry in M^n. - Antonio M. Oller-Marcén, May 21 2007
If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n > 0, a(n) is equal to the number of (n+1)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007
Number of n-permutations of 3 objects u,v,w, with repetition allowed, containing exactly one u. Example: a(2)=4 because we have uv, vu, uw and wu. - Zerinvary Lajos, Dec 27 2007
A member of the family of sequences defined by a(n) = n*[c(1)*...*c(r)]^(n-1); c(i) integer. This sequence has c(1)=2, A027471 has c(1)=3. - Ctibor O. Zizka, Feb 23 2008
a(n) is the number of ways to split {1,2,...,n-1} into two (possibly empty) complementary intervals {1,2,...,i} and {i+1,i+2,...,n-1} and then select a subset from each interval. - Geoffrey Critzer, Jan 31 2009
Equals the Jacobsthal sequence A001045 convolved with A003945: (1, 3, 6, 12, ...). - Gary W. Adamson, May 23 2009
Starting with offset 1 = A059570: (1, 2, 6, 14, 34, ...) convolved with (1, 2, 2, 2, ...). - Gary W. Adamson, May 23 2009
Equals the first left hand column of A167591. - Johannes W. Meijer, Nov 12 2009
The number of tatami tilings of an n X n square with n monomers is n*2^(n-1). - Frank Ruskey, Sep 25 2010
Under T. D. Noe's variant of the hypersigma function, this sequence gives hypersigma(2^n): a(n) = A191161(A000079(n)). - Alonso del Arte, Nov 04 2011
Number of Dyck (n+2)-paths with exactly one valley at height 1 and no higher valley. - David Scambler, Nov 07 2011
Equals triangle A059260 * A016777 as a vector, where A016777 = (3n + 1): [1, 4, 7, 10, 13, ...]. - Gary W. Adamson, Mar 06 2012
Main transitions in systems of n particles with spin 1/2 (see A212697 with b=2). - Stanislav Sykora, May 25 2012
Let T(n,k) be the triangle with (first column) T(n,1) = 2*n-1 for n >= 1, otherwise T(n,k) = T(n,k-1) + T(n-1,k-1), then a(n) = T(n,n). - J. M. Bergot, Jan 17 2013
Sum of all parts of all compositions (ordered partitions) of n. The equivalent sequence for partitions is A066186. - Omar E. Pol, Aug 28 2013
Starting with a(1)=1: powers of 2 (A000079) self-convolved. - Bob Selcoe, Aug 05 2015
Coefficients of the series expansion of the normalized Schwarzian derivative -S{p(x)}/6 of the polynomial p(x) = -(x-x1)*(x-x2) with x1 + x2 = 1 (cf. A263646). - Tom Copeland, Nov 02 2015
a(n) is the number of North-East lattice paths from (0,0) to (n+1,n+1) that have exactly one east step below y = x-1 and no east steps above y = x+1. Details can be found in Pan and Remmel's link. - Ran Pan, Feb 03 2016
Also the number of maximal and maximum cliques in the n-hypercube graph for n > 0. - Eric W. Weisstein, Dec 01 2017
Let [n]={1,2,...,n}; then a(n-1) is the total number of elements missing in proper subsets of [n] that contain n to form [n]. For example, for n = 3, a(2) = 4 since the proper subsets of [3] that contain 3 are {3}, {1,3}, {2,3} and the total number of elements missing in these subsets to form [3] is 4:  2 in the first subset, 1 in the second, and 1 in the third. - Enrique Navarrete, Aug 08 2020
Number of 3-permutations of n elements avoiding the patterns 132, 231. See Bonichon and Sun. - Michel Marcus, Aug 19 2022

			a(2)=4 since 2314, 2341,3124 and 4123 are the only 132-avoiding permutations of 1234 containing exactly one increasing subsequence of length 3.
x + 4*x^2 + 12*x^3 + 32*x^4 + 80*x^5 + 192*x^6 + 448*x^7 + ...
a(5) = 1*0 + 5*1 + 10*2 + 10*3 + 5*4 + 1*5 = 80, with 1,5,10,10,5,1 the 5th row of Pascal's triangle. - _J. M. Bergot_, Apr 29 2014

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 131.
Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 282.
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).

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..500
Rémi Abgrall and Wasilij Barsukow, Extensions of Active Flux to arbitrary order of accuracy, arXiv:2208.14476 [math.NA], 2022.
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].
F. S. Al-Kharousi, A. Umar, and M. M. Zubairu, On injective partial Catalan monoids, arXiv:2501.00285 [math.GR], 2024. See p. 9.
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, arXiv:1803.06706 [math.CO], 2018.
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Douglas W. Bass and I. Hal Sudborough, Hamilton decompositions and (n/2)-factorizations of hypercubes, J. Graph Algor. Appl., Vol. 7, No. 1 (2003), pp. 79-98.
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
Harlan J. Brothers, Pascal's Prism: Supplementary Material.
David Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Frank Ellermann, Illustration of binomial transforms
Mohamed Elkadi and Bernard Mourrain, Symbolic-numeric methods for solving polynomial equations and applications, Chap 3. of A. Dickenstein and I. Z. Emiris, eds., Solving Polynomial Equations, Springer, 2005, pp. 126-168. See p. 152.
Alejandro Erickson, Frank Ruskey, Mark Schurch and Jennifer Woodcock, Auspicious Tatami Mat Arrangements, The 16th Annual International Computing and Combinatorics Conference (COCOON 2010), July 19-21, Nha Trang, Vietnam. LNCS 6196 (2010) 288-297.
Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, Advances in Applied Mathematics, Vol. 77 (2016), pp. 1-42, arXiv preprint, arXiv:1603.01040 [math.CO], 2016.
Frank A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424.
Frank A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
Frank A. Haight, Letter to N. J. A. Sloane, n.d.
V. E. Hoggatt, Jr., Letter to N. J. A. Sloane, Jul 06, 1976
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=4, q=-4.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 408. (Dead link)
Milan Janjić, Two Enumerative Functions.
Milan Janjić and Boris Petković, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
Milan Janjić and Boris Petković, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
C. W. Jones, J. C. P. Miller, J. F. C. Conn, and R. C. Pankhurst, Tables of Chebyshev polynomials, Proc. Roy. Soc. Edinburgh. Sect. A. 62, (1946). 187-203.
Kenji Kimura and Saburo Higuchi, Monte Carlo estimation of the number of tatami tilings, International Journal of Modern Physics C, Vol. 27, No. 11 (2016), 1650128, arXiv preprint, arXiv:1509.05983 [cond-mat.stat-mech], 2015-2016, eq. (1).
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004). (Dead link)
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012.
T. Y. Lam, On the diagonalization of quadratic forms, Math. Mag., 72 (1999), 231-235 (see page 234).
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. See Eq.(3).
Duško Letić, Nenad Cakić, Branko Davidović, Ivana Berković and Eleonora Desnica, Some certain properties of the generalized hypercubical functions, Advances in Difference Equations, 2011, 2011:60.
Toufik Mansour and Armend Sh. Shabani, Bargraphs in bargraphs, Turkish Journal of Mathematics (2018) Vol. 42, Issue 5, 2763-2773.
Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Michael Penn, on the alternating sum of subsets, YouTube video, 2021.
Michael Penn, Rare proof of well-known sum, YouTube video, 2023.
Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
Maxwell Phillips, Ahmed Ammar, and Firas Hassan, A Generalized Multi-Level Structure for High-Precision Binary Decoders, IEEE 67th Int'l Midwest Symp. Circ. Sys. (MWSCAS 2024), 42-46.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Lara Pudwell, Nathan Chenette and Manda Riehl, Statistics on Hypercube Orientations, AMS Special Session on Experimental and Computer Assisted Mathematics, Joint Mathematics Meetings (Denver 2020).
Lara Pudwell, Connor Scholten, Tyler Schrock and Alexa Serrato, Noncontiguous pattern containment in binary trees, ISRN Comb. 2014, Article ID 316535, 8 p. (2014), chapter 5.2.
Aaron Robertson, Permutations containing and avoiding 123 and 132 patterns, Discrete Math. and Theoret. Computer Sci., 3 (1999), 151-154.
Aaron Robertson, Herbert S. Wilf and Doron Zeilberger, Permutation patterns and continued fractions, Electr. J. Combin. 6, 1999, #R38.
Thomas Selig and Haoyue Zhu, Complete non-ambiguous trees and associated permutations: connections through the Abelian sandpile model, arXiv:2303.15756 [math.CO], 2023, see p. 16.
Jeffrey Shallit, Letter to N. J. A. Sloane Mar 14, 1979, concerning A001787, A005209, A005210, A005211.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
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Eric Weisstein's World of Mathematics, Hypercube Graph.
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Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Three other versions, essentially identical, are A085750, A097067, A118442.
Partial sums of A001792.
A058922(n+1) = 4*A001787(n).
Equals A090802(n, 1).
Column k=1 of A038207.
Row sums of A003506, A322427, A322428.
Cf. A053109, A001788, A001789, A000337, A130300, A134083, A002064, A027471, A003945, A059670, A167591, A059260, A016777, A212697, A000079, A263646.

a001787 n = n * 2 ^ (n - 1)
a001787_list = zipWith (*) [0..] $ 0 : a000079_list
-- Reinhard Zumkeller, Jul 11 2014

[n*2^(n-1): n in [0..40]]; // Vincenzo Librandi, Feb 04 2016

spec := [S, {B=Set(Z, 0 <= card), S=Prod(Z, B, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..29); # Zerinvary Lajos, Oct 09 2006
A001787:=1/(2*z-1)^2; # Simon Plouffe in his 1992 dissertation, dropping the initial zero

Table[Sum[Binomial[n, i] i, {i, 0, n}], {n, 0, 30}] (* Geoffrey Critzer, Mar 18 2009 *)
f[n_] := n 2^(n - 1); f[Range[0, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
Array[# 2^(# - 1) &, 40, 0] (* Harvey P. Dale, Jul 26 2011 *)
Join[{0}, Table[n 2^(n - 1), {n, 20}]] (* Eric W. Weisstein, Dec 01 2017 *)
Join[{0}, LinearRecurrence[{4, -4}, {1, 4}, 20]] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[x/(-1 + 2 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

PARI
```
{a(n) = if( n<0, 0, n * 2^(n-1))}
```

concat(0, Vec(x/(1-2*x)^2 + O(x^50))) \\ Altug Alkan, Nov 03 2015

def A001787(n): return n*(1<Chai Wah Wu, Nov 14 2022

a(n) = Sum_{k=1..n} k*binomial(n, k). - Benoit Cloitre, Dec 06 2002
E.g.f.: x*exp(2x). - Paul Barry, Apr 10 2003
G.f.: x/(1-2*x)^2.
G.f.: x / (1 - 4*x / (1 + x / (1 - x))). - Michael Somos, Apr 07 2012
A108666(n) = Sum_{k=0..n} binomial(n, k)^2 * a(n). - Michael Somos, Apr 07 2012
PSumSIGN transform of A053220. PSumSIGN transform is A045883. Binomial transform is A027471(n+1). - Michael Somos, Jul 10 2003
Starting at a(1)=1, INVERT transform is A002450, INVERT transform of A049072, MOBIUS transform of A083413, PSUM transform is A000337, BINOMIAL transform is A081038, BINOMIAL transform of A005408. - Michael Somos, Apr 07 2012
a(n) = 2*a(n-1)+2^(n-1).
a(2*n) = n*4^n, a(2*n+1) = (2*n+1)4^n.
G.f.: x/det(I-x*M) where M=[1,i;i,1], i=sqrt(-1). - Paul Barry, Apr 27 2005
Starting 1, 1, 4, 12, ... this is 0^n + n2^(n-1), the binomial transform of the 'pair-reversed' natural numbers A004442. - Paul Barry, Jul 24 2003
Convolution of [1, 2, 4, 8, ...] with itself. - Jon Perry, Aug 07 2003
The signed version of this sequence, n(-2)^(n-1), is the inverse binomial transform of n(-1)^(n-1) (alternating sign natural numbers). - Paul Barry, Aug 20 2003
a(n-1) = (Sum_{k=0..n} 2^(n-k-1)*C(n-k, k)*C(1,(k+1)/2)*(1-(-1)^k)/2) - 0^n/4. - Paul Barry, Oct 15 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)(n-2k)^2. - Paul Barry, May 13 2005
a(n+2) = A049611(n+2) - A001788(n).
a(n) = n! * Sum_{k=0..n} 1/((k - 1)!(n - k)!). - Paul Barry, Mar 26 2003
a(n+1) = Sum_{k=0..n} 4^k * A109466(n,k). - Philippe Deléham, Nov 13 2006
Row sums of A130300 starting (1, 4, 12, 32, ...). - Gary W. Adamson, May 20 2007
Equals row sums of triangle A134083. Equals A002064(n) + (2^n - 1). - Gary W. Adamson, Oct 07 2007
a(n) = 4*a(n-1) - 4*a(n-2), a(0)=0, a(1)=1. - Philippe Deléham, Nov 16 2008
Sum_{n>0} 1/a(n) = 2*log(2). - Jaume Oliver Lafont, Feb 10 2009
a(n) = A000788(A000225(n)) = A173921(A000225(n)). - Reinhard Zumkeller, Mar 04 2010
a(n) = n * A011782(n). - Omar E. Pol, Aug 28 2013
a(n-1) = Sum_{t_1+2*t_2+...+n*t_n=n} (t_1+t_2+...+t_n-1)*multinomial(t_1+t_2 +...+t_n,t_1,t_2,...,t_n). - Mircea Merca, Dec 06 2013
a(n+1) = Sum_{r=0..n} (2*r+1)*C(n,r). - J. M. Bergot, Apr 07 2014
a(n) = A007283(n)*n/6. - Enxhell Luzhnica, Apr 16 2016
a(n) = (A000225(n) + A000337(n))/2. - Anton Zakharov, Sep 17 2016
Sum_{n>0} (-1)^(n+1)/a(n) = 2*log(3/2) = 2*A016578. - Ilya Gutkovskiy, Sep 17 2016
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (i+1) * C(k,i). - Wesley Ivan Hurt, Sep 21 2017
a(n) = Sum_{i=1..n} Sum_{j=1..n} phi(i)*binomial(n, i*j). - Ridouane Oudra, Feb 17 2024

A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)(n+k+1)binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

Original entry on oeis.org

1, -2, 6, 3, -24, 30, -4, 60, -180, 140, 5, -120, 630, -1120, 630, -6, 210, -1680, 5040, -6300, 2772, 7, -336, 3780, -16800, 34650, -33264, 12012, -8, 504, -7560, 46200, -138600, 216216, -168168, 51480, 9, -720, 13860, -110880, 450450, -1009008, 1261260, -823680, 218790

Offset: 0

N. J. A. Sloane, Jan 17 2020

Tables I, III, IV on pages 92 and 93 of Ser have integer entries and are A331430, A331431 (the present sequence), and A331432.

Given the system of equations 1 = Sum_{j=0..n} H(i, j) * x(j) for i = 2..n+2 where H(i,j) = 1/(i+j-1) for 1 <= i,j <= n is the n X n Hilbert matrix, then the solutions are x(j) = T(n, j). - Michael Somos, Mar 20 2020 [Corrected by Petros Hadjicostas, Jul 09 2020]

			Triangle begins:
   1;
  -2,    6;
   3,  -24,    30;
  -4,   60,  -180,     140;
   5, -120,   630,   -1120,     630;
  -6,  210, -1680,    5040,   -6300,     2772;
   7, -336,  3780,  -16800,   34650,   -33264,   12012;
  -8,  504, -7560,   46200, -138600,   216216, -168168,   51480;
   9, -720, 13860, -110880,  450450, -1009008, 1261260, -823680, 218790;
  ...

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93. See Table III.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Buhl, Book review: J. Ser - Les calculs formels des séries de factorielles, L'Enseignement Mathématique, 32 (1933), p. 275.
L. A. MacColl, Review: J. Ser, Les calculs formels des séries de factorielles, Bull. Amer. Math. Soc., 41(3) (1935), p. 174.
L. M. Milne-Thomson, Review of Les calculs formels des séries de factorielles. By J. Ser. Pp. vii, 98. 20 fr. 1933. (Gauthier-Villars), The Mathematical Gazette, Vol. 18, No. 228 (May, 1934), pp. 136-137.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages.)

Columns 1 is A331433 or equally A007531, column 2 is A331434 or equally A054559; the last three diagonals are A002738, A002736, A002457.

Cf. A000290 (row sums), A002457,, A100071, A108666 (alternating row sums), A109188 (diagonal sums), A331322, A331323, A331430, A331432.

[(-1)^(n+k)*(k+1)*(2*k+1)*Binomial(n+k+1,n-k)*Catalan(k): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 22 2022

gf := k -> (1+x)^(-2*(k+1)): ser := k -> series(gf(k), x, 32):
T := (n, k) -> ((2*k+1)!/(k!)^2)*coeff(ser(k), x, n-k):
seq(seq(T(n,k), k=0..n),n=0..7); # Peter Luschny, Jan 18 2020
S:=(n,k)->(-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!);
rho:=n->[seq(S(n,k),k=0..n)];
for n from 0 to 14 do lprint(rho(n)); od: # N. J. A. Sloane, Jan 18 2020

Table[(-1)^(n+k)*(n+k+1)*Binomial[2*k,k]*Binomial[n+k,n-k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 22 2022 *)

flatten([[(-1)^(n+k)*(2*k+1)*binomial(2*k,k)*binomial(n+k+1,n-k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Mar 22 2022

T(n, 0) = (-1)^n*A000027(n+1).
T(n, 1) = A331433(n-1) = (-1)^(n+1)*A007531(n+2).
T(n, 2) = A331434(n-2) = (-1)^n*A054559(n+3).
T(n, n-2) = A002738(n-2).
T(n, n-1) = (-1)*A002736(n).
T(n, n) = A002457(n).
T(2*n, n) = (-1)^n*(3*n+1)!/(n!)^3 = (-1)^n*A331322(n).
Sum_{k=0..n} T(n, k) = A000290(n+1) (row sums).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A108666(n+1) (alternating row sums).
Sum_{k=0..n} T(n-k, k) = (-1)^n*A109188(n+1) (diagonal sums).
2^n*Sum_{k=0..n} T(n, k)/2^k = (-1)^floor(n/2)*A100071(n+1) (positive half sums).
(-2)^n*Sum_{k=0..n} T(n, k)/(-2)^k = A331323(n) (negative half sums).
T(n, k) = ((2*k+1)!/(k!)^2)*[x^(n-k)] (1+x)^(-2*(k+1)). - Georg Fischer and Peter Luschny, Jan 18 2020
T(n,k) = (-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!), for n >= k >= 0. - N. J. A. Sloane, Jan 18 2020
From Petros Hadjicostas, Jul 09 2020: (Start)
Michael Somos's formulas above can be restated as
Sum_{k=0..n} T(n,k)/(i+k) = 1 for i = 1..n+1.
These are special cases of the following formula that is alluded to (in some way) in Ser's book:
1 - Sum_{k=0..n} T(n,k)/(x + k) = (x-1)*...*(x-(n + 1))/(x*(x+1)*...*(x+n)).
Because T(n,k) = (-1)^(n+1)*(n + k + 1)*A331430(n,k) and Sum_{k=0..n} A331430(n,k) = (-1)^(n+1), one may derive this formula from Ser's second formula stated in A331430. (End)
T(2*n+1, n) = (-2)*(-27)^n*Pochhammer(4/3, n)*Pochhammer(5/3, n)/(n!*(n+1)!). - G. C. Greubel, Mar 22 2022

Several typos in the data corrected by Georg Fischer and Peter Luschny, Jan 18 2020

Definition changed by N. J. A. Sloane, Jan 18 2020

A109187 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k (1,0)-steps.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 6, 0, 1, 6, 0, 12, 0, 1, 0, 30, 0, 20, 0, 1, 20, 0, 90, 0, 30, 0, 1, 0, 140, 0, 210, 0, 42, 0, 1, 70, 0, 560, 0, 420, 0, 56, 0, 1, 0, 630, 0, 1680, 0, 756, 0, 72, 0, 1, 252, 0, 3150, 0, 4200, 0, 1260, 0, 90, 0, 1, 0, 2772, 0, 11550, 0, 9240, 0, 1980, 0, 110, 0, 1

Offset: 0

Emeric Deutsch, Jun 21 2005

A Grand Motzkin path is a path in the half-plane x >= 0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).
From Peter Bala, Feb 11 2017: (Start)
Consider an infinite 1-dimensional integer lattice with an oriented self-loop at each vertex. Then T(n,k) equals the number of walks of length n from a vertex to itself having k loops. There is a bijection between such walks and Grand Motzkin paths which takes a right step and a left step on the lattice to an up step U and a down step D of a Grand Motzkin path respectively, and takes traversing a loop on the lattice to the horizontal step H. See A282252 for the corresponding triangle of walks on a 2-dimensional lattice with self-loops. (End)

			T(3,1)=6 because we have hud,hdu,udh,duh,uhd,dhu, where u=(1,1),d=(1,-1), h=(1,0).
Triangle begins:
n\k   [0]  [1]   [2]   [3]   [4]   [5]   [6]  [7]  [8]  [9] [10]
[0]    1;
[1]    0,   1;
[2]    2,   0,    1;
[3]    0,   6,    0,    1;
[4]    6,   0,   12,    0,    1;
[5]    0,  30,    0,   20,    0,    1;
[6]   20,   0,   90,    0,   30,    0,    1;
[7]    0, 140,    0,  210,    0,   42,    0,   1;
[8]   70,   0,  560,    0,  420,    0,   56,   0,   1;
[9]    0, 630,    0, 1680,    0,  756,    0,  72,   0,   1;
[10] 252,   0, 3150,    0, 4200,    0, 1260,   0,  90,   0,   1;
[11] ...
From _Peter Bala_, Feb 11 2017: (Start)
The infinitesimal generator begins
      0
      0    0
      2    0     0
      0    6     0     0
     -6    0    12     0     0
      0  -30     0    20     0   0
     80    0   -90     0    30   0   0
      0  560     0  -210     0  42   0  0
  -2310    0  2240     0  -420   0  56  0  0
  ....
and equals the generalized exponential Riordan array [log(Bessel_I(0,2x)),x], and so has integer entries. (End)

Gheorghe Coserea, Rows n = 0..100, flattened
Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.

Diagonal of rational function R(x, y, t) = 1/(1 - (x^2 + t*x*y + y^2)) with respect to x,y, i.e., T(n,k) = [(xy)^n*t^k] R(x,y,t). For t=0..7 we have the diagonals: A126869(t=0, column 0), A002426(t=1, row sums), A000984(t=2), A026375(t=3), A081671(t=4), A098409(t=5), A098410(t=6), A104454(t=7).

Cf. A089627, A109188, A105868, A171128, A108666, A282252, A298608.

G:=1/sqrt((1-t*z)^2-4*z^2):Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 13 do seq(coeff(t*P[n],t^k),k=1..n+1) od;
with(PolynomialTools): CL := p -> CoefficientList(simplify(p), x):
C := (n,x) -> binomial(2*n,n)*hypergeom([-n,-n],[-n+1/2],1/2-x/4):
seq(print(CL(C(n,x))), n=0..11); # Peter Luschny, Jan 23 2018

p[0] := 1; p[n_] := GegenbauerC[n, -n , -x/2];
Flatten[Table[CoefficientList[p[n], x], {n, 0, 11}]] (* Peter Luschny, Jan 23 2018 *)

T(n,k) = if ((n-k)%2, 0, binomial(n,k)*binomial(n-k, (n-k)/2));
concat(vector(12, n, vector(n, k, T(n-1, k-1)))) \\ Gheorghe Coserea, Sep 06 2018

G.f.: 1/sqrt((1-tz)^2-4z^2).
Row sums yield the central trinomial coefficients (A002426).
T(2n+1, 0) = 0.
T(2n, 0) = binomial(2n,n) (A000984).
Sum_{k=0..n} k*T(n,k) = A109188(n).
Except for the order, same rows as those of A105868.
Column k has e.g.f. (x^k/k!)*Bessel_I(0,2x). - Paul Barry, Mar 11 2006
T(n,k) = binomial((n+k)/2,k)*binomial(n,(n+k)/2)*(1+(-1)^(n-k))/2. - Paul Barry, Sep 18 2007
Coefficient array of the polynomials P(n,x) = x^n*hypergeom([1/2-n/2,-n/2], [1], 4/x^2). - Paul Barry, Oct 04 2008
G.f.: 1/(1-xy-2x^2/(1-xy-x^2/(1-xy-x^2/(1-xy-x^2/(1-.... (continued fraction). - Paul Barry, Jan 28 2009
From Paul Barry, Apr 21 2010: (Start)
Exponential Riordan array [Bessel_I(0,2x), x].
Coefficient array of the polynomials P(n,x) = Sum_{k=0..floor(n/2)} C(n,2k)*C(2k, k)*x^(n - 2k).
Diagonal sums are the aerated central Delannoy numbers (A001850 with interpolated zeros). (End)
From Peter Bala, Feb 11 2017: (Start)
T(n,k) = binomial(n,k)*binomial(n-k,floor((n-k)/2))*(1 + (-1)^(n-k))/2.
T(n,k) = (n/k) * T(n-1,k-1).
T(n,k) = the coefficient of H^k in the expansion of (H + U + 1/U)^n.
n-th row polynomial R(n,t) = Sum_{k = 0..floor(n/2)} binomial(n,2*k) * binomial(2*k,k) * t^(n-2*k) = coefficient of x^n in the expansion of (1 + t*x + x^2)^n.
R(n,t) = Sum_{k = 0..n} binomial(n,k)*binomial(2*k,k)*(t - 2)^(n-k).
d/dt(R(n,t)) = n*R(n-1,t).
R(n,t) = (1/Pi) * Integral_{x = 0..Pi} (t + 2*cos(x))^n dx.
Moment representation on a finite interval: R(n,t) = 1/Pi * Integral_{x = t-2 .. t+2} x^n/sqrt((t + 2 - x)*(x - t + 2)) dx.
Recurrence: n*R(n,t) = t*(2*n - 1)*R(n-1,t) - (t^2 - 4)*(n - 1)*R(n-2,t) with R(0,t) = 1 and R(1,t) = t.
R(n,t) = A002426 (t = 1), A000984 (t = 2), A026375 (t = 3), A081671 (t = 4), A098409 (t = 5), A098410 (t = 6) and A104454(t = 7).
The zeros of the row polynomials appear to lie on the imaginary axis in the complex plane. Also, the zeros of R(n,t) and R(n+1,t) appear to interlace on the imaginary axis.
The polynomials R(n,1 + t) are the row polynomials of A171128. (End)
From Peter Luschny, Jan 23 2018: (Start)
These are the coefficients of the polynomials G(n, -n , -x/2) where G(n, a, x) denotes the n-th Gegenbauer polynomial.
These polynomials can also be expressed as C(n, x) = binomial(2*n,n)*hypergeom([-n, -n], [-n+1/2], 1/2-x/4). (End)

A331511 Square array T(n,k), n >= 0, k >= 0, read by descending antidiagonals, where column k is the expansion of (1 - (k-3)x)/(1 - 2(k-1)x + ((k-3)x)^2)^(3/2).

Original entry on oeis.org

1, 1, 0, 1, 2, -15, 1, 4, -6, 32, 1, 6, 9, -12, 105, 1, 8, 30, 16, 30, -576, 1, 10, 57, 140, 25, 60, 105, 1, 12, 90, 384, 630, 36, -140, 5760, 1, 14, 129, 772, 2505, 2772, 49, -280, -13167, 1, 16, 174, 1328, 6430, 16008, 12012, 64, 630, -30400

Offset: 0

Seiichi Manyama, Jan 18 2020

			Square array begins:
      1,   1,  1,    1,     1,     1, ...
      0,   2,  4,    6,     8,    10, ...
    -15,  -6,  9,   30,    57,    90, ...
     32, -12, 16,  140,   384,   772, ...
    105,  30, 25,  630,  2505,  6430, ...
   -576,  60, 36, 2772, 16008, 52524, ...
.
From _Peter Luschny_, Jan 20 2020: (Start)
Read by ascending antidiagonals gives:
[0]      1
[1]      0,    1
[2]    -15,    2,  1
[3]     32,   -6,  4,     1
[4]    105,  -12,  9,     6,     1
[5]   -576,   30, 16,    30,     8,    1
[6]    105,   60, 25,   140,    57,   10,    1
[7]   5760, -140, 36,   630,   384,   90,   12,   1
[8] -13167, -280, 49,  2772,  2505,  772,  129,  14,  1
[9] -30400,  630, 64, 12012, 16008, 6430, 1328, 174, 16, 1 (End)

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A331551, A331552, A000290(n+1), A002457, A108666(n+1), A331323.
T(n,n+3) gives A331512.
Cf. A307883, A331513, A331514.

T := (n, k) -> (n + 1)^2*hypergeom([-n, -n], [2], k - 2):
seq(lprint(seq(simplify(T(n,k)), k=0..7)), n=0..6) # Peter Luschny, Jan 20 2020

T[n_, k_] := (n + 1)^2 * HypergeometricPFQ[{-n, -n}, {2}, k - 2];  Table[Table[T[n, k - n], {n, 0, k}], {k, 0, 9}] //Flatten (* Amiram Eldar, Jan 20 2020 *)

T(n,k) = Sum_{j=0..n} (k-3)^(n-j) * (n+j+1) * binomial(n,j) * binomial(n+j,j).
T(n,k) = Sum_{j=0..n} (k-2)^j * (j+1) * binomial(n+1,j+1)^2.
T(n,k) = (n + 1)^2*hypergeom([-n, -n], [2], k - 2). - Peter Luschny, Jan 20 2020
n * (2*n-1) * T(n,k) = 2 * (2 * (k-1) * n^2 - k + 2) * T(n-1,k) - (k-3)^2 * n * (2*n+1) * T(n-2,k) for n>1. - Seiichi Manyama, Jan 25 2020

A331512 a(n) = Sum_{k=0..n} n^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).

Original entry on oeis.org

1, 8, 90, 1328, 24150, 520272, 12926004, 363233600, 11376760230, 392615960600, 14791582824876, 603743206301424, 26528443526357500, 1248071683342913184, 62576263671773466600, 3330116426356595493120, 187430800395881065513734

Offset: 0

Seiichi Manyama, Jan 19 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..380

Cf. A001850, A108666, A331511, A331513.

a[n_] := Sum[If[n == n-k == 0, 1, n^(n-k)] * (n+k+1) * Binomial[n, k] * Binomial[n + k, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, May 05 2021 *)

{a(n) = sum(k=0, n, n^(n-k)*(n+k+1)*binomial(n, k)*binomial(n+k, k))}

{a(n) = polcoef((1-n*x)/(1-2*(n+2)*x+(n*x)^2)^(3/2), n)}

{a(n) = sum(k=0, n, (n+1)^k*(k+1)*binomial(n+1, k+1)^2)}

a(n) = [x^n] (1 - n*x)/(1 - 2*(n+2)*x + (n*x)^2)^(3/2).

a(n) = Sum_{k=0..n} (n+1)^k * (k+1) * binomial(n+1,k+1)^2.

A361743 Central circular Delannoy numbers: a(n) is the number of Delannoy loops on an n X n toroidal grid.

Original entry on oeis.org

1, 2, 16, 114, 768, 5010, 32016, 201698, 1257472, 7777314, 47800080, 292292946, 1779856128, 10799942322, 65336473104, 394246725570, 2373580947456, 14262064668738, 85546366040592, 512323096241714, 3063932437123840, 18300660294266322, 109183694129335056

Offset: 0

Noah Snyder, Mar 22 2023

nonn

An (n,m) Delannoy loop is an oriented unbased loop on a toroidal grid with points labeled by Z/n x Z/m composed of steps of the form (1,0), (0,1), and (1,1), and which loops around the torus exactly once in each of the x-direction and the y-direction. The central circular Delannoy numbers count the number of (n,n) Delannoy loops. This is a modification of the ordinary central Delannoy numbers A001850.

Dimensions of endomorphism algebras End(S^{{n}}) in the circular Delannoy category attached to the oligomorphic group of order-preserving self-bijections of the circle.

			When n=2 see Figure 3 of "The circular Delannoy Category".

Winston de Greef, Table of n, a(n) for n = 0..1296
Nate Harman, Andrew Snowden, and Noah Snyder, The circular Delannoy Category, arxiv: 2303.10814 [math.RT], 2023.

Circular analog of A001850.
Main diagonal of A361745.
Cf. A008288, A108666, A002003, A047781.

a[n_Integer?Positive] := Sum[k Binomial[n, k] Binomial[n, k] 2^k, {k, 1, n}]

a(n) = if(n == 0, 1, sum(k=0, n, binomial(n, k)^2*k*2^k)) \\ Winston de Greef, Mar 22 2023

from math import comb
def A361743(n): return sum(comb(n,k)**2*k<Chai Wah Wu, Mar 22 2023

a(n) = Sum_{k=0..n} binomial(n,k)^2*k*2^k for n >= 1.
a(n) = n*(D(n,n-1) + D(n-1,n-1)) = n*(D(n,n) - D(n-1,n)) for n >= 1, where D(i,j) = A008288(i,j) are the Delannoy numbers.
a(n) = 2*A108666(n) for n >= 1.
From Alois P. Heinz, Mar 22 2023: (Start)
G.f.: 1 + 2*(1-x)*x/sqrt(x^2-6*x+1)^3.
a(n) = n*A002003(n) for n >= 1.
a(n) = 2*n*A047781(n) for n >= 1. (End)
a(n) = 2*n^2*hypergeom([1 - n, 1 - n], [2], 1) for n >= 1. - Peter Luschny, Mar 22 2023

A110184 Number of (1,1)-steps on the lines y=x, y=x+1 and y=x-1 in all Delannoy paths of length n.

Original entry on oeis.org

0, 1, 8, 55, 354, 2205, 13484, 81523, 489158, 2919481, 17356752, 102884271, 608460330, 3591886293, 21172419444, 124649246955, 733107494286, 4307974826097, 25296523200920, 148448166035239, 870665283937010

Offset: 0

Emeric Deutsch, Jul 14 2005

nonn

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).

			a(3)=55 because on the 63 (=A001850(3)) Delannoy paths of length 3 we have altogether A108666(3)=57 D-steps; however 2 of these, namely the D's in NNDEE and EEDNN, are not on the lines y=x, y=x+1, y=x-1.

Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Cf. A001850, A108666, A110183.

r:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*(1-z-z*r+z^2+z^2*r)/(1-6*z+z^2)/(1-3*z+z^2-z*r+z^2*r): Gser:=series(G,z=0,27): 0,seq(coeff(Gser,z^n),n=1..24);

a(n) = sum(k*A110183(n,k),k=0..n).
G.f.: z(1-z-zr+z^2+z^2*r)/[(1-6z+z^2)(1-3z+z^2-zr+z^2*r)], where r=1+zr+zr^2=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318).
D-finite with recurrence n*a(n) +(-13*n+11)*a(n-1) +10*(5*n-9)*a(n-2) +10*(-5*n+16)*a(n-3) +(13*n-54)*a(n-4) +(-n+5)*a(n-5)=0. - R. J. Mathar, Jul 24 2022

A156052 Triangle read by rows: T(n, k) = binomial(n, k)/Beta(n+1, n-k+1) + binomial(n, n-k)/Beta(n+1, k+1).

Original entry on oeis.org

2, 8, 8, 33, 48, 33, 144, 240, 240, 144, 635, 1240, 1260, 1240, 635, 2778, 6510, 6720, 6720, 6510, 2778, 12019, 33600, 38430, 33600, 38430, 33600, 12019, 51488, 168672, 223776, 184800, 184800, 223776, 168672, 51488, 218799, 824400, 1275120, 1119888, 900900, 1119888, 1275120, 824400, 218799

Offset: 0

Roger L. Bagula, Feb 02 2009

Row sums are 2*A108666(n+1): {2, 16, 114, 768, 5010, 32016, 201698, 1257472, 7777314, 47800080, 292292946, ...}.

			Triangle begins as:
      2;
      8,      8;
     33,     48,     33;
    144,    240,    240,    144;
    635,   1240,   1260,   1240,    635;
   2778,   6510,   6720,   6720,   6510,   2778;
  12019,  33600,  38430,  33600,  38430,  33600,  12019;
  51488, 168672, 223776, 184800, 184800, 223776, 168672, 51488;

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

B:=Binomial;; Flat(List([0..10], n-> List([0..n], k-> (k+1)*B(n+1, k+1)*( B(2*n-k+1, n+1) + B(n+k+1, n+1) ) ))); # G. C. Greubel, Dec 01 2019

B:=Binomial; [(k+1)*B(n+1, k+1)*( B(2*n-k+1, n+1) + B(n+k+1, n+1) ): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 01 2019

b:=binomial; seq(seq( (k+1)*b(n+1, k+1)*( b(2*n-k+1, n+1) + b(n+k+1, n+1) ), k=0..n), n=0..10); # G. C. Greubel, Dec 01 2019

Table[Binomial[n, k]/Beta[n+1, n-k+1] + Binomial[n, n-k]/Beta[n+1, k+1], {n, 0, 10}, {k, 0, n}]//FlattenTable[(k+1)*Binomial[n+1, k+1]*(Binomial[n+k+1, n+1] + Binomial[2*n-k+1, n+1]), {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 01 2019 *)

T(n, k) = my(b=binomial); (k+1)*b(n+1, k+1)*( b(2*n-k+1, n+1) + b(n+k+1, n+1) ); \\ G. C. Greubel, Dec 01 2019

b=binomial; [[(k+1)*b(n+1, k+1)*( b(2*n-k+1, n+1) + b(n+k+1, n+1) ) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 01 2019

T(n, k) = binomial(n, k)/Beta(n+1, n-k+1) + binomial(n, n-k)/Beta(n+1, k+1).

T(n, k) = (k+1)*binomial(n+1, k+1)*( binomial(2*n-k+1, n+1) + binomial(n+k+1, n+1) ). - G. C. Greubel, Dec 01 2019

A132884 Triangle read by rows: T(n,k) is the number of paths in the right half-plane from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k h=(1,0) steps (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 0, 8, 0, 1, 13, 0, 15, 0, 1, 0, 57, 0, 24, 0, 1, 63, 0, 156, 0, 35, 0, 1, 0, 384, 0, 340, 0, 48, 0, 1, 321, 0, 1380, 0, 645, 0, 63, 0, 1, 0, 2505, 0, 3800, 0, 1113, 0, 80, 0, 1, 1683, 0, 11145, 0, 8855, 0, 1792, 0, 99, 0, 1, 0, 16008, 0, 37065, 0, 18368, 0, 2736, 0

Offset: 0

Emeric Deutsch, Sep 03 2007

T(2n,0)=A001850(n) (the central Delannoy numbers); T(2n+1,0)=0. T(2n,1)=0; T(2n-1,1)=A108666(n). T(n,k)=0 if n+k is odd. Row sums yield A059345. See A132277 for the same statistic on paths restricted to the first quadrant.

			Triangle starts:
   1;
   0,  1;
   3,  0,  1;
   0,  8,  0,  1;
  13,  0, 15,  0,  1;
   0, 57,  0, 24,  0,  1;
T(3,1)=8 because we have hH, Hh, hUD, UhD, UDh, hDU, DhU and DUh.

Cf. A001850, A108666, A059345, A132277.

G:=1/sqrt((1-t*z-z^2)^2-4*z^2): Gser:=simplify(series(G,z=0,15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) end do; # yields sequence in triangular form

G.f. = 1/sqrt((1-tz-z^2)^2-4z^2).

cp's OEIS Frontend

A001787 a(n) = n*2^(n-1).

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A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)*(n+k+1)*binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

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A109187 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k (1,0)-steps.

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A331511 Square array T(n,k), n >= 0, k >= 0, read by descending antidiagonals, where column k is the expansion of (1 - (k-3)*x)/(1 - 2*(k-1)*x + ((k-3)*x)^2)^(3/2).

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A331512 a(n) = Sum_{k=0..n} n^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).

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A361743 Central circular Delannoy numbers: a(n) is the number of Delannoy loops on an n X n toroidal grid.

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A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)(n+k+1)binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

A331511 Square array T(n,k), n >= 0, k >= 0, read by descending antidiagonals, where column k is the expansion of (1 - (k-3)x)/(1 - 2(k-1)x + ((k-3)x)^2)^(3/2).