A050143 -id:A050143 - cp's OEIS frontend

A050146 a(n) = T(n,n), array T as in A050143.

1, 1, 4, 18, 88, 450, 2364, 12642, 68464, 374274, 2060980, 11414898, 63521352, 354870594, 1989102444, 11180805570, 63001648608, 355761664002, 2012724468324, 11406058224594, 64734486343480, 367891005738690, 2093292414443164, 11923933134635298, 67990160422313808

Offset: 0

Clark Kimberling

nonn

Also main diagonal of array : m(i,1)=1, i>=1; m(1,j)=2, j>1; m(i,j)=m(i,j-1)+m(i-1,j-1)+m(i-1,j): 1 2 2 2 ... / 1 4 8 12 ... / 1 6 18 38 ... / 1 8 32 88 ... / - Benoit Cloitre, Aug 05 2002
a(n) is also the number of order-preserving partial transformations (of an n-element chain) of waist n (waist(alpha) = max(Im(alpha))). - Abdullahi Umar, Aug 25 2008
Define a finite triangle T(r,c) with T(r,0) = binomial(n,r) for 0<=r<=n, and the other terms recursively with T(r,c) = T(r,c-1) + 2*T(r-1,c-1). The sum of the last terms in each row is Sum_{r=0..n} T(r,r)=a(n+1). For n=4 the triangle is 1; 4 6; 6 14 26; 4 16 44 96; 1 9 41 129 321 with the sum of the last terms being 1 + 6 + 26 + 96 + 321 = 450 = a(5). - J. M. Bergot, Jan 29 2013
It may be better to define a(0) = 0 for formulas without exceptions. - Michael Somos, Nov 25 2016
a(n) is the number of points at L1 distance n-1 from any point in Z^n, for n>=1. - Shel Kaphan, Mar 24 2023

			G.f. = 1 + x + 4*x^2 + 18*x^3 + 88*x^4 + 450*x^5 + 2364*x^6 + 12642*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra, 278 (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq., 7 (2004), 04.3.8.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 4.
Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.

Cf. A002003, A006318, A050151, A008288, A002002, A026002.

-1-diagonal of A266213 for n>=1.

a050146 n = if n == 0 then 1 else a035607 (2 * n - 2) (n - 1)
-- Reinhard Zumkeller, Nov 05 2013, Jul 20 2013

Flatten[{1,RecurrenceTable[{(n-3)*(n-1)*a[n-2]-3*(n-2)*(2*n-3)*a[n-1]+(n-2)*(n-1)*a[n]==0,a[1]==1,a[2]==4},a,{n,20}]}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[ n_] := If[ n == 0, 1, Sum[ Binomial[n, k] Binomial[n + k - 2, k - 1], {k, n}]]; (* Michael Somos, Nov 25 2016 *)
a[ n_] := If[ n == 0, 1, n Hypergeometric2F1[1 - n, n, 2, -1]]; (* Michael Somos, Nov 25 2016 *)

taylor(-(x^4+sqrt(x^2-6*x+1)*(x^3-5*x^2+5*x+1)-8*x^3+16*x^2-6*x+1)/(x^3+sqrt(x^2-6*x+1)*(x^2-4*x-1)-7*x^2+7*x-1),x,0,10); /* Vladimir Kruchinin, Nov 25 2016 */

a(n)=if(n==0, 1, sum(k=1,n, binomial(n, k)*binomial(n+k-2, k-1)) ); \\ Joerg Arndt, May 04 2013

A050146 = lambda n : n*hypergeometric([1-n, n], [2], -1) if n>0 else 1
[round(A050146(n).n(100)) for n in (0..24)] # Peter Luschny, Sep 17 2014

From Vladeta Jovovic, Mar 31 2004: (Start)
Coefficient of x^(n-1) in expansion of ((1+x)/(1-x))^n, n > 0.
a(n) = Sum_{k=1..n} binomial(n, k)*binomial(n+k-2, k-1), n > 0. (End)
D-finite with recurrence (n-1)*(n-2)*a(n) = 3*(2*n-3)*(n-2)*a(n-1) - (n-1)*(n-3)*a(n-2) for n > 2. - Vladeta Jovovic, Jul 16 2004
a(n+1) = Jacobi_P(n, 1, -1, 3); a(n+1) = Sum{k=0..n} C(n+1, k)*C(n-1, n-k)*2^k. - Paul Barry, Jan 23 2006
a(n) = n*A006318(n-1) - Abdullahi Umar, Aug 25 2008
a(n) ~ sqrt(3*sqrt(2)-4)*(3+2*sqrt(2))^n/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
a(n+1) = A035607(2*n,n). - Reinhard Zumkeller, Jul 20 2013
a(n) = n*hypergeometric([1-n, n], [2], -1) for n >= 1. - Peter Luschny, Sep 17 2014
O.g.f.: -(x^4 + sqrt(x^2 - 6*x + 1)*(x^3 - 5*x^2 + 5*x + 1) - 8*x^3 + 16*x^2 - 6*x + 1)/(x^3 + sqrt(x^2 - 6*x + 1)*(x^2 - 4*x - 1)- 7*x^2 + 7*x - 1). - Vladimir Kruchinin, Nov 25 2016
0 = a(n)*(a(n+1) - 18*a(n+2) + 65*a(n+3) - 12*a(n+4)) + a(n+1)*(54*a(n+2) - 408*a(n+3) + 81*a(n+4)) + a(n+2)*(72*a(n+2) + 334*a(n+3) - 90*a(n+4)) + a(n+3)*(-24*a(n+3) + 9*a(n+4)) for all integer n if a(0) = 0 and a(n) = -2*A050151(-n) for n < 0. - Michael Somos, Nov 25 2016
O.g.f: (2 - x + x*(3 - x)/sqrt(x^2 - 6*x + 1))/2. - Petros Hadjicostas, Feb 14 2021
a(n) = A002002(n) - A026002(n-1) for n>=2. - Shel Kaphan, Mar 24 2023

A050151 a(n) = T(n,n+2), array T as in A050143.

Original entry on oeis.org

0, 1, 6, 33, 180, 985, 5418, 29953, 166344, 927441, 5188590, 29113953, 163786428, 923511849, 5217709266, 29532022785, 167417253648, 950453221153, 5402869685334, 30748881013153, 175186193208900, 999071379620601, 5702750629608186, 32578618535692033, 186257611786501080

Offset: 0

Clark Kimberling

nonn

Form an array having the first column all 1's and the first row the squares: m(n,1) = 1 and m(1,n) = n^2 for n = 1, 2, 3, .... Define interior terms m(i,j) = m(i,j-1) + m(i-1,j-1) + m(i-1,j). Then the terms on the main diagonal are the terms of this sequence. - J. M. Bergot, Nov 16 2012

Form an array with first row m(1,j)=1 and first column m(n,1) = n*(n-1)+1 for n=1,2,3... The remaining terms m(i,j) = m(i,j-1) + m(i-1,j-1) + m(i-1,j); Sum_{n=1,2,3,...} T(n,n) = a(n). The first five terms in the main diagonal are 1, 5, 27, 147, 805 with partial sums 1, 6, 33, 180, 985. - J. M. Bergot, Jan 26 2013

			G.f. = x + 6*x^2 + 33*x^3 + 180*x^4 + 985*x^5 + 5418*x^6 + 29953*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions

Cf. A001003, A001850, A006318, A050143, A050146.

A050151 := n -> (-1)^n*n*hypergeom([-n, n+1], [2], 2):
seq(simplify(A050151(n)), n=0..21); # Peter Luschny, Nov 09 2017

CoefficientList[Series[((1 - 3*x)/(1 - 6*x + x^2)^(1/2) - 1)/(4*x), {x, 0, 100}], x] (* Vincenzo Librandi, Feb 02 2013 *)
a[ n_] := n / 2 Hypergeometric2F1[1 + n, -n, 2, -1]; (* Michael Somos, Nov 25 2016 *)

a(n) = n*hypergeom([-n, n+1], [2], 2)\/(-1)^n \\ Charles R Greathouse IV, Oct 23 2023

From Vladeta Jovovic, Mar 28 2004: (Start)
G.f.: ((1-3*x)/sqrt(1-6*x+x^2)-1)/(4*x).
E.g.f.: exp(3*x)*BesselI(1, 2*sqrt(2)*x)/sqrt(2). (End)
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(n+1, k+1)/2. - Paul Barry, Sep 20 2004
a(n) = n*R(n)/2, where R(n)=A006318(n) are the large Schroeder numbers. - Emeric Deutsch, Jul 14 2005
From David Callan, Aug 16 2006: (Start)
a(n) = Sum_{k=0..n} 2^(k-1)*binomial(n, k)*binomial(n, k-1).
a(n) = (CentralDelannoy(n+1) - 3*CentralDelannoy(n))/4 where CentralDelannoy(n) is A001850. (End)
a(n) = (1/Pi)*Integral_{x=3-2*sqrt(2)..3+2*sqrt(2)} x^n*(x-3)/(4*sqrt(-x^2+6x-1)). - Paul Barry, Sep 16 2006
D-finite with recurrence (n+1)*a(n) - 9*n*a(n-1) + (19*n-27)*a(n-2) + 3*(-n+2)*a(n-3) = 0. - R. J. Mathar, Nov 16 2012
a(n) ~ sqrt(3*sqrt(2)-4) * (1+sqrt(2))^(2*n+2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 13 2016
a(n) = a(n)*(+9*a(n+1) - 90*a(n+2) + 81*a(n+3) - 12*a(n+4)) + a(n+1)*(-24*a(n+1) + 334*a(n+2) - 408*a(n+3) + 65*a(n+4)) + a(n+2)*(+72*a(n+2) + 54*a(n+3) - 18*a(n+4)) + a(n+3)*(+a(n+4)) for all n in Z where a(n) = -(1/2) * A050146(-n) if n < 0. - Michael Somos, Nov 25 2016
From Peter Luschny, Nov 09 2017: (Start)
a(n) = (-1)^n*n*hypergeom([-n, n+1], [2], 2).
a(n) = n*A001003(n). (End)

A050147 a(n) = T(n,n-1), array T as in A050143. Also T(2n+1,n), array T as in A055807.

Original entry on oeis.org

1, 3, 12, 56, 280, 1452, 7700, 41456, 225648, 1238420, 6840988, 37986984, 211842696, 1185635388, 6655993380, 37463920608, 211350457824, 1194706644516, 6765300359468, 38370431711000, 217931108199672

Offset: 1

Clark Kimberling

nonn

Cf. A002003, A006318, A050143, A055807.

a[n_]:=Binomial[2*n-3,n-1]*Hypergeometric2F1[-n+1,-n,-2*n+3,-1];
Table[a[n],{n,1,21}] (* Detlef Meya, Dec 04 2023 *)

a(n):=if n=1 then 1 else sum((binomial(n,i+1))*binomial(n+i-2,n-2),i,0,n-2)+binomial(2*n-3,n-2); /* Vladimir Kruchinin, Nov 25 2014 */

From Vladimir Kruchinin, Nov 25 2014: (Start)
G.f.: x*((-x^2 + 4*x + 1)/(2*sqrt(x^2 - 6*x + 1)) -x/2 + 1/2).
For n >= 2, a(n) = C(2*n-3,n-2) + Sum_{i=0..n-2} C(n,i+1)*C(n+i-2,n-2). (End)
a(n) ~ (1 + sqrt(2))^(2*n-2) / (2^(1/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 14 2021
a(n) = binomial(2*n-3, n-1)*hypergeom([-n+1, -n], [-2*n+3], -1). - Detlef Meya, Dec 04 2023
D-finite with recurrence (-n+1)*a(n) +(2*n+3)*a(n-1) +3*(8*n-29)*a(n-2) +(-10*n+49)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Jul 31 2025

A050148 a(n) = T(n,n-2), array T as in A050143.

Original entry on oeis.org

1, 7, 32, 160, 832, 4424, 23872, 130176, 715520, 3957448, 21998176, 122786976, 687736896, 3863423880, 21758302848, 122811081216, 694533774336, 3934558067336, 22323615621280, 126832567223712, 721500116473664

Offset: 2

Clark Kimberling

nonn

Also T(2*n-2,n-2), array T as in A055807.

Cf. A050143, A055807.

a(n) = Sum_{s=1..n} binomial(n,s)*binomial(s+n-4,n-3) = n*hypergeom([-n + 1, n-2], [2], -1) for n >= 3. - Petros Hadjicostas, Feb 13 2021

D-finite with recurrence (n-1)*(n-3)*a(n) +(-8*n^2+41*n-48)*a(n-1) +(13*n^2-95*n+162)*a(n-2) -2*(n-2)*(n-6)*a(n-3)=0 for n-3>=3. - R. J. Mathar, Jul 31 2025

A050152 a(n) = T(n,n+3), array T as in A050143.

Original entry on oeis.org

0, 1, 7, 42, 242, 1375, 7773, 43876, 247684, 1399293, 7913955, 44812878, 254064726, 1442131899, 8195232633, 46621424520, 265490365448, 1513290869881, 8633347134975, 49293941140402, 281670686307130

Offset: 0

Clark Kimberling, Dec 11 1999

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjic, Two Enumerative Functions

Cf. A050143.

a := n -> (-1)^n*(n+1)*(n/2)*hypergeom([-n,n+2], [3], 2);
seq(round(evalf(a(n),32)), n=0..20); # Peter Luschny, May 24 2014

Table[JacobiP[n-1, 1, 2, 3], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 12 2008 *)

a(n):=sum(2^(i-2)*(-1)^(n-i)*binomial(n,n-i+2)*binomial(n+i-1,n-1),i,0,n+2); /* Vladimir Kruchinin, May 24 2014 */

G.f.: (x*sqrt(x^2-6*x+1)-x^2+3*x)/(-x^4+sqrt(x^2-6*x+1)*(x^3-3*x^2-3*x+1)+6*x^3-2*x^2+6*x-1)+1/(4*x)-1/(4*x^2). - Vladimir Kruchinin, May 24 2014
a(n) = Sum_{i=0..n+2} 2^(i-2)*(-1)^(n-i)*binomial(n,n-i+2)*binomial(n+i-1,n-1). - Vladimir Kruchinin, May 24 2014
a(n) ~ sqrt(48+34*sqrt(2)) * (3+2*sqrt(2))^n / (8*sqrt(Pi*n)). - Vaclav Kotesovec, May 24 2014
a(n) = (-1)^n*(n+1)*(n/2)*hypergeom([-n, n+2], [3], 2). - Peter Luschny, May 24 2014
n^2*(n+1)*a(n-1) = Sum_{k=0..n-1} (2*k^3+k^2+k)*binomial(n-1,k)*binomial(n+k,k) for all n > 0. This follows from the Zeilberger algorithm. - Zhi-Wei Sun, Aug 30 2014
a(n) = Sum_{k=0..n} (binomial(n,k)*binomial(2*n-k+1,n-k-1)). - Vladimir Kruchinin, Oct 26 2016

Typo in Mathematica code fixed by Vincenzo Librandi, May 26 2013

A050149 a(n) = T(n,n-3), array T as in A050143.

Original entry on oeis.org

1, 15, 80, 432, 2352, 12896, 71136, 394400, 2196128, 12273648, 68811184, 386838480, 2179890000, 12309739968, 69641542848, 394643939904, 2239678552640, 12727572969680, 72415319422992, 412470467298032

Offset: 3

Clark Kimberling

nonn

a(n) = Sum_{s=1..n} binomial(n,s)*binomial(s+n-5,n-4) = n*hypergeom([-n + 1, n-3], [2], -1) for n >= 4 and a(1) = 1. - Petros Hadjicostas, Feb 14 2021

D-finite with recurrence 3*(n-1)*(n-4)*a(n) +(-23*n^2+139*n-180)*a(n-1) +3*(11*n^2-91*n+170)*a(n-2) -5*(n-2)*(n-7)*a(n-3)=0 for n-3>=4. - R. J. Mathar, Jul 31 2025

A002002 a(n) = Sum_{k=0..n-1} binomial(n,k+1) * binomial(n+k,k).

Original entry on oeis.org

0, 1, 5, 25, 129, 681, 3653, 19825, 108545, 598417, 3317445, 18474633, 103274625, 579168825, 3256957317, 18359266785, 103706427393, 586889743905, 3326741166725, 18885056428537, 107347191941249, 610916200215241

Offset: 0

N. J. A. Sloane, Simon Plouffe

From Benoit Cloitre, Jan 29 2002: (Start)
Array interpretation (first row and column are the natural numbers):
  1 2 3 ..j ... if b(i,j) = b(i-1,j) + b(i-1,j-1) + b(i,j-1) then a(n+1) = b(n,n)
  2 5 .........
  .............
  i........... b(i,j)
(End)
Number of ordered trees with 2n edges, having root of even degree, nonroot nodes of outdegree at most 2 and branches of odd length. - Emeric Deutsch, Aug 02 2002
Coefficient of x^n in ((1-x)/(1-2x))^n, n>0. - Michael Somos, Sep 24 2003
Number of peaks in all Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0). Example: a(2)=5 because HH, HU*D, U*DH, UHD, U*DU*D, UU*DD contain 5 peaks (indicated by *). - Emeric Deutsch, Dec 06 2003
a(n) is the total number of HHs in all Schroeder (n+1)-paths. Example: a(2)=5 because UH*HD, H*H*H, UDH*H, H*HUD contain 5 HHs (indicated by *) and the other 18 Schroeder 3-paths contain no HHs. - David Callan, Jul 03 2006
a(n) is the total number of Hs in all Schroeder n-paths. Example: a(2)=5 as the Schroeder 2-paths are HH, DUH, DHU, HDU, DUDU and DDUU, and there are 5 H's. In general, a(n) is the total number of H..Hs (m+1 H's) in all Schroeder (n+m)-paths. - FUNG Cheok Yin, Jun 19 2021
a(n) is the number of points in Z^(n+1) that are L1 (Manhattan) distance <= n from the origin, or the number of points in Z^n that are L1 distance <= n+1 from the origin. These terms occur in the crystal ball sequences: a(n) here is the n-th term in the sequence for the (n+1)-dimensional cubic lattice as well as the (n+1)-st term in the sequence for the n-dimensional cubic lattice. See A008288 for a list of crystal ball sequences (rows or columns of A008288). - Shel Kaphan, Dec 25 2022 [Edited by Peter Munn, Jan 05 2023]

			G.f. = x + 5*x^2 + 25*x^3 + 129*x^4 + 681*x^5 + 3653*x^6 + 19825*x^7 + 108545*x^8 + ...

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).

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0 through 100 were computed by Vincenzo Librandi)
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, Correlators for the Wigner-Smith time-delay matrix of chaotic cavities, arXiv:1601.06690 [math-ph], 2016.
L. Ericksen, Lattice path combinatorics for multiple product identities, J. of Statistical Planning and Inference 140 (2010) 2113-2226, see p. 2219.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 19.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32.

Bisection of A002003, Cf. A047781, A001003.
a(n)=T(n, n+1), array T as in A050143.
a(n)=T(n, n+1), array T as in A064861.
Half the first differences of central Delannoy numbers (A001850).
a(n)=T(n, n+1), array T as in A008288.
Cf. A026002, A190666, A259554.

[&+[Binomial(n,k+1)*Binomial(n+k,k): k in [0..n]]: n in [0..21]];  // Bruno Berselli, May 19 2011

A064861 := proc(n,k) option remember; if n = 1 then 1; elif k = 0 then 0; else A064861(n,k-1)+(3/2-1/2*(-1)^(n+k))*A064861(n-1,k); fi; end; seq(A064861(i,i+1),i=1..40);

CoefficientList[Series[((1-x)/Sqrt[1-6x+x^2]-1)/2, {x,0,30}],x]  (* Harvey P. Dale, Mar 17 2011 *)
a[ n_] := n Hypergeometric2F1[ n + 1, -n + 1, 2, -1] (* Michael Somos, Aug 09 2011 *)
a[ n_] := With[{m = Abs@n}, Sign[n] Sum[ Binomial[ m, k] Binomial[ m + k - 1, m], {k, m}]]; (* Michael Somos, Aug 09 2011 *)

makelist(sum(binomial(n,k+1)*binomial(n+k,k), k, 0, n), n, 0, 21); /* Bruno Berselli, May 19 2011 */

{a(n) = my(m = abs(n)); sign( n) * sum( k=0, m-1, binomial( m, k+1) * binomial( m+k, k))}; /* Michael Somos, Aug 09 2011 */

/* L.g.f.: Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*(1-x)^(-n)/n! */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A=(sum(m=1, n+1, Dx(m-1, x^(2*m-1)/(1-x)^m/m!)+x*O(x^n))); n*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, May 17 2015

a = lambda n: hypergeometric([1-n, -n], [1], 2) if n>0 else 0
[simplify(a(n)) for n in range(22)] # Peter Luschny, Nov 19 2014

G.f.: ((1-x)/sqrt(1-6*x+x^2)-1)/2. - Emeric Deutsch, Aug 02 2002
E.g.f.: exp(3*x)*(BesselI(0, 2*sqrt(2)*x)+sqrt(2)*BesselI(1, 2*sqrt(2)*x)). - Vladeta Jovovic, Mar 28 2004
a(n) = Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k+1). - Paul Barry, Sep 20 2004
a(n) = n * hypergeom([n + 1, -n + 1], [2], -1) = ((n+1)*LegendreP(n+1,3) - (5*n+3)*LegendreP(n,3))/(2*n) for n > 0. - Mark van Hoeij, Jul 12 2010
G.f.: x*d/dx log(1/(1-x*A006318(x))). - Vladimir Kruchinin, Apr 19 2011
a(n) = -a(-n) for all n in Z. - Michael Somos, Aug 09 2011
G.f.: -1 + 1 / ( 1 - x / (1 - 4*x / (1 - x^2 / (1 - 4*x / (1 - x^2 / (1 - 4*x / ...)))))). - Michael Somos, Jan 03 2013
a(n) = Sum_{k=0..n} A201701(n,k)^2 = Sum_{k=0..n} A124182(n,k)^2 for n > 0. - Philippe Deléham, Dec 05 2011
D-finite with recurrence: 2*(6*n^2-12*n+5)*a(n-1)-(n-2)*(2*n-1)*a(n-2)-n*(2*n-3)*a(n)=0. - Vaclav Kotesovec, Oct 04 2012
a(n) ~ (3+2*sqrt(2))^n/(2^(5/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 04 2012
D-finite (an alternative): n*a(n) = (6-n)*a(n-6) + (14*n-72)*a(n-5) + (264-63*n)*a(n-4) + 100*(n-3)*a(n-3) + (114-63*n)*a(n-2) + 2*(7*n-6)*a(n-1), n >= 7. - Fung Lam, Feb 05 2014
a(n) = (-1)^(n-1)*Sum_{k=0..n-1} (-2)^k*binomial(n-1,k)*binomial(n+k,k) and n^3*a(n) = Sum_{k=0..n-1} (4*k^3+4*k^2+4*k+1)*binomial(n-1,k)*binomial(n+k,k). For each of the two equalities, both sides satisfy the same recurrence -- this follows from the Zeilberger algorithm. - Zhi-Wei Sun, Aug 30 2014
a(n) = hypergeom([1-n, -n], [1], 2) for n >= 1. - Peter Luschny, Nov 19 2014
Logarithmic derivative of A001003 (little Schroeder numbers). - Paul D. Hanna, May 17 2015
L.g.f.: L(x) = Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1-x)^(-n) / n! = Sum_{n>=1} a(n)*x^n/n where exp(L(x)) = g.f. of A001003. - Paul D. Hanna, May 17 2015
a(n+1) = (1/2^(n+1)) * Sum_{k >= 0} (1/2^k) * binomial(n + k, n)*binomial(n + k, n + 1). - Peter Bala, Mar 02 2017
2*a(n) = A110170(n), n > 0. - R. J. Mathar, Feb 10 2022
a(n) = (LegendreP(n,3) - LegendreP(n-1,3))/2. - Mark van Hoeij, Jul 14 2022
D-finite with recurrence n*a(n) +(-7*n+5)*a(n-1) +(7*n-16)*a(n-2) +(-n+3)*a(n-3)=0. - R. J. Mathar, Aug 01 2022
From Peter Bala, Nov 08 2022: (Start)
a(n) = (-1)^(n+1)*hypergeom( [n+1, -n+1], [1], 2) for n >= 1.
The Gauss congruences hold: a(n*p^r) == a(n^p^(r-1)) (mod p^r) for all primes p and all positive integers n and r. (End)
From Peter Bala, Apr 18 2024: (Start)
G.f.: Sum_{n >= 1} binomial(2*n-1, n)*x^n/(1 - x)^(2*n) = x + 5*x^2 + 25*x^3 + 129*x^4 + ....
Row sums of A253283. (End)

More terms from Clark Kimberling

A055807 Triangle T read by rows: T(i,j) = R(i-j,j), where R(i,0) = 1 for i >= 0, R(0,j) = 0 for j >= 1, and R(i,j) = Sum_{h=0..i-1, k=0..j} R(h,k) for i >= 1 and j >= 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 4, 1, 0, 1, 15, 12, 5, 1, 0, 1, 31, 32, 18, 6, 1, 0, 1, 63, 80, 56, 25, 7, 1, 0, 1, 127, 192, 160, 88, 33, 8, 1, 0, 1, 255, 448, 432, 280, 129, 42, 9, 1, 0, 1, 511, 1024, 1120, 832, 450, 180, 52, 10, 1, 0, 1, 1023

Offset: 0

Clark Kimberling, May 28 2000

Formatted as a triangular array, it is [1, 0, 1, 1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 0, -1, 1, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 05 2006

The square array (R(n,k): n,k >= 0) referred to in the name of the sequence is actually A050143. - Petros Hadjicostas, Feb 13 2021

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,   0;
  1,   1,   0;
  1,   3,   1,   0;
  1,   7,   4,   1,   0;
  1,  15,  12,   5,   1,   0;
  1,  31,  32,  18,   6,   1,  0;
  1,  63,  80,  56,  25,   7,  1, 0;
  1, 127, 192, 160,  88,  33,  8, 1, 0;
  1, 255, 448, 432, 280, 129, 42, 9, 1, 0;
  ...
Florez et al. (2019) give the triangle in this form:
    1,    0,    0,   0,   0,   0,  0,  0, 0, ...
    3,    1,    0,   0,   0,   0,  0,  0, 0, ...
    7,    4,    1,   0,   0,   0,  0,  0, 0, ...
   15,   12,    5,   1,   0,   0,  0,  0, 0, ...
   31,   32,   18,   6,   1,   0,  0,  0, 0, ...
   63,   80,   56,  25,   7,   1,  0,  0, 0, ...
  127,  192,  160,  88,  33,   8,  1,  0, 0, ...
  255,  448,  432, 280, 129,  42,  9,  1, 0, ...
  511, 1024, 1120, 832, 450, 180, 52, 10, 1, ...
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics Vol. 342, Issue 11 (2019), 3079-3097. See page 3091. Gives the triangle in a slightly different form (see the Examples section).
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40(4) (2002) 328-338, Example 3A.

Rows sums: A001519 (odd-indexed Fibonacci numbers).
Cf. A050143, A050147, A050148, A111516.
Cf. A055809, A055810, A055811, A055815, A055816, A055817.

T:= function(i,j)
    if j=0 then return 1;
    elif i=0 then return 0;
    else return Sum([0..i-1], h-> Sum([0..j], m-> T(h,m) ));
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n-k,k) ))); # G. C. Greubel, Jan 23 2020

function T(i,j)
  if j eq 0 then return 1;
  elif i eq 0 then return 0;
  else return (&+[(&+[T(h,m): m in [0..j]]): h in [0..i-1]]);
  end if; return T; end function;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 23 2020

T:= proc(i, j) option remember;
      if j=0 then 1
    elif i=0 then 0
    else add(add(T(h,m), m=0..j), h=0..i-1)
      fi; end:
seq(seq(T(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Jan 23 2020

T[i_, j_]:= T[i, j]= If[j==0, 1, If[i==0, 0, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]]; Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 23 2020 *)

T(i,j) = if(j==0, 1, if(i==0, 0, sum(h=0,i-1, sum(m=0,j, T(h,m) ))));
for(n=0,12, for(k=0, n, print1(T(n-k,k), ", "))) \\ G. C. Greubel, Jan 23 2020

@CachedFunction
def T(i, j):
    if j==0: return 1
    elif i==0: return 0
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[[T(n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 23 2020

T(2*n,n) = A050146(n).
G.f.: (1-2*x)*(1-x*y)/((1-x)*(1-x*y-2*x+x^2*y)). - R. J. Mathar, Aug 11 2015
From Petros Hadjicostas, Feb 13 2021: (Start)
T(n,k) = A050143(n-k, k) for 0 <= k <= n.
T(n,k) = (n-k)*hypergeom([-n + k + 1, k], [2], -1) = Sum_{s=1..n-k} binomial(n-k,s)*binomial(s+k-2,k-1) for 1 <= k <= n.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 2 <= k <= n-1 with initial conditions T(n,0) = 1 for n >= 0, T(n,n) = 0 for n >= 1, and T(n,1) = 2^(n-1) - 1 for n >= 2. (End)

A062110 A(n,k) is the coefficient of x^k in (1-x)^n/(1-2*x)^n for n, k >= 0; Table A read by descending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 5, 3, 1, 0, 8, 12, 9, 4, 1, 0, 16, 28, 25, 14, 5, 1, 0, 32, 64, 66, 44, 20, 6, 1, 0, 64, 144, 168, 129, 70, 27, 7, 1, 0, 128, 320, 416, 360, 225, 104, 35, 8, 1, 0, 256, 704, 1008, 968, 681, 363, 147, 44, 9, 1, 0, 512, 1536, 2400, 2528, 1970

Offset: 0

Henry Bottomley, May 30 2001

The triangular version of this square array is defined by T(n,k) = A(k,n-k) for 0 <= k <= n. Conversely, A(n,k) = T(n+k,n) for n,k >= 0. We have [o.g.f of T](x,y) = [o.g.f. of A](x*y, x) and [o.g.f. of A](x,y) = [o.g.f. of T](y,x/y). - Petros Hadjicostas, Feb 11 2021
From Paul Barry, Nov 10 2008: (Start)
As number triangle, Riordan array (1, x(1-x)/(1-2x)). A062110*A007318 is A147703.
[0,1,1,0,0,0,....] DELTA [1,0,0,0,.....]. (Philippe Deléham's DELTA is defined in A084938.) (End)
Modulo 2, this triangle T becomes triangle A106344. - Philippe Deléham, Dec 18 2008

			Table A(n,k) (with rows n >= 0 and columns k >= 0) begins:
  1, 0,  0,   0,   0,    0,    0,     0,     0,     0, ...
  1, 1,  2,   4,   8,   16,   32,    64,   128,   256, ...
  1, 2,  5,  12,  28,   64,  144,   320,   704,  1536, ...
  1, 3,  9,  25,  66,  168,  416,  1008,  2400,  5632, ...
  1, 4, 14,  44, 129,  360,  968,  2528,  6448, 16128, ...
  1, 5, 20,  70, 225,  681, 1970,  5500, 14920, 39520, ...
  1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, ...
  ... - _Petros Hadjicostas_, Feb 15 2021
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,   1;
  0,   4,   5,   3,   1;
  0,   8,  12,   9,   4,   1;
  0,  16,  28,  25,  14,   5,   1;
  0,  32,  64,  66,  44,  20,   6,   1;
  0,  64, 144, 168, 129,  70,  27,   7,   1;
  0, 128, 320, 416, 360, 225, 104,  35,   8,   1;
  ... - _Philippe Deléham_, Nov 30 2008

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

Rows of A include A000007, A011782, A045623, A058396, A062109, A169792, A169793, A169794, A169795, A169796, A169797.
Columns of A include A000012, A001477, A000096, A000297.
Main diagonal of A is A002002.
Table A(n, k) is a multiple of 2^(k-n); dividing by this gives a table similar to A050143 except at the edges.
Essentially the same array as A105306, A160232.

t[n_, n_] = 1; t[n_, k_] := 2^(n-2*k)*k*Hypergeometric2F1[1-k, n-k+1, 2, -1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 30 2013, after Philippe Deléham + symbolic sum *)

a(i,j)=if(i<0 || j<0,0,polcoeff(((1-x)/(1-2*x)+x*O(x^j))^i,j))

Formulas for the square array (A(n,k): n,k >= 0):
A(n, k) = A(n-1, k) + Sum_{0 <= j < k} A(n, j) for n >= 1 and k >= 0 with A(0, k) = 0^k for k >= 0.
G.f.: 1/(1-x*(1-y)/(1-2*y)) = Sum_{i, j >= 0} A(i, j) x^i*y^j.
From Petros Hadjicostas, Feb 15 2021: (Start)
A(n,k) = 2^(k-n)*n*hypergeom([1-n, k+1], [2], -1) for n >= 0 and k >= 1.
A(n,k) = 2*A(n,k-1) + A(n-1,k) - A(n-1,k-1) for n,k >= 1 with A(n,0) = 1 for n >= 0 and A(0,k) = 0 for k >= 1. (End)
Formulas for the triangle (T(n,k): 0 <= k <= n):
From Philippe Deléham, Aug 01 2006: (Start)
T(n,k) = A121462(n+1,k+1)*2^(n-2*k) for 0 <= k < n.
T(n,k) = 2^(n-2*k)*k*hypergeom([1-k, n-k+1], [2], -1) for 0 <= k < n. (End)
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 = -7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Dec 09 2008
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 1 <= k <= n-1 with T(0,0) = T(1,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, and T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 30 2013
G.f.: Sum_{n.k>=0} T(n,k)*x^n*y^k = (1 - 2*x)/(x^2*y - x*y - 2*x + 1). - Petros Hadjicostas, Feb 15 2021

Various sections edited by Petros Hadjicostas, Feb 15 2021

cp's OEIS Frontend

A050146 a(n) = T(n,n), array T as in A050143.

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A050151 a(n) = T(n,n+2), array T as in A050143.

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A050147 a(n) = T(n,n-1), array T as in A050143. Also T(2n+1,n), array T as in A055807.

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A050148 a(n) = T(n,n-2), array T as in A050143.

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A050152 a(n) = T(n,n+3), array T as in A050143.

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A050149 a(n) = T(n,n-3), array T as in A050143.

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A002002 a(n) = Sum_{k=0..n-1} binomial(n,k+1) * binomial(n+k,k).

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