A110110 -id:A110110 - cp's OEIS frontend

A002003 a(n) = 2 * Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k).

0, 2, 8, 38, 192, 1002, 5336, 28814, 157184, 864146, 4780008, 26572086, 148321344, 830764794, 4666890936, 26283115038, 148348809216, 838944980514, 4752575891144, 26964373486406, 153196621856192, 871460014012682, 4962895187697048, 28292329581548718

Offset: 0

N. J. A. Sloane

nonn

a(n) is the number of order-preserving partial self maps of {1,...,n}. For example, a(2) = 8 because there are 8 order-preserving partial self maps of {1,2}: (1 2), (1 1), (2 2), (1 -), (2 -), (- 1), (- 2), (- -). Here for example (2 -) represents the partial map which maps 1 to 2 but does not include 2 in its domain. - James East, Oct 25 2005
From Peter Bala, Mar 02 2020: (Start)
For fixed m = 1,2,3,..., we conjecture that the sequence b(n) := a(m*n) satisfies a recurrence of the form P(2*m,n)*b(n+1) + P(2*m,-n)*b(n-1) = Q(2*m,n)*b(n), where the polynomials P(2*m,n) and Q(2*m,n) have degree 2*m. Conjecturally, the polynomial Q(2*m,n) is an even function of n; its 2*m zeros seem to belong to the interval [-1, 1] and 2*m - 2 of these zeros appear to lie close to the rational numbers of the form +-(2*k + 1)/(2*m), where 0 <= k <= m - 2. Cf. A103885. (End)
a(n), n>0, is the number of points at L1 distance = n from any given point in Z^n.  The sequence is also the difference between the central diagonal (A001850) and +-1 diagonal (A002002) of the Delannoy number triangle (A008288).  - Shel Kaphan, Feb 15 2023

			G.f. = 2*x + 8*x^2 + 38*x^3 + 192*x^4 + 1002*x^5 + 5336*x^6 + 28814*x^7 + ...

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).

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
Peter Bala, A supercongruence for A002003
J. Brzozowski, M. Szykula, Large Aperiodic Semigroups, arXiv preprint arXiv:1401.0157 [cs.FL], 2013-2014.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32.

Cf. A002002, A006318, A027307, A108424, A103885.

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);

Flatten[{0,Table[SeriesCoefficient[((1+x)/Sqrt[1-6*x+x^2]-1)/2,{x,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 04 2012 *)
a[ n_] := If[ n < 1, 0, Hypergeometric2F1[ n, -n, 1, -1]]; (* Michael Somos, Aug 24 2014 *)
Table[2*Sum[Binomial[n-1,k]Binomial[n+k,k],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Sep 18 2024 *)

{a(n) = if( n<1, 0, polcoeff( ((1 - x^2) / (1 - x)^2 + x * O(x^n))^n, n))} /* Michael Somos, Sep 24 2003 */

from math import comb
def A002003(n): return sum(comb(n,k)**2*k<Chai Wah Wu, Mar 22 2023

a(n) = 2*A047781(n).
From Vladeta Jovovic, Mar 28 2004: (Start)
G.f.: ((1+x)/sqrt(1-6*x+x^2)-1)/2.
E.g.f.: exp(3*x)*(2*BesselI(0, 2*sqrt(2)*x)+sqrt(2)*BesselI(1, 2*sqrt(2)*x)). (End)
a(n) = T(n, n-1), array T as in A064861.
a(n) = T(n, n-2), array T as in A049600.
a(n+1) = A110110(2n+1). - Tilman Neumann, Feb 05 2009
a(n) = 2 * JacobiP(n-1,0,1,3) = ((7*n+3)*LegendreP(n,3) - (n+1)*LegendreP(n+1,3)) /(2*n) for n > 0. - Mark van Hoeij, Jul 12 2010
Logarithmic derivative of A006318, the large Schroeder numbers. - Paul D. Hanna, Oct 25 2010
D-finite with recurrence: 4*(3*n^2-6*n+2)*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^(3/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 04 2012
Recurrence (an alternative): n*a(n) = (6-n)*a(n-6) + 2*(5*n-27)*a(n-5) + (84-15*n)*a(n-4) + 52*(3-n)*a(n-3) + 3*(2-5*n)*a(n-2) + 2*(5*n-3)*a(n-1), n>=7. - Fung Lam, Feb 05 2014
a(n) = Hyper2F1([-n, n], [1], -1) for n > 0. - Peter Luschny, Aug 02 2014
a(n) = [x^n] ((1+x)/(1-x))^n for n > 0. - Seiichi Manyama, Jun 07 2018
From Peter Bala, Mar 13 2020: (Start)
a(n) = 2 * Sum_{k = 0..n-1}  2^k*C(n,k+1)*C(n-1,k).
a(n) = 2 * (-1)^(n+1) * Sum_{k = 0..n-1} (-2)^k*C(n+k,n-1)*C(n-1,k).
a(n) = Sum_{k = 0..n} C(n,k)*C(2*n-k-1,n-1).
Conjecture: a(n) = - [x^n] (1 - F(x))^n, where F(x) = 2*x + 6*x^2 + 34*x^3 + 238*x^4 + ... is the o.g.f. of A108424. Equivalently, a(n) = -[x^n](G(x))^(-n), where G(x) = 1 + 2*x + 10*x^2 + 66*x^3 + 498*x^4 + ... is the o.g.f. of A027307.
a(p) == 2 ( mod p^3 ) for prime p >= 5. (End)
a(n) = Sum_{k = 1..n} C(n, k) * C(n-1, k-1) * 2^k. - Michael Somos, May 23 2021
a(n) = A001850(n) - A002002(n), for n > 0. - Shel Kaphan, Feb 15 2023

More terms from Barbara Haas Margolius (b.margolius(AT)csuohio.edu), Oct 10 2001

A378939 Number of Schroeder paths of semilength n up to reversal.

Original entry on oeis.org

1, 2, 5, 15, 54, 216, 947, 4375, 21018, 103550, 520041, 2649391, 13655190, 71053780, 372727751, 1968880111, 10463765490, 55909445082, 300160457453, 1618364548591, 8759315367894, 47574840887024, 259215969470139, 1416461749625543, 7760734001872842, 42624971709868054

Offset: 0

Andrew Howroyd, Dec 19 2024

nonn

A Schroeder path of semilength n is a path from (0,0) to (2n,0) using only steps U = (1,1), H = (2,0) and D = (1,-1). This sequence considers a path and its reversal to be the same.

			The a(1)..a(3) paths are:
a(1) = 1: H, UD;
a(2) = 5: HH, UHD, UDUD, UUDD, HUD=UDH;
a(3) = 15: HHH, HUDH, UHHD, UDHUD, UDUDUD, UUHDD, UUDUDD, UUUDDD, HHUD=UDHH, HUHD=UHDH, HUDUD=UDUDH, UHDUD=UDUHD, HUUDD=UDUDH, UHUDD=UUDHD, UDUUDD=UUDDUD.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A006318, A110110, A007123 (similar for Dyck paths), A378941 (similar for Motzkin paths).

seq(n) = { my(A=O(x^(n+2))); Vec(( -2*x - sqrt(1 - 6*x + x^2 + A) + sqrt(1 - 6*x^2 + x^4 + A)*(1 + x)/(1 - 2*x - x^2) ) / (4*x)) }

a(n) = (A006318(n) + A110110(n))/2.

G.f.: ( -2*x - sqrt(1 - 6*x + x^2) + sqrt(1 - 6*x^2 + x^4)*(1 + x)/(1 - 2*x - x^2) ) / (4*x).

A214868 Triangle T read by rows: T(n,0) = T(n,n) = 1 for n>=0, for n>=2 and 1<=k<=n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) if k = [n/2] or k = [(n+1)/2], else T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 5, 1, 1, 7, 11, 11, 7, 1, 1, 9, 23, 22, 23, 9, 1, 1, 11, 39, 45, 45, 39, 11, 1, 1, 13, 59, 107, 90, 107, 59, 13, 1, 1, 15, 83, 205, 197, 197, 205, 83, 15, 1

Offset: 0

Philippe Deléham, Mar 10 2013

			Triangle begins
1
1, 1
1, 2, 1
1, 3, 3, 1
1, 5, 6, 5, 1
1, 7, 11, 11, 7, 1
1, 9, 23, 22, 23, 9, 1
1, 11, 39, 45, 45, 39, 11, 1
1, 13, 59, 107, 90, 107, 59, 13, 1
1, 15, 83, 205, 197, 197, 205, 83, 15, 1
1, 17, 111, 347, 509, 394, 509, 347, 111, 17, 1
1, 19, 143, 541, 1061, 903, 903, 1061, 541, 143, 19, 1
1, 21, 179, 795, 1949, 2473, 1806, 2473, 1949, 795, 179, 21, 1
...

Cf. A001003, A006318, A065096, A110110

Sum_{k, 0<=k<=n} T(n,k) = A110110(n), number of symmetric Schroeder paths of length 2n.
Sum_{k, 0<=k<=n-2} T(n+k,k) = A065096(n-1), n>=2.
T(2n,n) = A006318(n), large Schroeder numbers.
T(2n+1,n) = A001003(n+1), little Schroeder numbers.
T(n,0) = A000012(n).
T(n,1) = A004280(n).
T(n+2,2) = A142463(n) = A132209(n), n>0.

cp's OEIS Frontend

A002003 a(n) = 2 * Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k).

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A378939 Number of Schroeder paths of semilength n up to reversal.

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A214868 Triangle T read by rows: T(n,0) = T(n,n) = 1 for n>=0, for n>=2 and 1<=k<=n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) if k = [n/2] or k = [(n+1)/2], else T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k).

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