Nachum Dershowitz - cp's OEIS frontend

User: Nachum Dershowitz

Nachum Dershowitz has authored 5 sequences.

A337500 a(n) is the number of ballot sequences of length n tied or won by at most 3 votes.

1, 2, 4, 8, 14, 30, 50, 112, 182, 420, 672, 1584, 2508, 6006, 9438, 22880, 35750, 87516, 136136, 335920, 520676, 1293292, 1998724, 4992288, 7696444, 19315400, 29716000, 74884320, 115000920, 290845350, 445962870

Offset: 0

Nachum Dershowitz, Aug 30 2020

Also the number of n-step walks on a path graph ending within 3 steps of the origin.

Also the number of monotonic paths of length n ending within 3 steps of the diagonal.

Cf. A337499, A024483.

Bisections give A162551 (odd part, starting from second element), A051924 (even part).

a(n) = A337499(n) + (n mod 2)*A024483(floor((n+3)/2)).

Conjecture: D-finite with recurrence -(n+3)*(n-4)*a(n) +2*(n^2-2*n-11)*a(n-1) +4*(n-1)^2*a(n-2) -8*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 27 2020

A337499 a(n) is the number of ballot sequences of length n tied or won by at most 2 votes.

Original entry on oeis.org

1, 2, 4, 6, 14, 20, 50, 70, 182, 252, 672, 924, 2508, 3432, 9438, 12870, 35750, 48620, 136136, 184756, 520676, 705432, 1998724, 2704156, 7696444, 10400600, 29716000, 40116600, 115000920, 155117520, 445962870

Offset: 0

Nachum Dershowitz, Aug 29 2020

Also the number of n-step walks on a path graph ending within 2 steps of the origin. Also the number of monotonic paths of length n ending within 2 steps of the diagonal.

Robert Israel, Table of n, a(n) for n = 0..3323

Cf. A001791, A128014.

Bisections give A000984 (odd part, starting from second element), A051924 (even part).

f:= gfun:-rectoproc({(4 + 4*n)*a(n) + (-12 - 4*n)*a(1 + n) + (-22 - 5*n)*a(2 + n) + (n + 4)*a(n + 3) + (6 + n)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 6},a(n),remember):
map(f, [$0..100]); # Robert Israel, Oct 08 2020

a(n) = A128014(n+1) + ((n+1) mod 2)*2*A001791(ceiling(n/2)).
D-finite with recurrence +(n+2)*a(n) +n*a(n-1) +(-5*n-2)*a(n-2) +4*(-n+1)*a(n-3) +4*(n-3)*a(n-4)=0. - Conjectured by R. J. Mathar, Sep 27 2020, verified by Robert Israel, Oct 08 2020
G.f.: ((4*x + 2)*sqrt(-4*x^2 + 1) + 4*x^2 + 4*x + 2)/(sqrt(-4*x^2 + 1)*(1 + sqrt(-4*x^2 + 1))^2). - Robert Israel, Oct 08 2020
a(n) ~ 2^(n - 1/2) * (5 + (-1)^n) / sqrt(Pi*n). - Vaclav Kotesovec, Mar 08 2023

A336675 Number of paths of length n starting at initial node of the path graph P_10.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 251, 460, 911, 1690, 3327, 6225, 12190, 22950, 44744, 84626, 164407, 312019, 604487, 1150208, 2223504, 4239225, 8181175, 15621426, 30108147, 57556155, 110820165, 212037241, 407946421, 781074572, 1501844193, 2877011660, 5529362694

Offset: 0

Nachum Dershowitz, Jul 30 2020

Also the number of paths along a corridor width 10, starting from one side.

In general, a(n,m) = (2^n/(m+1))*Sum_{r=1..m} (1-(-1)^r)*cos(Pi*r/(m+1))^n*(1+cos(Pi*r/(m+1))) gives the number of paths of length n starting at the initial node on the path graph P_m. Here we have m=10. - Herbert Kociemba, Sep 14 2020

Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-3,1).

This is row 10 of A094718. Bisections give A224514 (even part), A216710 (odd part).

Cf. A000004 (row 0), A000007 (row 1), A000012 (row 2), A016116 (row 3), A000045 (row 4), A038754 (row 5), A028495 (row 6), A030436 (row 7), A061551 (row 8), A178381 (row 9), this sequence (row 10), A336678 (row 11), A001405 (limit).

X := j -> (-1)^(j/11) - (-1)^(1-j/11):
a := k -> add((2 + X(j))*X(j)^k, j in [1, 3, 5, 7, 9])/11:
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 17 2020

a[n_,m_]:=2^(n+1)/(m+1) Module[{x=(Pi r)/(m+1)},Sum[Cos[x]^n (1+Cos[x]),{r,1,m,2}]]
Table[a[n,10],{n,0,40}]//Round (* Herbert Kociemba, Sep 14 2020 *)

my(x='x+O('x^44)); Vec((1 - 3*x^2 + x^4)/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5)) \\ Joerg Arndt, Jul 31 2020

From Stefano Spezia, Jul 30 2020: (Start)
G.f.: (1 - 3*x^2 + x^4)/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5).
a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 3*a(n-4) + a(n-5) for n > 4. (End)
a(n) = (2^n/11)*Sum_{r=1..10} (1-(-1)^r)*cos(Pi*r/11)^n*(1+cos(Pi*r/11)). - Herbert Kociemba, Sep 14 2020

A336678 Number of paths of length n starting at initial node of the path graph P_11.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 461, 922, 1702, 3404, 6315, 12630, 23494, 46988, 87533, 175066, 326382, 652764, 1217483, 2434966, 4542526, 9085052, 16950573, 33901146, 63255670, 126511340, 236063915, 472127830, 880983606, 1761967212, 3287837741

Offset: 0

Nachum Dershowitz, Jul 30 2020

Also the number of paths along a corridor width 11, starting from one side.

Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-9,0,2).

This is row 11 of A094718. Bisections give A087944 (even part), A087946 (odd part).
Cf. A000004 (row 0), A000007 (row 1), A000012 (row 2), A016116 (row 3), A000045 (row 4), A038754 (row 5), A028495 (row 6), A030436 (row 7), A061551 (row 8),
A178381 (row 9), A336675 (row 10), this sequence (row 11), A001405 (limit).

X := j -> (-1)^(j/12) - (-1)^(1-j/12):
a := k -> add((2 + X(j))*X(j)^k, j in [1, 3, 5, 7, 9, 11])/12:
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 17 2020

LinearRecurrence[{0, 6, 0, -9, 0, 2}, {1, 1, 2, 3, 6, 10}, 40] (* Harvey P. Dale, Sep 08 2020 *)
a[n_,m_]:=2^(n+1)/(m+1) Module[{x=(Pi r)/(m+1)},Sum[Cos[x]^n (1+Cos[x]),{r,1,m,2}]]
Table[a[n,11], {n,0,40}]//Round (* Herbert Kociemba, Sep 14 2020 *)

my(x='x+O('x^44)); Vec(-(x^5+3*x^4-3*x^3-4*x^2+x+1)/((2*x^2-1)*(x^4-4*x^2+1))) \\ Joerg Arndt, Jul 31 2020

G.f.: -(x^5+3*x^4-3*x^3-4*x^2+x+1)/((2*x^2-1)*(x^4-4*x^2+1)).

a(n) = (2^n/12)*Sum_{r=1..11} (1-(-1)^r)*cos(Pi*r/12)^n*(1+cos(Pi*r/12)). - Herbert Kociemba, Sep 14 2020

A114997 Number of ordered trees with n edges and no unary or binary nodes.

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 8, 13, 31, 71, 144, 318, 729, 1611, 3604, 8249, 18803, 42907, 98858, 228474, 528735, 1228800, 2865180, 6693712, 15676941, 36807239, 86584783, 204060509, 481823778, 1139565120, 2699329341, 6403500057, 15211830451, 36183117255, 86171536894, 205459894230, 490417795075

Offset: 1

Nachum Dershowitz, Feb 23 2006

nonn

Also counts sequences of n natural numbers, excluding 1 and 2, such that the sum of every prefix is no more than its length.

a(n) is the number of Dyck paths of semilength n with all ascents of length >= 3. For example, a(6) = 4 counts U^6.D^6, U^3.D.U^3.D^5, U^3.D^2.U^3.D^4, U^3.D^3.U^3.D^3 where ^ denotes repetition and a dot denotes concatenation. - David Callan, Dec 08 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
Nachum Dershowitz and Shmuel Zaks, More patterns in trees: Up and down, young and old, odd and even, SIAM J. Discrete Mathematics, 23 (2009), 447-465.

Cf. A000108 (rev. of x/(1+1*Sum_{k>=1} x^k) ), A005043 (rev. of x/(1+x*Sum_{k>=1} x^k) ), A215341 (rev. of x/(1+x^3*Sum_{k>=1} x^k) ).

eq := x^3*A^3+x*A^2-(1+x)*A+1 = 0: A := RootOf(eq, A): Aser := series(A, x = 0, 40): seq(coeff(Aser, x, n), n = 1 .. 38); # Emeric Deutsch, Jan 13 2015

Table[Sum[1/(n+1)*Binomial[n+1,k]*Binomial[2*k-n-3,n-k],{k,Ceiling[(n+3)/2],n}],{n,1,20}] (* Vaclav Kotesovec, Mar 22 2014 *)

a(n)=sum(k=ceil((n+3)/2), n, (1/(n+1) * binomial(n+1, k) * binomial(2*k-n-3, n-k)) ); \\ Joerg Arndt, Aug 19 2012

N=66; gf=serreverse(x/(1+x^2*sum(k=1,N,x^k))+O(x^N)) / x;
/* = 1 + x^3 + x^4 + x^5 + 4*x^6 + 8*x^7 + 13*x^8 + 31*x^9 + ... */
v114997=Vec(gf) /* = [1, 0, 0, 1, 1, 1, 4, 8, 13, 31, ...] */  \\ Joerg Arndt, Aug 19 2012

a(n) = Sum_{(n+3)/2 <= k <= n} (1/(n+1) * binomial(n+1, k) * binomial(2*k-n-3, n-k)).
If A(x) is the g.f. for the sequence with a(0)=1, then x^3*A^3+x*A^2-(1 + x)*A+1 = 0. - Emeric Deutsch, Jan 13 2015
Let A(x) be the g.f. for the sequence with a(0)=1, then x*A(x) is the reversion of x/(1+x^2*sum(k>=1,x^k)). - Joerg Arndt, Aug 19 2012 (proved by Emeric Deutsch, Jan 13 2015)
Recurrence: (n+1)*(n+2)*(28*n^2 - 38*n - 15)*a(n) = -4*(n+1)*(14*n^3 - 12*n^2 + 7*n - 15)*a(n-1) + (n-2)*(140*n^3 + 90*n^2 - 221*n + 45)*a(n-2) + 6*(n-2)*(28*n^3 - 24*n^2 - 75*n + 95)*a(n-3) + 23*(n-3)*(n-2)*(28*n^2 + 18*n - 25)*a(n-4). - Vaclav Kotesovec, Mar 22 2014
a(n) ~ c / (n^(3/2) * r^n), where r = (4*sqrt(2) - 3 + 23*sqrt((344*sqrt(2))/529 - 235/529))/46 = 0.402505948621022106992... is the root of the equation 23*r^4+6*r^3+5*r^2-2*r-1 = 0 and c = sqrt((280 + 133*sqrt(2) - 25*sqrt(14*(11 + 8*sqrt(2)))) / (7*Pi))/4 = 0.273007516... - Vaclav Kotesovec, Mar 22 2014, updated Jan 14 2015

Offset set to 1 by Joerg Arndt, Aug 19 2012

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User: Nachum Dershowitz

A337500 a(n) is the number of ballot sequences of length n tied or won by at most 3 votes.

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A337499 a(n) is the number of ballot sequences of length n tied or won by at most 2 votes.

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A336675 Number of paths of length n starting at initial node of the path graph P_10.

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A336678 Number of paths of length n starting at initial node of the path graph P_11.

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A114997 Number of ordered trees with n edges and no unary or binary nodes.

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