A114464 -id:A114464 - cp's OEIS frontend

A160637 Hankel transform of A114464(n+1).

1, 1, -2, -8, -32, -128, 1024, 16384, 262144, 4194304, -134217728, -8589934592, -549755813888, -35184372088832, 4503599627370496, 1152921504606846976, 295147905179352825856, 75557863725914323419136, -38685626227668133590597632, -39614081257132168796771975168

Offset: 0

Paul Barry, May 21 2009

Hankel transform of A114464(n) is A160636.

This is a generalized Somos-4 sequence. - Michael Somos, Mar 14 2020

G. C. Greubel, Table of n, a(n) for n = 0..114

Cf. A011848, A114464, A160636.

[(-2)^Floor(Binomial(n+1,2)/2): n in [0..50]]; // G. C. Greubel, May 03 2018

Table[(-2)^Floor[Binomial[n + 1, 2]/2], {n, 0, 50}] (* G. C. Greubel, May 03 2018 *)
a[ n_] := (-2)^Quotient[n (n + 1), 4]; (* Michael Somos, Mar 14 2020 *)

for(n=0, 50, print1((-2)^floor(binomial(n+1,2)/2), ", ")) \\ G. C. Greubel, May 03 2018

a(n) = (-2)^floor(C(n+1,2)/2) = (-2)^A011848(n+1).

0 = a(n)*a(n+4) - 2*a(n+1)*a(n+3) + 4*a(n+2)^2 = a(n)*a(n+5) - 4*a(n+1)*a(n+4) for all n in Z. - Michael Somos, Mar 14 2020

A160636 Hankel transform of A114464.

Original entry on oeis.org

1, 0, -1, -2, -8, 0, 128, 1024, 16384, 0, -4194304, -134217728, -8589934592, 0, 35184372088832, 4503599627370496, 1152921504606846976, 0, -75557863725914323419136, -38685626227668133590597632

Offset: 0

Paul Barry, May 21 2009

Hankel transform of A114464(n+1) is A160637.

G. C. Greubel, Table of n, a(n) for n = 0..115

Cf. A004524, A160637.

R:= RealField(); [Round(2^Floor(Binomial(n,2)/2)*((Sqrt(2)/2 -1/2)*Sin(3*Pi(R)*n/4+Pi(R)/4)+(Sqrt(2)/2+1/2)*Cos(Pi(R)*n/4+Pi(R)/4))): n in [0..50]]; // G. C. Greubel, May 03 2018

Table[Round[2^Floor[Binomial[n, 2]/2]*((Sqrt[2]-1)*Sin[(3*n+1)*Pi/4]/2 + (Sqrt[2]+1)*Cos[(n+1)*Pi/4]/2)], {n, 0, 50}] (* G. C. Greubel, May 03 2018 *)
a[ n_] := -Sign[Mod[n - 1, 4]]*(-1)^Quotient[n - 1, 4]*2^Quotient[n (n - 1), 4]; (* Michael Somos, Mar 14 2020 *)

for(n=0,50, print1(round(2^floor(binomial(n,2)/2)*((sqrt(2)-1)*sin((3*n+1)*Pi/4)/2 +(sqrt(2)+1)*cos((n+1)*Pi/4)/2)), ", ")) \\ G. C. Greubel, May 03 2018

A160636(n)=if(n%4!=1,(-1)^((n+2)\4)<<(binomial(n,2)\2),0) \\ M. F. Hasler, May 09 2018

a(n) = 2^floor(C(n,2)/2)*((sqrt(2)-1)*sin((3*n+1)*Pi/4)/2 +(sqrt(2)+1)*cos((n+1)*Pi/4)/2).
a(4k+1) = 0, a(n) = (-1)^floor((n+2)/4) * 2^A011848(n) if n !== 1 (mod 4), where A011848(n) = floor(C(n,2)/2). - M. F. Hasler, May 09 2018
a(n) = -a(2-n) * 2^A004524(n) for all n in Z. - Michael Somos, Mar 14 2020

Comment with an incorrect formula deleted by M. F. Hasler, May 09 2018

A114465 Number of Dyck paths of semilength n having no ascents of length 2 that start at an odd level.

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 105, 317, 982, 3105, 9981, 32520, 107157, 356481, 1195662, 4038909, 13728369, 46919812, 161143157, 555857157, 1924956954, 6689953057, 23325404153, 81567552320, 286009944649, 1005371062561, 3542175587306

Offset: 0

Emeric Deutsch, Nov 29 2005

nonn

Column 0 of A114463.

			a(4)=13 because among the 14 Dyck paths of semilength 4 only UUD(UU)DDD has an ascent of length 2 that starts at an odd level (shown between parentheses).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril and Paul Barry, Two kinds of partial Motzkin paths with air pockets, arXiv:2212.12404 [math.CO], 2022.
Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, Consecutive pattern-avoidance in Catalan words according to the last symbol, Univ. Bourgogne (France 2023).
Jean-Luc Baril, Rigoberto Flórez, and José L. Ramírez, Counting symmetric and asymmetric peaks in motzkin paths with air pockets, Univ. Bourgogne (France, 2023).

Cf. A114463, A114462, A114464.

g:=-1/2/z/(1+z^2-z)*(z^2-1+sqrt((z^2+1)*(z^2-4*z+1))): gser:=series(g,z=0,33): 1,seq(coeff(gser,z^n),n=1..30);

CoefficientList[Series[(1-x^2-Sqrt[(1+x^2)*(1-4*x+x^2)])/(2*x*(1-x+x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

Vec((1 - x^2 - sqrt((1+x^2)*(1-4*x+x^2)))/(2*x*(1-x+x^2)) + O(x^50)) \\ G. C. Greubel, Jan 28 2017

G.f.: [1 - z^2 - sqrt((1+z^2)*(1-4z+z^2))]/[2*z*(1-z+z^2)].
(n+1)*a(n) = (5*n-1)*a(n-1) - (7*n-5)*a(n-2) + 10*(n-2)*a(n-3) - (7*n-23)*a(n-4) + (5*n-19)*a(n-5) - (n-5)*a(n-6). - Vaclav Kotesovec, Mar 20 2014
a(n) ~ sqrt(24+14*sqrt(3)) * (2+sqrt(3))^n / (6 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

A114462 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an even level (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 7, 1, 18, 19, 5, 54, 59, 18, 1, 166, 191, 65, 7, 522, 631, 242, 34, 1, 1670, 2123, 906, 154, 9, 5418, 7247, 3395, 680, 55, 1, 17786, 25011, 12746, 2932, 300, 11, 58974, 87071, 47931, 12414, 1540, 81, 1, 197226, 305275, 180439, 51878, 7552

Offset: 0

Emeric Deutsch, Nov 29 2005

Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). Sum(kT(n,k), k=0..floor(n/2)) = binomial(2n-3,n-1)-binomial(2n-4,n) = A077587(n-2) (n>=2). Column 0 yields A114464.

			T(4,1) = 7 because we have (UU)DDUDUD, UD(UU)DDUD, UDUD(UU)DD, (UU)DUDDUD,
UD(UU)DUDD, (UU)DUDUDD and (UU)DUUDDD, where U=(1,1), D=(1,-1) (the ascents of length 2 starting at an even level are shown between parentheses; note that the last path has an ascent of length 2 that starts at an odd level).
Triangle starts:
1;
1;
1,   1;
2,   3;
6,   7,  1;
18, 19,  5;
54, 59, 18, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A077587, A000108, A114463, A114464, A114465, A102402.

G:= 1/2/z*(3*z^2+2*z^3*t+1-z^3*t^2-3*z^2*t-z^3+t*z-z -sqrt(1+20*z^3*t-18*z^5*t^2+15*z^4*t^2+18*z^5*t+6*z^5*t^3-2*z^4*t^3-12*z^2*t -12*z^3 -6*z-24*z^4*t-8*z^3*t^2+z^6-6*z^5+11*z^4 +z^2*t^2+6*z^6*t^2 -4*z^6*t^3 -4*z^6*t+z^6*t^4+2*t*z +11*z^2)): Gser:=simplify(series(G,z=0,17)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 14 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0,
      `if`(t=2, z, 1), expand(b(x-1, y-1, min(3, t+1))+
      `if`(t=2 and irem(y, 2)=0, z, 1)*b(x-1, y+1, 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Mar 12 2014

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, If[t==2, z, 1], Expand[ b[x-1, y-1, Min[3, t+1]] + If[t==2 && Mod[y, 2]==0, z, 1]*b[x-1, y+1, 0]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

G.f.: G(t,z) satisfies zG^2-(1-z+tz-3tz^2+3z^2-z^3-t^2z^3+2tz^3)G+1-z+z^2+tz-tz^2=0.

A114463 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an odd level (0<=k<=floor(n/2)-1 for n>=2; k=0 for n=0,1).

Original entry on oeis.org

1, 1, 2, 5, 13, 1, 36, 6, 105, 26, 1, 317, 104, 8, 982, 402, 45, 1, 3105, 1522, 225, 10, 9981, 5693, 1052, 69, 1, 32520, 21144, 4698, 412, 12, 107157, 78188, 20319, 2249, 98, 1, 356481, 288340, 85864, 11522, 679, 14, 1195662, 1061520, 356535, 56360, 4230

Offset: 0

Emeric Deutsch, Nov 29 2005

Row n (n>=2) has floor(n/2) terms. Row sums are the Catalan numbers (A000108). Sum(kT(n,k),k=0..floor(n/2)-1)=binomial(2n-4,n) (A002694). Column 0 yields A114465.

			T(5,1) = 6 because we have UUD(UU)DUDDD, UUD(UU)DDUDD, UUD(UU)DDDUD,
UDUUD(UU)DDD, UUDUD(UU)DDD and UUUDD(UU)DDD, where U=(1,1), D=(1,-1) (the ascents of length 2 starting at an odd level are shown between parentheses; note that the fourth path has an ascent of length 2 that starts at an even level).
Triangle starts:
:  0 :    1;
:  1 :    1;
:  2 :    2;
:  3 :    5;
:  4 :   13,    1;
:  5 :   36,    6;
:  6 :  105,   26,    1;
:  7 :  317,  104,    8;
:  8 :  982,  402,   45,  1;
:  9 : 3105, 1522,  225, 10;
: 10 : 9981, 5693, 1052, 69, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A114462, A114464, A114465, A000108, A002694, A243752.

G:=-1/2*(1-z^2+z^2*t-sqrt((z^2*t-z^2+4*z-1)*(z^2*t-z^2-1)))/z/(-z^2+z^2*t+z-z*t-1): Gser:=simplify(series(G,z=0,18)): P[0]:=1: for n from 1 to 15 do P[n]:=coeff(Gser,z^n) od: 1; 1; for n from 2 to 15 do seq(coeff(t*P[n],t^j),j=1..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 2, 5, 2][t])
      *`if`(t=5, z, 1) +b(x-1, y-1, [1, 3, 4, 1, 3][t]))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..15);  # Alois P. Heinz, Jun 10 2014

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, 1, Expand[b[x-1, y+1, {2, 2, 2, 5, 2}[[t]]]*If[t==5, z, 1] + b[x-1, y-1, {1, 3, 4, 1, 3}[[t]]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

G.f.: G=G(t, z) satisfies z[(1-t)z^2-(1-t)z+1]G^2-[1-(1-t)z^2]G+1=0.

cp's OEIS Frontend

A160637 Hankel transform of A114464(n+1).

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A160636 Hankel transform of A114464.

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A114465 Number of Dyck paths of semilength n having no ascents of length 2 that start at an odd level.

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A114462 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an even level (0<=k<=floor(n/2)).

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A114463 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an odd level (0<=k<=floor(n/2)-1 for n>=2; k=0 for n=0,1).

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