A230823 -id:A230823 - cp's OEIS frontend

A274372 Number T(n,k) of modified skew Dyck paths of semilength n such that the area between the x-axis and the path is k; triangle T(n,k), n>=0, n<=k<=n^2, read by rows.

1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 0, 3, 2, 3, 1, 3, 2, 2, 1, 1, 0, 1, 1, 0, 4, 3, 7, 4, 7, 5, 8, 6, 6, 3, 5, 4, 3, 2, 2, 1, 1, 0, 1, 1, 0, 5, 4, 12, 10, 17, 12, 20, 18, 22, 14, 19, 16, 18, 14, 14, 12, 12, 7, 8, 7, 5, 4, 3, 2, 2, 1, 1, 0, 1

Offset: 0

Alois P. Heinz, Jun 19 2016

A modified skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and A=(-1,1) (anti-down) so that A and D steps do not overlap.

			T(3,3) = 1:   /\/\/\
.
                 /\         /\
T(3,5) = 2:   /\/  \   ,   /  \/\
.
              /\
              \ \
T(3,6) = 1:   /  \
.
               /\/\
T(3,7) = 1:   /    \
.
                /\
               /  \
T(3,9) = 1:   /    \
.
Triangle T(n,k) begins:
n\k: 0 1 2 3 4 5 6 7 8 9 . . . . . .16 . . . . . . . .25
---+----------------------------------------------------
00 : 1
01 :   1
02 :     1 0 1
03 :       1 0 2 1 1 0 1
04 :         1 0 3 2 3 1 3 2 2 1 1 0 1
05 :           1 0 4 3 7 4 7 5 8 6 6 3 5 4 3 2 2 1 1 0 1

Alois P. Heinz, Rows n = 0..40, flattened

Row sums give: A230823.
Column sums give: A274376.
Cf. A000217, A002061 (number of terms in row n), A129172, A274054, A274373.

b:= proc(x, y, t, n) option remember; expand(`if`(y>n, 0,
      `if`(n=y, `if`(t=2, 0, 1), b(x+1, y+1, 0, n-1)*z^
      (2*y+1)+`if`(t<>1 and x>0, b(x-1, y+1, 2, n-1), 0)+
      `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=n..n^2))(b(0$3, 2*n)):
seq(T(n), n=0..8);

b[x_, y_, t_, n_] := b[x, y, t, n] = Expand[If[y>n, 0, If[n == y, If[t == 2, 0, 1], b[x+1, y+1, 0, n-1]*z^(2*y+1) + If[t != 1 && x>0, b[x-1, y+1, 2, n-1], 0] + If[t != 2 && y>0, b[x+1, y-1, 1, n-1], 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, n, n^2}]][b[0, 0, 0, 2*n]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jan 25 2017, translated from Maple *)

Sum_{k=n..n^2} k * T(n,k) = A274373(n).
T(n,n) = T(n,n^2) = 1.
T(n,n+1) = T(n,n^2-1) = 0.
T(n,n*(n+1)/2) = T(n,A000217(n)) = A274054(n).

A274404 Number T(n,k) of modified skew Dyck paths of semilength n with exactly k anti-down steps; triangle T(n,k), n>=0, 0<=k<=n-floor((1+sqrt(max(0,8n-7)))/2), read by rows.

Original entry on oeis.org

1, 1, 2, 5, 1, 14, 6, 42, 28, 3, 132, 120, 28, 1, 429, 495, 180, 20, 1430, 2002, 990, 195, 10, 4862, 8008, 5005, 1430, 165, 4, 16796, 31824, 24024, 9009, 1650, 117, 1, 58786, 125970, 111384, 51688, 13013, 1617, 70, 208012, 497420, 503880, 278460, 89180, 16016, 1386, 35

Offset: 0

Alois P. Heinz, Jun 20 2016

A modified skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and A=(-1,1) (anti-down) so that A and D steps do not overlap.

			              /\
              \ \
T(3,1) = 1:   /  \
.
Triangle T(n,k) begins:
:     1;
:     1;
:     2;
:     5,     1;
:    14,     6;
:    42,    28,     3;
:   132,   120,    28,    1;
:   429,   495,   180,   20;
:  1430,  2002,   990,  195,   10;
:  4862,  8008,  5005, 1430,  165,   4;
: 16796, 31824, 24024, 9009, 1650, 117, 1;

Alois P. Heinz, Rows n = 0..160, flattened

Columns k=0-3 give: A000108, A002694(n-1), A074922(n-2), A232224(n-3).
Row sums give A230823.
Last elements of rows give A092392(n-1) for n>0.
Cf. A083920, A274405.

b:= proc(x, y, t, n) option remember; expand(`if`(y>n, 0,
      `if`(n=y, `if`(t=2, 0, 1), b(x+1, y+1, 0, n-1)+
      `if`(t<>1 and x>0, b(x-1, y+1, 2, n-1)*z, 0)+
      `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(0$3, 2*n)):
seq(T(n), n=0..14);

b[x_, y_, t_, n_] := b[x, y, t, n] = Expand[If[y > n, 0,
     If[n == y, If[t == 2, 0, 1], b[x + 1, y + 1, 0, n - 1] +
     If[t != 1 && x > 0, b[x - 1, y + 1, 2, n - 1] z, 0] +
     If[t != 2 && y > 0, b[x + 1, y - 1, 1, n - 1], 0]]]];
T[n_] := CoefficientList[b[0, 0, 0, 2n], z];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)

Sum_{k>0} k * T(n,k) = A274405(n).

A274373 Sum of the areas of all modified skew Dyck paths of semilength n.

Original entry on oeis.org

0, 1, 6, 35, 188, 989, 5131, 26411, 135229, 689814, 3509014, 17811637, 90256685, 456719880, 2308440442, 11656409995, 58809893357, 296500180806, 1493924791698, 7523064390774, 37866103978109, 190510720248534, 958122016323881, 4816944544836927, 24209532464417585

Offset: 0

Alois P. Heinz, Jun 19 2016

nonn

A modified skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and A=(-1,1) (anti-down) so that A and D steps do not overlap.

a(n)^(1/n) tends to 5. - Vaclav Kotesovec, Jun 26 2016

			a(3) = 35 = 9+7+5+6+5+3 = sum of the areas of UUUDDD, UUDUDD, UUDDUD, UAUDDD, UDUUDD, UDUDUD, respectively.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A230823, A274372.

b:= proc(x, y, t, n) option remember; `if`(y>n, 0, `if`(n=y,
     `if`(t=2, 0, [1, 0]), (p-> p+[0, p[1]*(2*y+1)])(b(x+1, y
      +1, 0, n-1))+`if`(t<>1 and x>0, b(x-1, y+1, 2, n-1), 0)
      +`if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0)))
    end:
a:= n-> b(0$3, 2*n)[2]:
seq(a(n), n=0..30);

b[x_, y_, t_, n_] := b[x, y, t, n] = If[y > n, 0, If[n == y, If[t == 2, {0, 0}, {1, 0}], Function[p, p + {0, p[[1]] (2y + 1)}][b[x + 1, y + 1, 0, n - 1]] + If[t != 1 && x > 0, b[x - 1, y + 1, 2, n - 1], 0] + If[t != 2 && y > 0, b[x + 1, y - 1, 1, n - 1], 0]]];
a[n_] := b[0, 0, 0, 2 n][[2]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

a(n) = Sum_{k=n..n^2} k * A274372(n,k).

A274405 Number of anti-down steps in all modified skew Dyck paths of semilength n.

Original entry on oeis.org

0, 0, 0, 1, 6, 34, 179, 915, 4607, 22988, 114090, 564359, 2785921, 13735074, 67665208, 333211828, 1640575047, 8077199130, 39770520844, 195852723348, 964689515033, 4752800817185, 23422061819883, 115456855588378, 569293729146929, 2807864888917275

Offset: 0

Alois P. Heinz, Jun 20 2016

nonn

A modified skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and A=(-1,1) (anti-down) so that A and D steps do not overlap.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A230823, A274404.

b:= proc(x, y, t, n) option remember; `if`(y>n, 0, `if`(n=y,
      `if`(t=2, 0, [1, 0]), b(x+1, y+1, 0, n-1)+`if`(t<>1
       and x>0, (p-> p+[0, p[1]])(b(x-1, y+1, 2, n-1)), 0)+
      `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0)))
    end:
a:= n-> b(0$3, 2*n)[2]:
seq(a(n), n=0..30);

b[x_, y_, t_, n_] := b[x, y, t, n] = If[y > n, 0, If[n == y, If[t == 2, {0, 0}, {1, 0}], b[x + 1, y + 1, 0, n - 1] + If[t != 1 && x > 0, Function[p, p + {0, p[[1]]}][b[x - 1, y + 1, 2, n - 1]], 0] + If[t != 2 && y > 0, b[x + 1, y - 1, 1, n - 1], 0]]];
a[n_] := b[0, 0, 0, 2 n][[2]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

a(n) = Sum_{k>0} k * A274404(n,k).

a(n) ~ c * 5^n / sqrt(n), where c = 0.0554525135364274199547478570703521322323... . - Vaclav Kotesovec, Jun 26 2016

A355040 Expansion of the continued fraction 1 / (1-q / (1-q-q^2 / (1-q-q^2-q^3 / (1-q-q^2-q^3-q^4 / (...))))).

Original entry on oeis.org

1, 1, 2, 5, 13, 35, 96, 266, 742, 2079, 5843, 16457, 46423, 131099, 370527, 1047858, 2964698, 8390837, 23754234, 67260645, 190478213, 539484321, 1528094423, 4328632609, 12262352881, 34738763766, 98416624789, 278825903115, 789961599608, 2238129694407, 6341171821627, 17966261019890, 50903653156245

Offset: 0

Joerg Arndt, Jun 16 2022

nonn

Cf. A230823, A355043, A355046.

Cf. A088354.

nmax = 40; CoefficientList[Series[1/(1 - x/(1 - x + ContinuedFractionK[-x^k, 1 - x*(1 - x^k)/(1 - x), {k, 2, nmax}])), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 16 2022 *)
A355040List[nmax_] := Module[{a, b, q},
a = ContinuedFractionK[-q^k, (q^(k + 1) - 2 q + 1)/(q - 1), {k, 0, nmax}];
b = Series[a, {q, 0, nmax}]; CoefficientList[b, q] ];
A355040List[32] (* Peter Luschny, Jun 20 2022 *)

N=44; q='q+O('q^N);
f(n) = 1 - sum(k=1,n-1,q^k);
s=1; forstep(j=N, 1, -1, s = q^j/s; s = f(j) - s ); s = 1/s;
Vec(s)

a(n) ~ c * d^n, where d = 2.8333645039948803621658813720055524872119... and c = 0.1710130167563241590871776261530008679... - Vaclav Kotesovec, Jun 16 2022

a(n) = [q^n] K_{k>=0} -q^k / ((q^(k + 1) - 2*q + 1)/(q - 1)), where K is Gauss's notation for continued fractions. - Peter Luschny, Jun 20 2022

A274054 Number of modified skew Dyck paths of semilength n such that the area between the x-axis and the path is n*(n+1)/2.

Original entry on oeis.org

1, 1, 0, 1, 3, 6, 14, 40, 140, 422, 1346, 4487, 15234, 52632, 183913, 651948, 2336751, 8438406, 30712379, 112603500, 415459873, 1541646225, 5750112809, 21548036621, 81096740799, 306404247854, 1161863199131, 4420429256826, 16869986745367, 64567073382731

Offset: 0

Alois P. Heinz, Jun 19 2016

nonn

A modified skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and A=(-1,1) (anti-down) so that A and D steps do not overlap.

			a(1) = 1:   /\
.
            /\
            \ \
a(3) = 1:   /  \
.
                /\                          /\
               /  \         /\/\/\         /  \
a(4) = 3:   /\/    \   ,   /      \   ,   /    \/\   .

Cf. A000217, A230823, A274372.

a(n) = A274372(n,n*(n+1)/2) = A274372(n,A000217(n)).

A274376 Number of modified skew Dyck paths such that the area between the x-axis and the path is n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 8, 12, 20, 32, 52, 84, 135, 219, 353, 572, 924, 1495, 2419, 3912, 6331, 10240, 16570, 26807, 43374, 70178, 113546, 183721, 297258, 480974, 778220, 1259184, 2037389, 3296554, 5333923, 8630446, 13964340, 22594740, 36559034, 59153708, 95712668

Offset: 0

Alois P. Heinz, Jun 19 2016

nonn

A modified skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and A=(-1,1) (anti-down) so that A and D steps do not overlap.

			               /\    /\
a(5) = 3:   /\/  \  /  \/\  /\/\/\/\/\  .
.
            /\
            \ \    /\          /\          /\
a(6) = 5:   /  \  /  \/\/\  /\/  \/\  /\/\/  \  /\/\/\/\/\/\  .

Column sums of A274372.

Cf. A230823.

a(n) = Sum_{k=0..n} A274372(k,n).

cp's OEIS Frontend

A274372 Number T(n,k) of modified skew Dyck paths of semilength n such that the area between the x-axis and the path is k; triangle T(n,k), n>=0, n<=k<=n^2, read by rows.

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A274404 Number T(n,k) of modified skew Dyck paths of semilength n with exactly k anti-down steps; triangle T(n,k), n>=0, 0<=k<=n-floor((1+sqrt(max(0,8n-7)))/2), read by rows.

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A274373 Sum of the areas of all modified skew Dyck paths of semilength n.

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A274405 Number of anti-down steps in all modified skew Dyck paths of semilength n.

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A355040 Expansion of the continued fraction 1 / (1-q / (1-q-q^2 / (1-q-q^2-q^3 / (1-q-q^2-q^3-q^4 / (...))))).

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A274054 Number of modified skew Dyck paths of semilength n such that the area between the x-axis and the path is n*(n+1)/2.

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A274376 Number of modified skew Dyck paths such that the area between the x-axis and the path is n.

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