Original entry on oeis.org
1, 1, 3, 7, 24, 74, 277, 997, 4016, 16029, 68802, 296740, 1347175, 6185975, 29530010, 143008050, 714469780, 3625572745, 18884279461, 99936069760, 540947985741, 2974463266900, 16686653393208, 95053009906135, 551356966419818, 3245644584299434, 19425857465136193
Offset: 0
-
b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (x+k*y)/y, 1)
+b(x-1, y, false, k) +b(x-1, y+1, true, k)))
end:
A:= (n, k)-> b(n, 0, false, k):
T:= proc(n, k) option remember;
add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!
end:
a:= n-> add(T(n, k), k=0..n/2):
seq(a(n), n=0..30);
-
b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0, If[x == 0, 1,
b[x - 1, y - 1, False, k]*If[t, (x + k*y)/y, 1] +
b[x - 1, y, False, k] +
b[x - 1, y + 1, True, k]]];
A[n_, k_] := b[n, 0, False, k];
T[n_, k_] := T[n, k] =
Sum[A[n, i]*(-1)^(k-i)*Binomial[k, i], {i, 0, k}]/k!;
a[n_] := Sum[T[n, k], {k, 0, n/2}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 30 2022, after Alois P. Heinz *)
A258306
A(n,k) is the sum over all Motzkin paths of length n of products over all peaks p of (x_p+k*y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 7, 14, 1, 1, 5, 9, 23, 43, 1, 1, 6, 11, 34, 71, 141, 1, 1, 7, 13, 47, 105, 255, 490, 1, 1, 8, 15, 62, 145, 411, 911, 1785, 1, 1, 9, 17, 79, 191, 615, 1496, 3535, 6789, 1, 1, 10, 19, 98, 243, 873, 2269, 6169, 13903, 26809
Offset: 0
Square array A(n,k) begins:
: 1, 1, 1, 1, 1, 1, 1, ...
: 1, 1, 1, 1, 1, 1, 1, ...
: 2, 3, 4, 5, 6, 7, 8, ...
: 5, 7, 9, 11, 13, 15, 17, ...
: 14, 23, 34, 47, 62, 79, 98, ...
: 43, 71, 105, 145, 191, 243, 301, ...
: 141, 255, 411, 615, 873, 1191, 1575, ...
-
b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (x+k*y)/y, 1)
+b(x-1, y, false, k) +b(x-1, y+1, true, k)))
end:
A:= (n, k)-> b(n, 0, false, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
-
b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (x + k*y)/y, 1] + b[x - 1, y, False, k] + b[x - 1, y + 1, True, k]]]; A[n_, k_] := b[n, 0, False, k]; Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 23 2017, translated from Maple *)
A258310
T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258309(n,i); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.
Original entry on oeis.org
1, 1, 2, 1, 4, 3, 9, 14, 3, 21, 50, 15, 51, 204, 122, 15, 127, 784, 644, 105, 323, 3212, 4115, 1310, 105, 835, 13068, 22587, 9270, 945, 2188, 55475, 137503, 85109, 16764, 945, 5798, 238073, 787127, 614779, 149754, 10395
Offset: 0
Triangle T(n,k) begins:
: 1;
: 1;
: 2, 1;
: 4, 3;
: 9, 14, 3;
: 21, 50, 15;
: 51, 204, 122, 15;
: 127, 784, 644, 105;
: 323, 3212, 4115, 1310, 105;
: 835, 13068, 22587, 9270, 945;
: 2188, 55475, 137503, 85109, 16764, 945;
-
b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)
+b(x-1, y, false, k) +b(x-1, y+1, true, k)))
end:
A:= (n, k)-> b(n, 0, false, k):
T:= proc(n, k) option remember;
add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!
end:
seq(seq(T(n, k), k=0..n/2), n=0..14);
-
b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0,
If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (k*x + y)/y, 1]
+ b[x - 1, y, False, k] + b[x - 1, y + 1, True, k]]];
A[n_, k_] := b[n, 0, False, k];
T[n_, k_] := Sum[A[n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}]/k!;
Table[Table[T[n, k], {k, 0, n/2}], {n, 0, 14}] // Flatten (* Jean-François Alcover, May 01 2022, after Alois P. Heinz *)
A258312
Sum over all Motzkin paths of length n of products over all peaks p of x_p/y_p, where x_p and y_p are the coordinates of peak p.
Original entry on oeis.org
1, 1, 2, 5, 14, 43, 141, 490, 1785, 6789, 26809, 109632, 462755, 2012441, 8997402, 41297927, 194306557, 936082502, 4612095475, 23219012907, 119328025012, 625545408219, 3342370197206, 18190297736313, 100768960522871, 567886743369378, 3253833477309093
Offset: 0
-
b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, b(x-1, y-1, false) *`if`(t, x/y, 1)
+b(x-1, y, false)+b(x-1, y+1, true)))
end:
a:= n-> b(n, 0, false):
seq(a(n), n=0..30);
-
b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x == 0, 1, b[x-1, y-1, False]*If[t, x/y, 1] + b[x-1, y, False] + b[x-1, y+1, True]]];
a[n_] := b[n, 0, False];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 10 2017, translated from Maple *)
Showing 1-4 of 4 results.