Original entry on oeis.org
1, 1, 3, 7, 26, 86, 392, 1660, 9065, 46705, 297984, 1805926, 13186497, 91788477, 754481662, 5924676900, 54092804430, 472512732558, 4739696836485, 45540919862179, 497377234156959, 5208759709993591, 61475622078245542, 696384168181553136, 8825761698420052542
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, (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:
a:= proc(n) option remember; add(T(n, k), k=0..n/2) end:
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, (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!;
a[n_] := Sum[T[n, k], {k, 0, n/2}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 01 2022, after Alois P. Heinz *)
A258307
T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258306(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, 5, 2, 14, 9, 1, 43, 28, 3, 141, 114, 21, 1, 490, 421, 82, 4, 1785, 1750, 442, 38, 1, 6789, 7114, 1941, 180, 5, 26809, 30854, 9868, 1210, 60, 1, 109632, 134239, 46337, 6191, 335, 6, 462755, 609276, 235035, 37321, 2700, 87, 1, 2012441, 2800134, 1157603, 199424, 15806, 560, 7
Offset: 0
Triangle T(n,k) begins:
: 1;
: 1;
: 2, 1;
: 5, 2;
: 14, 9, 1;
: 43, 28, 3;
: 141, 114, 21, 1;
: 490, 421, 82, 4;
: 1785, 1750, 442, 38, 1;
: 6789, 7114, 1941, 180, 5;
: 26809, 30854, 9868, 1210, 60, 1;
-
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:
seq(seq(T(n, k), k=0..n/2), n=0..13);
-
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!;
Table[T[n, k], {n, 0, 13}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Jun 06 2018, from Maple *)
A258309
A(n,k) is the sum over all Motzkin paths of length n of products over all peaks p of (k*x_p+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, 4, 1, 1, 4, 7, 9, 1, 1, 5, 10, 23, 21, 1, 1, 6, 13, 43, 71, 51, 1, 1, 7, 16, 69, 151, 255, 127, 1, 1, 8, 19, 101, 261, 703, 911, 323, 1, 1, 9, 22, 139, 401, 1485, 2983, 3535, 835, 1, 1, 10, 25, 183, 571, 2691, 6973, 14977, 13903, 2188
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, ...
: 4, 7, 10, 13, 16, 19, 22, ...
: 9, 23, 43, 69, 101, 139, 183, ...
: 21, 71, 151, 261, 401, 571, 771, ...
: 51, 255, 703, 1485, 2691, 4411, 6735, ...
-
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):
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, (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];
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 04 2017, translated from Maple *)
Showing 1-3 of 3 results.