Original entry on oeis.org
1, 2, 13, 166, 3450, 105053, 4385297, 239389538, 16497800177, 1396841773631, 142194450687440, 17100401655609460, 2394468068218870494, 385647096554809325098, 70702689662684594772871, 14623755150209185924416598, 3385915623744083331349813602
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+1, true, k) ))
end:
A:= (n, k)-> b(2*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) end:
seq(a(n), n=0..20);
-
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 + 1, True, k]]];
A[n_, k_] := b[2*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}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 28 2022, after Alois P. Heinz *)
Original entry on oeis.org
1, 8, 1749, 1944225, 6439957299, 47971886252910, 677927299391810160, 16243385150174371081830, 609634394448842168438414483, 33797743985046745897969800271770, 2645657421035128682909045799293446355, 282144864134810484141733900449168244617439
Offset: 0
A258220
T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258219(n,i); triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 1, 1, 4, 6, 1, 25, 49, 15, 1, 208, 498, 217, 28, 1, 2146, 6016, 3360, 635, 45, 1, 26368, 84042, 56728, 13997, 1475, 66, 1, 375733, 1332661, 1046619, 316281, 43974, 2954, 91, 1, 6092032, 23660034, 21053089, 7479444, 1283817, 114576, 5334, 120, 1
Offset: 0
Triangle T(n,k) begins:
: 1;
: 1, 1;
: 4, 6, 1;
: 25, 49, 15, 1;
: 208, 498, 217, 28, 1;
: 2146, 6016, 3360, 635, 45, 1;
: 26368, 84042, 56728, 13997, 1475, 66, 1;
Column k=0 gives
A005411 (for n>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+1, true, k) ))
end:
A:= (n, k)-> b(2*n, 0, false, k):
T:= (n, k)-> add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!:
seq(seq(T(n, k), k=0..n), n=0..10);
-
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+1, True, k]]]; A[n_, k_] := b[2*n, 0, False, k]; T [n_, k_] := Sum[A[n, i]*(-1)^(k-i)*Binomial[k, i], {i, 0, k}]/k!; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 20 2017, translated from Maple *)
A258222
A(n,k) is the sum over all Dyck paths of semilength 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, 2, 2, 1, 3, 10, 5, 1, 4, 24, 74, 14, 1, 5, 44, 297, 706, 42, 1, 6, 70, 764, 4896, 8162, 132, 1, 7, 102, 1565, 17924, 100278, 110410, 429, 1, 8, 140, 2790, 47650, 527844, 2450304, 1708394, 1430, 1, 9, 184, 4529, 104454, 1831250, 18685164, 69533397, 29752066, 4862
Offset: 0
Square array A(n,k) begins:
: 1, 1, 1, 1, 1, 1, ...
: 1, 2, 3, 4, 5, 6, ...
: 2, 10, 24, 44, 70, 102, ...
: 5, 74, 297, 764, 1565, 2790, ...
: 14, 706, 4896, 17924, 47650, 104454, ...
: 42, 8162, 100278, 527844, 1831250, 4953222, ...
-
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+1, true, k) ))
end:
A:= (n, k)-> b(2*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 + 1, True, k]]];
A [n_, k_] := b[2*n, 0, False, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Apr 23 2016, translated from Maple *)
Showing 1-4 of 4 results.
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