A318115
Number of compositions of n into exactly n nonnegative parts <= seven.
Original entry on oeis.org
1, 1, 3, 10, 35, 126, 462, 1716, 6427, 24229, 91828, 349570, 1335698, 5119856, 19678452, 75814560, 292695291, 1132074847, 4385740683, 17015510820, 66102536360, 257103599280, 1001078753370, 3901733646660, 15220908863866, 59427261280126, 232201593611796
Offset: 0
-
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i=0, 0, add(b(n-j, i-1), j=0..min(n, 7))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
A318116
Number of compositions of n into exactly n nonnegative parts <= eight.
Original entry on oeis.org
1, 1, 3, 10, 35, 126, 462, 1716, 6435, 24301, 92278, 351990, 1347710, 5176640, 19938348, 76977360, 297811491, 1154300103, 4481325903, 17423296059, 67830758310, 264387659490, 1031636761290, 4029420952890, 15752622069630, 61634789550126, 241342568718696
Offset: 0
-
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i=0, 0, add(b(n-j, i-1), j=0..min(n, 8))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
A318117
Number of compositions of n into exactly n nonnegative parts <= ten.
Original entry on oeis.org
1, 1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352705, 1351934, 5199117, 20050460, 77512860, 300292131, 1165534689, 4531337778, 17642948675, 68785126410, 268497991905, 1049210943531, 4104103226699, 16068342973950, 62963565243876, 246913415221671
Offset: 0
-
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i=0, 0, add(b(n-j, i-1), j=0..min(n, 10))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
A348478
Number of compositions of n into exactly n nonnegative parts such that each positive i-th part has the same parity as i.
Original entry on oeis.org
1, 1, 1, 4, 7, 23, 55, 164, 407, 1235, 3051, 9432, 23431, 72989, 182624, 571384, 1436855, 4511979, 11387467, 35866100, 90782837, 286622226, 727226578, 2300578392, 5848776767, 18533394763, 47197285045, 149769168304, 381956145802, 1213526310665, 3098742448230
Offset: 0
a(0) = 1: [].
a(1) = 1: [1].
a(2) = 1: [0,2].
a(3) = 4: [1,2,0], [0,2,1], [3,0,0], [0,0,3].
a(4) = 7: [1,2,1,0], [1,0,1,2], [3,0,1,0], [1,0,3,0], [0,2,0,2], [0,4,0,0], [0,0,0,4].
-
b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),
add(`if`(j=0 or (t-j)::even, b(n-j, t-1), 0), j=0..n))
end:
a:= n-> b(n$2):
seq(a(n), n=0..33);
-
b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n],
Sum[If[j == 0 || EvenQ[t - j], b[n - j, t - 1], 0], {j, 0, n}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)