A336875
Number of parts, counted without multiplicity, in all compositions of n.
Original entry on oeis.org
0, 1, 2, 6, 13, 30, 66, 144, 308, 655, 1380, 2891, 6024, 12500, 25844, 53274, 109530, 224690, 460033, 940276, 1918979, 3911186, 7962194, 16191875, 32896364, 66776727, 135445212, 274532607, 556086916, 1125727954, 2277650681, 4605981879, 9310120876, 18810538092
Offset: 0
a(4) = 1 + 2 + 2 + 2 + 1 + 2 + 2 + 1 = 13: (1)111, (1)1(2), (1)(2)1, (2)(1)1, (2)2, (1)(3), (3)(1), (4).
-
b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],
`if`(i<1, 0, add((p-> [0, `if`(j=0, 0, p[1])]+p)(
b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..38);
-
b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0},
If[i<1, {0, 0}, Sum[{0, If[j == 0, 0, #[[1]]]}+#&[
b[n-i*j, i-1, p+j]/j!], {j, 0, n/i}]]];
a[n_] := b[n, n, 0][[2]];
a /@ Range[0, 38] (* Jean-François Alcover, Jun 13 2021, after Alois P. Heinz *)
A336511
Total sum of the left-to-right maxima in all compositions of n.
Original entry on oeis.org
0, 1, 3, 9, 22, 52, 117, 260, 565, 1217, 2593, 5487, 11538, 24146, 50316, 104490, 216337, 446754, 920506, 1892904, 3885719, 7964162, 16300646, 33321640, 68038796, 138784403, 282824924, 575866839, 1171612786, 2381938742, 4839331484, 9825841526, 19938975797
Offset: 0
a(4) = 1 + 1 + 2 + 1 + 2 + 2 + 2 + 1 + 3 + 3 + 4 = 22: (1)111, (1)1(2), (1)(2)1, (2)11, (2)2, (1)(3), (3)1, (4).
-
b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> [0,
`if`(j>m, j*p[1], 0)]+p)(b(n-j, max(m, j))), j=1..n))
end:
a:= n-> b(n, -1)[2]:
seq(a(n), n=0..50);
-
b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {0,
If[j > m, j*p[[1]], 0]} + p][b[n - j, Max[m, j]]], {j, 1, n}]];
a[n_] := b[n, -1][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)
A336512
Total sum of the left-to-right minima in all compositions of n.
Original entry on oeis.org
0, 1, 3, 8, 17, 38, 78, 162, 330, 672, 1355, 2736, 5503, 11058, 22191, 44507, 89198, 178697, 357852, 716440, 1434041, 2869935, 5742801, 11490298, 22988084, 45988166, 91995547, 184021931, 368093352, 736266262, 1472660452, 2945526806, 5891385159, 11783304479
Offset: 0
a(4) = 1 + 1 + 1 + 2 + 1 + 2 + 1 + 3 + 1 + 4 = 17: (1)111, (1)12, (1)21, (2)(1)1, (2)2, (1)3, (3)(1), (4).
-
b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> [0,
`if`(j b(n, n+1)[2]:
seq(a(n), n=0..50);
-
b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {0,
If[j < m, j*p[[1]], 0]} + p][b[n - j, Min[m, j]]], {j, 1, n}]];
a[n_] := b[n, n + 1][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 07 2022, after Alois P. Heinz *)
A336579
Sum of prime parts, counted without multiplicity, in all compositions of n.
Original entry on oeis.org
0, 0, 2, 7, 14, 38, 83, 193, 421, 917, 1969, 4210, 8908, 18763, 39287, 81940, 170270, 352726, 728663, 1501711, 3088326, 6339424, 12991312, 26583389, 54323352, 110876435, 226057023, 460432903, 936963134, 1905110662, 3870698364, 7858803605, 15945759386
Offset: 0
a(4) = 2 + 2 + 2 + 2 + 3 + 3 = 14: 1111, 11(2), 1(2)1, (2)11, (2)2, 1(3), (3)1, 4.
-
b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],
`if`(i<1, 0, add((p-> [0, `if`(j>0 and isprime(i),
p[1]*i, 0)]+p)(b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..38);
-
b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0},
If[i < 1, {0, 0}, Sum[Function[q, {0, If[j > 0 && PrimeQ[i],
q[[1]]*i, 0]} + q][b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
a[n_] := b[n, n, 0][[2]];
Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Mar 17 2022, after Alois P Heinz *)
Showing 1-4 of 4 results.