A339312 -id:A339312 - cp's OEIS frontend

A339006 Sum over all partitions lambda of n of binomial(|lambda|, |{lambda}|).

1, 1, 3, 5, 11, 20, 40, 72, 130, 227, 395, 671, 1124, 1864, 3040, 4909, 7830, 12394, 19388, 30145, 46395, 70977, 107661, 162383, 243108, 362037, 535684, 788677, 1154605, 1682402, 2439123, 3520706, 5058786, 7239027, 10315920, 14644309, 20709800, 29182353

Offset: 0

Alois P. Heinz, Nov 18 2020

nonn

|lambda| is the number of parts in lambda and |{lambda}| is the number of distinct parts.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A108492, A339011, A339312.

b:= proc(n, i, p, d) option remember; `if`(n=0, binomial(p, d),
     `if`(i<1, 0, add(b(n-i*j, i-1, p+j, `if`(j=0, d, d+1)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);

b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, Binomial[p, d],
    If[i<1, 0, Sum[b[n-i*j, i-1, p+j, If[j == 0, d, d+1]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)

A339011 Sum over all partitions of n of the product of the number of parts and the number of distinct parts.

Original entry on oeis.org

0, 1, 3, 8, 17, 34, 61, 107, 176, 284, 442, 676, 1007, 1483, 2140, 3055, 4299, 5993, 8255, 11284, 15272, 20529, 27373, 36274, 47735, 62484, 81293, 105251, 135555, 173818, 221836, 282003, 356980, 450256, 565765, 708537, 884296, 1100287, 1364736, 1687952, 2081724

Offset: 0

Alois P. Heinz, Nov 18 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A096541, A339006, A339312.

Essentially partial sums of A093694.

b:= proc(n, i, p, d) option remember; `if`(n=0, d*p, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, `if`(j=0, d, d+1)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n<=0 or i=0, [0$2],
     `if`(i=1, [1, n], b(n, i-1)+ (p-> p+[0, p[1]])(b(n-i, i))))
    end:
a:= proc(n) option remember; b(n$2)[2]+`if`(n<0, 0, a(n-1)) end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 25 2022

b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, d*p, If[i < 1, 0,
     Sum[b[n - i*j, i - 1, p + j, If[j == 0, d, d + 1]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 09 2021, after Alois P. Heinz *)

A339394 Sum over all partitions of n of the LCM of the number of parts and the number of distinct parts.

Original entry on oeis.org

0, 1, 3, 6, 15, 26, 43, 81, 138, 218, 320, 514, 751, 1131, 1570, 2319, 3159, 4457, 6077, 8344, 11224, 15337, 20297, 26908, 35773, 46434, 60711, 78433, 100987, 129222, 166590, 209719, 267120, 335842, 423341, 527739, 659974, 816805, 1015990, 1251686, 1543864

Offset: 0

Alois P. Heinz, Dec 02 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A003990, A051173, A339312.

b:= proc(n, i, p, d) option remember; `if`(n=0, ilcm(p, d),
      add(b(n-i*j, i-1, p+j, d+signum(j)), j=`if`(i>1, 0..n/i, n)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);

b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, LCM[p, d],
     Sum[b[n - i*j, i - 1, p + j, d + Sign[j]],
     {j, If[i > 1, Range[0, n/i], {n}]}]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 09 2021, after Alois P. Heinz *)

cp's OEIS Frontend

A339006 Sum over all partitions lambda of n of binomial(|lambda|, |{lambda}|).

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A339011 Sum over all partitions of n of the product of the number of parts and the number of distinct parts.

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Maple

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A339394 Sum over all partitions of n of the LCM of the number of parts and the number of distinct parts.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica