cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A178684 Partial sums of cardinalities of coalition sets A095941.

Original entry on oeis.org

0, 0, 1, 5, 18, 53, 138, 332, 757, 1661, 3546, 7424, 15328, 31336, 63618, 128531, 258811, 519956, 1042992, 2090009, 4185231, 8377158, 16762853, 33536516, 67086633, 134190278, 268401718, 536829625, 1073691505, 2147422558
Offset: 1

Views

Author

Jonathan Vos Post, Dec 25 2010

Keywords

Comments

Partial sums of number of subsets of {1,2,...,n} such that every number in the set is no larger than the sum of the other numbers in the set. See formula in A095944. The subsequence of primes begins: 5, 53, 757, 2090009, 16762853.

Examples

			a(9) = 0 + 0 + 1 + 4 + 13 + 35 + 85 + 194 + 425 = 757 is prime.

Crossrefs

Cf. A000009, A002623, A095941, A095944.

A095944 Number of subsets S of {1,2,...,n} which contain a number that is greater than the sum of the other numbers in S.

Original entry on oeis.org

1, 3, 6, 11, 18, 28, 42, 61, 86, 119, 162, 217, 287, 375, 485, 622, 791, 998, 1251, 1558, 1929, 2376, 2912, 3552, 4314, 5218, 6287, 7548, 9031, 10770, 12805, 15180, 17945, 21158, 24883, 29193, 34171, 39909, 46511, 54095, 62792, 72749, 84132, 97125
Offset: 1

Views

Author

W. Edwin Clark, Jul 13 2004

Keywords

Comments

Convolution of A000009 and A001477. - Vaclav Kotesovec, Mar 12 2016

Examples

			a(3) = 6 since the subsets {1},{2},{3},{1,2},{1,3},{2,3} are the only subsets of {1,2,3} which contain a number greater than the sum of the other numbers in the set.

Links

Crossrefs

Equals 2^n - 1 - A095941(n).
Column k=1 of A360634.

Programs

  • Mathematica
    r[s_, x_] := r[s,x] = 1 + Sum[r[s-i, i-1], {i, Min[x,s]}]; f[n_] := Sum[r[k-1, k-1], {k, n}]; Array[f, 50] (* Giovanni Resta, Mar 16 2006 *)
Accumulate[ Accumulate[q = PartitionsQ[ Range[1, 50]]]+1] - Accumulate[q] (* Jean-François Alcover, Nov 14 2011 *)

Formula

Second differences are A000009, partitions into distinct parts. Proof from Fred W. Helenius (fredh(AT)ix.netcom.com): Let k be the largest element (the "dictator") in S and let j be the sum of the remaining elements, so 0 <= j < k. For a given k and j, the number of subsets S is just the number of partitions j into distinct parts; call that a(j). Then b(n) = Sum_{1<=k<=n} Sum_{0<=jN. J. A. Sloane and proved by Michael Reid.
a(n) ~ 3^(3/4) * n^(1/4) * exp(sqrt(n/3)*Pi) / Pi^2. - Vaclav Kotesovec, Mar 12 2016
G.f.: (x/(1 - x)^2)*Product_{k>=1} (1 + x^k). - Ilya Gutkovskiy, Jan 03 2017

Extensions

More terms from John W. Layman, Aug 10 2004
More terms from Giovanni Resta, Mar 16 2006

A341507 Number of nonempty subsets S of {1,2,...,n} in which all elements are strictly less than the sum of the other elements of S.

Original entry on oeis.org

0, 0, 0, 0, 2, 9, 28, 74, 178, 402, 872, 1842, 3821, 7830, 15913, 32161, 64761, 130091, 260911, 522749, 1046667, 2094797, 4191414, 8385079, 16772926, 33549239, 67102603, 134210207, 268426453, 536860171, 1073729049, 2147468499, 4294949383, 8589913467, 17179844335
Offset: 0

Views

Author

Henry Bottomley, Feb 13 2021

Keywords

Comments

In other words, every element of S is strictly less than half the sum.

Examples

			For n = 5 the a(5)=9 subsets are {2,3,4}, {2,4,5}, {3,4,5}, {1,2,3,4}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}, and {1,2,3,4,5}.

Links

Crossrefs

Cf. A000009, A002623, A095941, A095944, A317910.

Programs

  • Maple
    b:= proc(n, s) option remember; `if`(s<1, 2^n,
     `if`(n*(n+1)/2 add(b(j-1, j+1), j=1..n):
seq(a(n), n=0..37);  # Alois P. Heinz, Feb 13 2021
  • Mathematica
    gf := (1 - x - x^2)/((1 - 2 x) (1 - x)^2) - QPochhammer[-1, x]/(2 (1 - x)^2);
CoefficientList[Series[gf, {x, 0, 34}], x] (* Peter Luschny, Feb 13 2021 *)

Formula

a(n) = A095941(n) - A317910(n).
G.f.: (1-x-x^2)/((1-x)^2*(1-2*x)) - (1/(1-x)^2)*Product_{k>=1} (1 + x^k).
Showing 1-3 of 3 results.

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