A000914 -id:A000914 - cp's OEIS frontend

A024172 Integer part of ((3rd elementary symmetric function of 1,2,..,n)/(2nd elementary symmetric function of 1,2,...,n)).

0, 0, 1, 2, 4, 6, 8, 10, 13, 16, 20, 24, 28, 33, 38, 43, 48, 54, 60, 67, 74, 81, 89, 97, 105, 113, 122, 131, 141, 151, 161, 172, 183, 194, 205, 217, 229, 242, 255, 268, 282, 296, 310, 324

Offset: 2

Clark Kimberling

nonn

			a(3) = floor(6/11) = 0; a(4) = floor(50/35) = 1. - _R. J. Mathar_, Sep 15 2009

Ivan Neretin, Table of n, a(n) for n = 2..10000

List([2..50],n->Int((1/2)*n*(n-2)*(n+1)/(3*n+2))); # Muniru A Asiru, May 19 2018

seq(floor((1/2)*n*(n-2)*(n+1)/(3*n+2)),n=2..50); # Muniru A Asiru, May 19 2018

Table[Floor[1/2 (n - 2) n (n + 1)/ (3 n + 2)], {n, 2, 45}] (* Ivan Neretin, May 19 2018 *)

a(n) = floor( A001303(n-2)/A000914(n-1) ). - R. J. Mathar, Sep 15 2009
Empirical g.f.: x^4*(x^4-x^3+x^2-x+1)*(x^5-x^3-x^2-x-1) / ((x-1)^3*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Aug 16 2014
a(n) = floor((1/2)*(n - 2)*n*(n + 1)/(3*n + 2)).

Offset set to 2 by R. J. Mathar, Sep 15 2009

A024173 Integer part of ((4th elementary symmetric function of 1,2,..,n)/(2nd elementary symmetric function of 1,2,...,n)).

Original entry on oeis.org

0, 0, 0, 3, 9, 21, 41, 72, 119, 185, 275, 395, 549, 744, 987, 1285, 1645, 2076, 2586, 3185, 3882, 4688, 5612, 6667, 7863, 9213, 10731, 12428, 14318, 16416, 18737, 21295, 24106, 27187, 30553, 34223, 38214, 42543, 47231, 52295, 57756, 63633, 69948, 76721

Offset: 2

Clark Kimberling

nonn

			a(4) = floor(24/35) = 0; a(5) = floor(274/85) = 3. - _R. J. Mathar_, Sep 15 2009

Ivan Neretin, Table of n, a(n) for n = 2..10000

List([2..50],n->Int((1/240)*(n-3)*(n-2)*(15*n^3+15*n^2-10*n-8)/(3*n+2))); # Muniru A Asiru, May 19 2018

seq(floor((1/240)*(n-3)*(n-2)*(15*n^3+15*n^2-10*n-8)/(3*n+2)),n=2..50); # Muniru A Asiru, May 19 2018

Table[Floor[1/240 (n - 3) (n - 2) (15 n^3 + 15 n^2 - 10 n - 8)/ (2 + 3 n)], {n, 2, 45}] (* Ivan Neretin, May 19 2018 *)

a(n)= floor(A000915(n-3)/A000914(n-1)). - R. J. Mathar, Sep 15 2009

a(n) = floor((1/240) * (n-3) * (n-2) * (15*n^3 + 15*n^2 - 10*n - 8) / (2 + 3*n)). - Ivan Neretin, May 19 2018

Offset changed to 2 by R. J. Mathar, Sep 15 2009

A027916 Least k such that 1+2+...+k >= E{1,2,...,n}, where E = 2nd elementary symmetric function.

Original entry on oeis.org

2, 5, 8, 13, 19, 25, 33, 42, 51, 62, 74, 86, 100, 115, 130, 147, 165, 183, 203, 224, 245, 268, 292, 316, 342, 369, 396, 425, 455, 485, 517, 550, 583, 618, 654, 690, 728, 767, 806, 847, 889, 931, 975, 1020, 1065, 1112, 1160, 1208, 1258, 1309, 1360, 1413, 1467

Offset: 2

Clark Kimberling

nonn

Cf. A000217, A000914, A049347.

Table[Total[Table[IntegerExponent[2^(n - k) 4^k, 8], {k, 0, n}]], {n, 2, 100}] (* Fred Daniel Kline, Jun 05 2012 *)

G.f.: x^2 * (x+2) / ((1-x^3)*(1-x)^2).
a(n) = A000217(n+1) + (A049347(n) - 4*(n+1))/3. - R. J. Mathar, Aug 18 2008
Conjecture: a(n) = n + (n^2 mod 3) + a(n-1). - Jon Maiga, Aug 02 2019
a(n) = ceiling((1/2)*(sqrt(3*n^4 + 2*n^3 - 3*n^2 - 2*n + 3)/sqrt(3) - 1)) = (3*n+4)*(n-1)/6 + ((n+2) mod 3)/3. - Rick Mabry, Jul 01 2023

Extended according to the g.f. by R. J. Mathar, Aug 18 2008

A027919 a(n) = least k such that 2nd elementary symmetric function of {1,2,...,k+1} >= 3rd elementary symmetric function of {1,2,...,n}.

Original entry on oeis.org

2, 4, 6, 8, 11, 13, 16, 19, 22, 25, 29, 32, 36, 39, 43, 47, 51, 56, 60, 64, 69, 74, 78, 83, 88, 93, 98, 103, 109, 114, 119, 125, 131, 136, 142, 148, 154, 160, 166, 172, 178, 185, 191, 198, 204, 211, 217, 224, 231, 238, 245, 252, 259, 266

Offset: 3

Clark Kimberling

nonn

Cf. A000914, A001303.

SymmPolyn := proc(L::list,n::integer)
    local c,a,sel;
    a :=0 ;
    sel := combinat[choose](nops(L),n) ;
    for c in sel do
        a := a+mul(L[e],e=c) ;
    end do:
    a;
end proc:
A027919 := proc(n)
     local k,i;
     [seq(i,i=1..n)] ;
    e3 := SymmPolyn(%,3) ;
    for k from 1 do
        [seq(i,i=1..k+1)] ;
        if SymmPolyn(%,2) >= e3 then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Sep 23 2016

a(n) = min{k: A000914(k) >= A001303(n-2)}. - Sean A. Irvine, Dec 10 2019

Definition modified by R. J. Mathar, Sep 23 2016

A067057 Let A(n) = {1,2,3,...n}. Let B(r) and C(n-r) be two subsets of A(n) having r and n-r elements respectively, such that B(r) U C(n-r) = A(n) and B and C are disjoint; then a(n) = sum of the products of all combination sums of elements of B and C for r =1 to n-1.

Original entry on oeis.org

0, 2, 22, 140, 680, 2800, 10304, 34944, 111360, 337920, 985600, 2782208, 7641088, 20500480, 53903360, 139264000, 354287616, 889061376, 2203975680, 5404098560, 13120307200, 31569477632, 75342282752, 178467635200, 419849830400

Offset: 1

Amarnath Murthy, Jan 02 2002, May 31 2003

nonn

In other words, consider the set N = {1,2,3,...,n}; let S and S' be subsets of N such that S union S' is N. Define prod(S) = ( sum of members of S)*( sum of members of S'); then a(n) = sum of all possible prod(S).

			For n = 4, N = {1,2,3,4}, the 5 columns below give S sum(S) S' sum(S') prod(S):
  { } 0 {1,2,3,4} 10 0
  {1} 1 {2,3,4} 9 9
  {2} 2 {1,3,4} 8 16
  {3} 3 {1,2,4} 7 21
  {4} 4 {1,2,3} 6 24
  {1,2} 3 {3,4} 7 21
  {1,3} 4 {2,4} 6 24
  {1,4} 5 {2,3} 5 25
Hence a(4) = 1*(2 + 3 + 4) + 2*(1 + 3 + 4) + 3*(1 + 2 + 4) + 4*(1 + 2 + 3) + (1 + 2)*(3 + 4) + (1 + 3)*(2 + 4) + (1 + 4)*(2 + 3) = 140.

Cf. A000914.

print1(0, ", "); LIMIT = 40; V = vector(LIMIT*(LIMIT + 1)/2); V[1] = 1; for (i = 2, LIMIT, forstep (j = i*(i - 1)/2, 1, -1, V[i + j] += V[j]); V[i]++; k = i*(i + 1)/2; s = sum(j = 1, (k - 1)\2, j*(k - j)*V[j]); if (!(k%2), s += k*k*V[k\2]/8); print1(s, ", ")); \\ David Wasserman, Dec 22 2004

For n>1, all listed values are given by a(n)=(2^(n-2))*s(n+1, n-1), where the s(n+1, n-1) are Stirling numbers of the first kind (A000914). - John W. Layman, Jan 05 2002

Conjecture: G.f.: (-2*x^2*(x+1))/(2*x-1)^5. [From Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009]

More terms and formula from John W. Layman, Jan 05 2002
Further terms from David Wasserman, Dec 22 2004
Edited by N. J. A. Sloane, Nov 01 2008 at the suggestion of R. J. Mathar

A134449 Sum of even products minus sum of odd products of different pairs of numbers from 1 to n.

Original entry on oeis.org

0, 2, 5, 29, 39, 129, 150, 374, 410, 860, 915, 1707, 1785, 3059, 3164, 5084, 5220, 7974, 8145, 11945, 12155, 17237, 17490, 24114, 24414, 32864, 33215, 43799, 44205, 57255, 57720, 73592, 74120, 93194, 93789, 116469, 117135, 143849, 144590, 175790

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jan 31 2008

			{1,2,3} -> 1*2-1*3+2*3 = 5.
{1,2,3,4} -> 1*2-1*3+1*4+2*3+2*4+3*4 = 29.
{1,2,3,4,5} -> 1*2-1*3+1*4-1*5+2*3+2*4+2*5+3*4-3*5+4*5 = 39.

Colin Barker, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A000914, A000915.

P:=proc(n) local a,i,j,k,w; for i from 1 by 1 to n do a:=0; for j from 1 by 1 to i do w:=j; k:=i; while k>w do a:=a+w*k*(-1)^(w*k); k:=k-1; od; od; print(a); od; end: P(100);

epop[n_]:=Module[{f=Times@@@Subsets[n,{2}]},Total[Select[f,EvenQ]]-Total[ Select[ f,OddQ]]]; Table[epop[Range[n]],{n,40}] (* Harvey P. Dale, Sep 17 2017 *)

a(n) = {s = 0; for (i=1, n, for (j=i+1, n, p = i*j; if (p % 2, s -= p, s += p););); s;} \\ Michel Marcus, Mar 20 2015

Empirical g.f.: x^2*(x^5-6*x^4+2*x^3-16*x^2-3*x-2) / ((x-1)^5*(x+1)^4). - Colin Barker, Sep 03 2013
Conjectures from Colin Barker, Mar 20 2015: (Start)
a(n) = (n^4+4*n^3-2*n^2-4*n)/16 for n even.
a(n) = (n^4-1)/16 for n odd. (End)
The above conjectures are true. - Sela Fried, Dec 08 2024
E.g.f.: (x*(1 + 17*x + 6*x^2 + x^3)*cosh(x) - (1 + x - 7*x^2 - 10*x^3 - x^4)*sinh(x))/16. - Stefano Spezia, Dec 09 2024

A253207 a(n) = number of permutations of (1,2,...,n) producible by an ordered quadruple of distinct transpositions.

Original entry on oeis.org

11, 59, 359, 1799, 7091, 22995, 64143, 159093, 359348, 752180, 1478204, 2754752, 4906202, 8402522, 13907394, 22337388, 34933761, 53348561, 79746821, 116926733, 168459797, 238853045, 333735545, 460071495, 626402322, 843120306, 1122776354

Offset: 4

Andrew Woods, Dec 28 2014

Colin Barker, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000914, for two transpositions, and A253171, for three.

Vec(-x^4*(2*x^8-18*x^7+72*x^6-168*x^5+254*x^4-232*x^3+224*x^2-40*x+11)/(x-1)^9 + O(x^100)) \\ Colin Barker, Dec 30 2014

a(n) = n!*(1/(384*(n-8)!)+1/(24*(n-7)!)+13/(72*(n-6)!)+1/(5*(n-5)!)+1/(8*(n-4)!)+1/(3*(n-3)!)) for n>=8.

A349645 Triangular array read by rows: T(n,k) is the number of square n-permutations possessing exactly k cycles; n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 11, 0, 1, 0, 24, 0, 35, 0, 1, 0, 0, 184, 0, 85, 0, 1, 0, 720, 0, 994, 0, 175, 0, 1, 0, 0, 9708, 0, 4249, 0, 322, 0, 1, 0, 40320, 0, 72764, 0, 14889, 0, 546, 0, 1, 0, 0, 648576, 0, 402380, 0, 44373, 0, 870, 0, 1

Offset: 0

Steven Finch, Nov 23 2021

A permutation p in S_n is a square if there exists q in S_n with q^2=p.

For such a p, the number of cycles of any even length in its disjoint cycle decomposition must be even.

			The three square 3-permutations are (1, 2, 3) with three cycles (fixed points) and (3, 1, 2) & (2, 3, 1), each with one cycle.
Among the twelve square 4-permutations are {1, 4, 2, 3} & {1, 3, 4, 2} and {3, 4, 1, 2} & {4, 3, 2, 1}, all with two cycles but differing types.
Triangle begins:
[0]   1;
[1]   0,   1;
[2]   0,   0,    1;
[3]   0,   2,    0,   1;
[4]   0,   0,   11,   0,    1;
[5]   0,  24,    0,  35,    0,   1;
[6]   0,   0,  184,   0,   85,   0,   1;
[7]   0, 720,    0, 994,    0, 175,   0,   1;
[8]   0,   0, 9708,   0, 4249,   0, 322,   0,   1;
...

Alois P. Heinz, Rows n = 0..200, flattened
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Columns k=0-1 give: A000007, A005359(n-1).
Row sums give A003483.
T(n+2,n) gives A000914.
Cf. A214851, A246945.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i, 2)=0 and irem(j, 2)=1, 0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1))*x^j, j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Nov 23 2021

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
     Sum[If[Mod[i, 2] == 0 && Mod[j, 2] == 1, 0, (i-1)!^j*multinomial[n,
     Join[{n-i*j}, Table[i, {j}]]]/j!*b[n-i*j, i-1]]*x^j, {j, 0, n/i}]]]];
T[n_] := With[{p = b[n, n]}, Table[Coefficient[p, x, i], {i, 0, n}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

A351760 a(n) = Sum_{1 <= i < j <= n} (i*j)^4.

Original entry on oeis.org

0, 0, 16, 1393, 26481, 247731, 1516515, 6978790, 26131686, 83684778, 237014778, 607915231, 1436816095, 3170754405, 6600189141, 13064343516, 24750198748, 45116627556, 79482515700, 135826148445, 225852708445, 366397514791, 581244702423, 903454469346, 1378306878690, 2066986566190

Offset: 0

Roudy El Haddad, Feb 18 2022

a(n) is the sum of all products of two distinct elements from the set {1^4, ..., n^4}.

Roudy El Haddad, Multiple Sums and Partition Identities, arXiv:2102.00821 [math.CO], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A000217 (for power 0), A000914 (for power 1), A000596 (for squares), A347107 (for cubes).

Cf. A000583 (fourth powers), A000538 (sum of fourth powers).

{a(n) = n*(n-1)*(n+1)*(2*n-1)*(2*n+1)*(9*n^5+20*n^4-15*n^3-50*n^2+n+30)/1800};

a(n) = sum(j=2, n, sum(i=1, j-1, i^4*j^4));

def A351760(n): return n*(n*(n*(n*(n*(n*(n*(n*(n*(9*n+20<<2)-105)-300)+88)+390)-20)-200)+1)+30)//1800 # Chai Wah Wu, Oct 03 2024

a(n) = Sum_{j=2..n} Sum_{i=1..j-1} j^4*i^4.
a(n) = n*(n - 1)*(n + 1)*(2*n - 1)*(2*n + 1)*(9*n^5 + 20*n^4 - 15*n^3 - 50*n^2 + n + 30)/1800.
a(n) = binomial(2*n+2, 5)*(9*n^5 + 20*n^4 - 15*n^3 - 50*n^2 + n + 30)/5!.
G.f.: x^2*(16 + 1217*x + 12038*x^2 + 30415*x^3 + 23364*x^4 + 5263*x^5 + 262*x^6 + x^7)/(1 - x)^11. - Stefano Spezia, Feb 18 2022

A027920 Least k such that 2nd elementary symmetric function of {1,2,...,k} >= 4th elementary symmetric function of {1,2,...,n}.

Original entry on oeis.org

3, 6, 10, 15, 20, 26, 33, 40, 49, 58, 68, 78, 90, 102, 115, 129, 143, 158, 174, 191, 208, 227, 246, 265, 286, 307, 329, 352, 376, 400, 425, 451, 477, 505, 533, 562, 591, 622, 653, 685, 717, 751, 785, 820, 856, 892, 929, 967, 1006, 1046, 1086, 1127, 1169, 1211

Offset: 4

Clark Kimberling

nonn

Cf. A000914, A000915.

a(n) = min{k: A000914(k) >= A000915(n-3)}. - Sean A. Irvine, Dec 10 2019

More terms from Sean A. Irvine, Dec 10 2019

cp's OEIS Frontend

A024172 Integer part of ((3rd elementary symmetric function of 1,2,..,n)/(2nd elementary symmetric function of 1,2,...,n)).

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A024173 Integer part of ((4th elementary symmetric function of 1,2,..,n)/(2nd elementary symmetric function of 1,2,...,n)).

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A027916 Least k such that 1+2+...+k >= E{1,2,...,n}, where E = 2nd elementary symmetric function.

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A027919 a(n) = least k such that 2nd elementary symmetric function of {1,2,...,k+1} >= 3rd elementary symmetric function of {1,2,...,n}.

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A067057 Let A(n) = {1,2,3,...n}. Let B(r) and C(n-r) be two subsets of A(n) having r and n-r elements respectively, such that B(r) U C(n-r) = A(n) and B and C are disjoint; then a(n) = sum of the products of all combination sums of elements of B and C for r =1 to n-1.

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A134449 Sum of even products minus sum of odd products of different pairs of numbers from 1 to n.

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A253207 a(n) = number of permutations of (1,2,...,n) producible by an ordered quadruple of distinct transpositions.

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A349645 Triangular array read by rows: T(n,k) is the number of square n-permutations possessing exactly k cycles; n >= 0, 0 <= k <= n.

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