A001303 -id:A001303 - cp's OEIS frontend

A259464 From higher-order arithmetic progressions.

75, 21875, 5512500, 1512630000, 484041600000, 184834742400000, 84715923600000000, 46534591303200000000, 30489464221856640000000, 23681690417572387200000000, 21660852835272876825600000000, 23175597788788462617600000000000, 28817200450516396946227200000000000

Offset: 0

N. J. A. Sloane, Jun 30 2015

nonn

"3 over n!" on page 15 in the Dienger article is A087047; A_3 is A001303. - Georg Fischer, Dec 16 2024

Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]

Cf. A001303, A087047, A259462, A259463.

rXI := proc(n, a, d)
        n*(n+1)*(n+2)/6*a+(n+2)*(n+1)*n*(n-1)/24*d;
end proc:
A259464 := proc(n)
        mul(rXI(i, a, d), i=1..n+3) ;
        coeftayl(%, d=0, 3) ;
        coeftayl(%, a=0, n) ;
end proc:
seq(A259464(n), n=1..25) ; # R. J. Mathar, Jul 15 2015

rXI[n_, a_, d_] := (n(n+1)(n+2)/6)*a+((n+2)(n+1)n(n-1)/24)*d;
A259464[n_] :=
   Product[rXI[i, a, d], {i, 1, n+4}]//
   SeriesCoefficient[#, {d, 0, 3}]&//
   SeriesCoefficient[#, {a, 0, n+1}]&;
Table[A259464[n], {n, 0, 12}] (* Jean-François Alcover, Apr 26 2023, after R. J. Mathar *)

D-finite with recurrence: -6*n*(n+2)*a(n) +(n+6)*(n+5)*(n+4)^3*a(n-1)=0. - R. J. Mathar, Jul 15 2015

a(n) = 2^(-n-5)*3^(-n-4)*(n+4)!*(n+5)!*(n+6)!*(n+4)^2*(n+3)^2*(n+2)*(n+1)/3072. - Georg Fischer, Dec 16 2024

A378234 From higher-order arithmetic progressions: Corrected version of A259461.

Original entry on oeis.org

40, 5000, 472500, 43218000, 4148928000, 432081216000, 49509306000000, 6275893932000000, 881135508052800000, 136878615942868800000, 23474682634201999200000, 4432282735129048800000000, 918537831584839065600000000, 208281986149676045967360000000, 51516317681413623440962560000000

Offset: 0

Georg Fischer, Dec 16 2024

nonn

Only the first 5 terms of A259461 are correct. - R. J. Mathar, Jul 14 2015

"2 over n!" on page 13 in the Dienger article is A006472; A_3 is A001303.

Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]

Cf. A001303, A006472, A259459, A259460, A259461.

rV := proc(n,a,d)
    n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
end proc:
A259461 := proc(n)
    mul(rV(i,a,d),i=1..n+3) ;
    coeftayl(%,d=0,3) ;
    coeftayl(%,a=0,n) ;
end proc:
seq(A259461(n),n=1..5) ; # R. J. Mathar, Jul 14 2015

D-finite with recurrence: -2*n*(n+2)*a(n) + (n+4)^3*(n+5)*a(n-1) = 0.

a(n) = (n+5)!*(n+4)!^3 / (1296*2^(n+4)*n!^2*(n+2)*(n+1)).

A191685 Eighth diagonal a(n) = s(n,n-7) of the unsigned Stirling numbers of the first kind with n>7.

Original entry on oeis.org

5040, 109584, 1172700, 8409500, 45995730, 206070150, 790943153, 2681453775, 8207628000, 23057159840, 60202693980, 147560703732, 342252511900, 756111184500, 1599718388730, 3256091103430, 6400590336096, 12191224980000, 22563937825000, 40681506808800

Offset: 8

Paul Weisenhorn, Jun 11 2011

The Maple programs under I generate the sequence. The Maple program under II generates explicit formulas for a(n+1) = s(n+1,n+1-c) with c>=1 and n>=c.

			c=1; a(n+1) = binomial(n+1,2)
c=2; a(n+1) = binomial(n+1,3)*(2+3*n)/4
c=3; a(n+1) = binomial(n+1,4)*(n+n^2)/2
c=4; a(n+1) = binomial(n+1,5)*(-8-10*n+15*n^2 +15*n^3)/48
c=5; a(n+1) = binomial(n+1,6)*(-6*n-7*n^2+2*n^3+ 3*n^4)/16
c=6; a(n+1) = binomial(n+1,7)*(96+140*n-224*n^2-315*n^3+63*n^5)/576
c=7; a(n+1) = binomial(n+1,8)*(80*n+114*n^2-23*n^3-75*n^4-9*n^5+9*n^6)/144
c=8; a(n+1) = binomial(n+1,9)*(-1152-1936*n+2820*n^2+
  5320*n^3+735*n^4-1575*n^5-315*n^6+135*n^7)/3840
c=9; a(n+1) = binomial(n+1,10)*(-1008*n-1676*n^2 +100*n^3+1295*n^4+392*n^5-210*n^6-60*n^7 +15*n^8)/768

K. Seidel, Variation der Binomialkoeffizienten, Bild
der Wissenschaft, 6 (1980), pp. 127-128.

T. D. Noe, Table of n, a(n) for n = 8..1000

Cf. A130534, A000012 (c=0; 1st diagonal), A000217 (c=1; 2nd diagonal), A000914 (c=2; 3rd diagonal), A001303 (c=3; 4th diagonal), A000915 (c=4; 5th diagonal), A053567 (c=5; 6th diagonal), A112002 (c=6; 7th diagonal), A191685 (c=7; 8th diagonal).

I: programs generate the sequence:
with(combinat): c:=7; a:= proc(n) a(n):=abs(stirling1(n,n-c)); end: seq(a(n), n=c+1..28);
for n from 7 to 27 do a(n+1) := binomial(n+1,8)*(80*n+ 114*n^2- 23*n^3- 75*n^4- 9*n^5+ 9*n^6)/144 end do: seq(a(n),n=8..28);
II: program generates explicit formulas for a(n+1) =  s(n+1,n+1-c):
k[1,0]:=1: v:=1:
for c from 2 to 10 do
  c1:=c-1: c2:=c-2: p0:=0:
  for j from 0 to c2 do p0:=p0+k[c1,j]*m^j: end do:
  f:=expand(2*c*(m+1)*p0/v):
  p1:=0: p2:=0:
  for j from 0 to c1 do
    p1:=p1+k[c,j]*(m+1)^j:
    p2:=p2+k[c,j]*m^j:
  end do:
  g:=collect((m+2)*p1-(m-c1)*p2-f,m):
  kh[0]:=rem(g,m,m): Mk:=[kh[0]]: Mv:=[k[c,0]]:
  for j from 1 to c1 do
    kh[j]:=coeff(g,m^j):
    Mk:=[op(Mk),kh[j]]: Mv:=[k[c,j],op(Mv)]:
  end do:
  sol:=solve(Mk,Mv):
  v:=1:
  for j from 1 to c do
    k[c,c-j]:=eval(k[c,c-j],sol[1,j]):
    nen[j]:=denom(k[c,c-j]):
    v:=ilcm(v,nen[j]):
  end do:
  for j from 0 to c1 do k[c,j]:=k[c,j]*v:
    printf("%8d",k[c,j]): end do:
  p3:=0:
  for j from 0 to c1 do p3:=p3+k[c,j]*n^j: end do:
  s[n+1,n+1-c]:=binomial(n+1,c+1)*(c+1)*p3/(2^c*k[c,c1]):
end do:
for c from 2 to 10 do print("%a\n",s[n+1,n+1-c]):
end do:

a(n+1) = A130534(T(n,n-7)) = s(n+1,n+1-7)

a(n+1) = binomial(n+1,8)*(80*n+114*n^2-23*n^3-75*n^4-9*n^5+9*n^6)/144

A352980 a(n) = Sum_{1 <= i < j < k <= n} (kji)^3.

Original entry on oeis.org

0, 0, 0, 216, 16280, 335655, 3587535, 25421200, 135459216, 584760870, 2145870870, 6918983280, 20073184560, 53334782501, 131555523645, 304453955520, 666698215360, 1390977293580, 2780695001196, 5351537889480, 9954554649480, 17957698726275

Offset: 0

Roudy El Haddad, Apr 13 2022

a(n) is the sum of all products of three distinct cubes of positive integers up to n, i.e., the sum of all products of three distinct elements from the set of cubes {1^3, ..., n^3}.

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. See Theorem 5.1 for m = 3 and p = 3.
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1). [Typo corrected by _Georg Fischer_, Sep 30 2022]

Cf. A352979 (for nondistinct cubes).
Cf. A001303 (for power 1), A000597 (for squares).
Cf. A000578 (cubes), A000537 (sum of first n cubes), A347107 (order 2).

{a(n) = n^2 * (n + 1)^2 * (n - 1) * (n - 2) * (35*n^6 + 5*n^5 - 237*n^4 - 77*n^3 + 502*n^2 + 148*n -336)/13440};

def A352980(n): return n**2*(n*(n*(n*(n*(n*(n*(n*(n*(n*(35*n - 30) - 347) + 180) + 1365) - 350) - 2541) + 240) + 2160) - 40) - 672)//13440 # Chai Wah Wu, May 15 2022

a(n) = Sum_{k=3..n} Sum_{j=2..k-1} Sum_{i=1..j-1} k^3*j^3*i^3.
a(n) = n^2 * (n + 1)^2 * (n - 1) * (n - 2) * (35*n^6 + 5*n^5 - 237*n^4 - 77*n^3 + 502*n^2 + 148*n -336)/13440.
a(n) = binomial(n+1,4)*binomial(n+1,2)*(35*n^6 + 5*n^5 - 237*n^4 - 77*n^3 + 502*n^2 + 148*n -336)/280.

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

Original entry on oeis.org

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

A024174 a(n) is floor((4th elementary symmetric function of 1,2,..,n)/(3rd elementary symmetric function of 1,2,...,n)).

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 8, 10, 13, 16, 19, 22, 25, 29, 33, 37, 42, 47, 52, 57, 62, 68, 74, 80, 87, 94, 101, 108, 115, 123, 131, 139, 148, 157, 166, 175, 184, 194, 204, 214, 225, 236, 247, 258, 269, 281, 293, 305, 318, 331, 344, 357, 370, 384, 398, 412, 427, 442

Offset: 3

Clark Kimberling

nonn

			G.f. = x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 6*x^9 + 8*x^10 + 10*x^11 + 13*x^12 + ...

Ivan Neretin, Table of n, a(n) for n = 3..10000

Cf. A000915, A001303, A011848, A000217.

Table[Floor[(n - 3) (15 n^3 + 15 n^2 - 10 n - 8)/(120 n (n + 1))], {n, 3, 45}] (* Ivan Neretin, Nov 25 2016 *)
Insert[Table[Floor[1/8 (-2 - 3 n + n^2)], {n, 4, 45}], 0, 1] (* Ralf Steiner, Oct 27 2021 *)

{a(n) = if( n<4, 0, (n-3) * (15*n^3 + 15*n^2 - 10*n - 8) \ (120 * n * (n+1)))}; /* Michael Somos, Nov 25 2016 */

Empirical g.f.: x^5*(x^7-2*x^6+2*x^5-2*x^4+x^3-x^2+x-1) / ((x-1)^3*(x^2+1)*(x^4+1)). - Colin Barker, Aug 16 2014
a(n) = floor( A000915(n-3)/A001303(n-2) ). - R. J. Mathar, Sep 23 2016
a(n) = floor((n - 3)*(15n^3 + 15n^2 - 10n - 8)/(120*n*(n + 1))). - Ivan Neretin, Nov 25 2016
a(n) = floor((A000217(n-2)/2 - 1)/2) = floor((n^2 - 3*n - 2)/8), n >= 4. - Ralf Steiner, Oct 25 2021

Offset set to 3 by R. J. Mathar, Sep 23 2016

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

A259458 From higher-order arithmetic progressions.

Original entry on oeis.org

18, 750, 20250, 463050, 9878400, 205752960, 4286520000, 90561240000, 1956122784000, 43410118752000, 992644715462400, 23427803599200000, 571192163942400000, 14391113340764160000, 374682915193466880000, 10078235746321526784000, 279950992953375744000000, 8026706333564126208000000

Offset: 0

N. J. A. Sloane, Jun 30 2015

nonn

A_2 in page 12 of the article is A001303. - Georg Fischer, Dec 06 2024

Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]

Cf. A001303, A259457.

rX := proc(n, a, d)
        n*a+(n-1)*n/2*d;
end proc:
A259458 := proc(n)
        mul(rX(i, a, d), i=1..n+3) ;
        coeftayl(%, d=0, 3) ;
        coeftayl(%, a=0, n) ;
end proc:
seq(A259458(n), n=1..25) ; # R. J. Mathar, Jul 15 2015

rX[n_, a_, d_] := n*a + (n-1)*n/2*d;
A259458[n_] :=
   Product[rX[i, a, d], {i, 1, n+4}] //
   SeriesCoefficient[#, {d, 0, 3}]& //
   SeriesCoefficient[#, {a, 0, n+1}]&;
Table[A259458[n], {n, 0, 16}] (* Jean-François Alcover, Apr 27 2023, after R. J. Mathar *)

D-finite with recurrence: -n*(n+2)*a(n) +(n+4)^3*a(n-1)=0. - R. J. Mathar, Jul 15 2015
Conjectured g.f.: 18*3F1(5,5,5;3;x). - R. J. Mathar, Aug 09 2015
a(n) = (n+4)!*(n+1)*(n+2)*(n+3)^2*(n+4)^2/384. - Georg Fischer, Dec 06 2024

A027917 a(n) = least k such that 1+2+...+k >= E{1,2,...,n}, where E is the 3rd elementary symmetric function.

Original entry on oeis.org

3, 10, 21, 38, 63, 95, 137, 191, 256, 334, 427, 535, 661, 805, 968, 1151, 1357, 1585, 1838, 2117, 2422, 2755, 3117, 3510, 3934, 4392, 4883, 5409, 5973, 6574, 7213, 7894, 8615, 9379, 10187, 11041, 11940, 12887, 13883, 14929, 16026, 17176, 18379, 19637, 20952, 22323, 23754, 25244, 26795, 28408

Offset: 3

Clark Kimberling

nonn

a(n) = min{k: A000217(k) >= A001303(n-2)}. - R. J. Mathar, Sep 23 2016

cp's OEIS Frontend

A259464 From higher-order arithmetic progressions.

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A378234 From higher-order arithmetic progressions: Corrected version of A259461.

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A191685 Eighth diagonal a(n) = s(n,n-7) of the unsigned Stirling numbers of the first kind with n>7.

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A352980 a(n) = Sum_{1 <= i < j < k <= n} (k*j*i)^3.

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

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A024174 a(n) is floor((4th elementary symmetric function of 1,2,..,n)/(3rd elementary symmetric function of 1,2,...,n)).

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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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Maple

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A259458 From higher-order arithmetic progressions.

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A352980 a(n) = Sum_{1 <= i < j < k <= n} (kji)^3.