A000537 -id:A000537 - cp's OEIS frontend

A269947 Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.

1, 0, 1, 0, 1, 1, 0, 8, 9, 1, 0, 216, 251, 36, 1, 0, 13824, 16280, 2555, 100, 1, 0, 1728000, 2048824, 335655, 15055, 225, 1, 0, 373248000, 444273984, 74550304, 3587535, 63655, 441, 1, 0, 128024064000, 152759224512, 26015028256, 1305074809, 25421200, 214918, 784, 1

Offset: 0

Peter Luschny, Mar 22 2016

			Triangle starts:
1,
0, 1,
0, 1,       1,
0, 8,       9,       1,
0, 216,     251,     36,     1,
0, 13824,   16280,   2555,   100,   1,
0, 1728000, 2048824, 335655, 15055, 225, 1.

Peter Luschny, The P-transform.

Variant: A249677.
Cf. A007318 (order 0), A132393 (order 1), A269944 (order 2).
Cf. A000442, A000537, A255433.

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + (n-1)^3*T(n-1, k))) end:
for n from 0 to 6 do seq(T(n,k), k=0..n) od;

T[n_, k_] := T[n, k] = Which[n == k, 1, k < 0 || k > n, 0, True, T[n - 1, k - 1] + (n - 1)^3 T[n - 1, k]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)

T(n,1) = ((n-1)!)^3 for n>=1 (cf. A000442).
T(n,n-1) = (n*(n-1)/2)^2 for n>=1 (cf. A000537).
Row sums: Product_{k=1..n} ((k-1)^3+1) for n>=0 (cf. A255433).

A269948 Triangle read by rows, Stirling set numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+k^3*T(n-1, k), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 9, 1, 0, 1, 73, 36, 1, 0, 1, 585, 1045, 100, 1, 0, 1, 4681, 28800, 7445, 225, 1, 0, 1, 37449, 782281, 505280, 35570, 441, 1, 0, 1, 299593, 21159036, 33120201, 4951530, 130826, 784, 1

Offset: 0

Peter Luschny, Mar 22 2016

Also called 3rd central factorial numbers.

			1,
0, 1,
0, 1, 1,
0, 1, 9,     1,
0, 1, 73,    36,     1,
0, 1, 585,   1045,   100,    1,
0, 1, 4681,  28800,  7445,   225,   1,
0, 1, 37449, 782281, 505280, 35570, 441, 1.

Variant: A098436.
Cf. A007318 (order 0), A048993 (order 1), A269945 (order 2).
Cf. A000537, A023001, A098437.

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + k^3*T(n-1, k))) end:
for n from 0 to 9 do seq(T(n,k), k=0..n) od;

T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + k^3*T[n - 1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 25 2019 *)

T(n,2) = (8^(n-1)-1)/7 for n>=1 (cf. A023001).
T(n,n-1) = (n*(n-1)/2)^2  for n>=1 (cf. A000537).
Row sums: A098437.

A293550 a(n) = Sum_{k=0..n} k^3binomial(2n-k,n).

Original entry on oeis.org

0, 1, 11, 69, 354, 1650, 7293, 31213, 130832, 540702, 2212550, 8989090, 36327810, 146228940, 586823265, 2349424125, 9389012160, 37467344310, 149345215290, 594753416790, 2366845396500, 9413555798556, 37423053793026, 148719333293394, 590842248405024, 2346813893147500

Offset: 0

Ilya Gutkovskiy, Oct 11 2017

nonn

Main diagonal of iterated partial sums array of cubes (starting with the first partial sums). For nonnegative integers see A002054, for squares see A265612.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for sequences related to sums of cubes

Cf. A000537, A000578, A002054, A024166, A101094, A101097, A101102, A265612, A302352.

Table[Sum[k^3 Binomial[2 n - k, n], {k, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[x (1 + 4 x + x^2)/(1 - x)^(n + 5), {x, 0, n}], {n, 0, 25}]
Table[2^(2 n + 1) n^2 (13 n + 7) Gamma[n + 3/2]/(Sqrt[Pi] Gamma[n + 5]), {n, 0, 25}]
CoefficientList[Series[(6 - 6 Sqrt[1 - 4 x] - 36 x + 24 Sqrt[1 - 4 x] x + 55 x^2 - 19 Sqrt[1 - 4 x] x^2 - 15 x^3 + Sqrt[1 - 4 x] x^3)/(2 Sqrt[1 - 4 x] x^4), {x, 0, 25}], x]
CoefficientList[Series[(E^(2 x) (36 - 24 x + 13 x^2) BesselI[0, 2 x])/x^2 + (E^(2 x) (-36 + 24 x - 31 x^2 + 13 x^3) BesselI[1, 2 x])/x^3, {x, 0, 25}], x]* Range[0, 25]!

a(n) = [x^n] x*(1 + 4*x + x^2)/(1 - x)^(n+5).
a(n) = 2^(2*n+1)*n^2*(13*n + 7)*Gamma(n+3/2)/(sqrt(Pi)*Gamma(n+5)).
a(n) ~ 26*4^n/sqrt(Pi*n).

A301912 Numbers k such that the decimal representation of k ends that of the sum of the first k cubes.

Original entry on oeis.org

0, 1, 5, 25, 76, 376, 500, 625, 876, 1876, 2500, 5001, 5625, 9376, 15625, 25001, 40625, 50001, 62500, 65625, 71876, 75001, 90625, 109376, 171876, 265625, 375001, 390625, 500001, 765625, 875001, 890625, 1171876, 2265625, 2890625, 4062500, 4375001, 5000001

Offset: 1

Robert Dawson, Mar 28 2018

For j >= 3, 1 + 5*10^j = A199685(j) is in the sequence, so the sequence is infinite. - Vaclav Kotesovec, Mar 29 2018
From Robert Dawson, Apr 12 2018: (Start)
This sequence is the union of the following ten subsequences.
Terms in  have fewer than d digits: they are always terms of the sequence, and always appear elsewhere, as an earlier term of the same subsequence or a related subsequence. (However, the d-th terms of the subsequences are always distinct for any d > 4.) Dashes replace certain solutions to the congruences for small values of d for which certain other divisibility criteria are not met. The integers n_0(d) and n_1(d) are the even and odd zeros of n^2+3n+4 (mod 2^d) (note that by Hensel's Lemma these always exist and each is unique).
(i)   p(d) satisfying 2^d| p(d) - n_0(d), 5^d |p(d):
  (0,<0>,500,2500,62500,62500,4062500,14062500,...)
(ii)  q(d) satisfying 2^{d-1}|q(d)-1, 5^d|q(d) for d != 3:
  (0,25,-,<625>,40625,390625,2890625,12890625,...)
(iii) q(d) + 5x10^{d-1} for d != 2:
  (5,-, 625,5625,90625, 890625,7890625, 62890625,...)
(iv)  q'(d) satisfying 2^{d-1}|q'(d) - n_1(d), 5^d|q'(d), for d != 1,3:
  (-,25,-,<625>,15625,265625,2265625,47265625,...)
(v)   q'(d) + 5x10^{d-1} for d != 2:
  (5,-,625,5625,65625,765625,7265625,97265625,...)
(vi)  r(d) satisfying 2^d|r(d), 5^d|r(d)-1 for d >= 2
  (-,76,376,9376,<9376>,109376,7109376,87109376,...) = A016090(d)
(vii) r'(d) satisfying 2^d|r'(d) - n_0(d), 5^d|r'(d)-1 for d >= 2:
  (-,76,876,1876,71876,171876,1171876,<1171876>,...)
(viii)s(d) := 5x10^{d-1}+1 for d >= 4:
  (-,-,-,5001,50001,500001,5000001,50000001,...) = A199685(d-1)
(ix)  t(d) satisfying 2^{d-1}|t(d)-n_0(d), 5^d|t(d)-1:
  (1,<1>,<1>,<1>,25001,375001,4375001,34375001,...)
(x)   t(d) + 5x10^{d-1} for d >= 4:
  (-,-,-,5001,75001,875001,9375001,84375001,...)
For d > 4, the sequence A301912 has at most 10 and at least 5 terms with d digits. The maximum is first attained for d=7. The minimum is first attained for d=168. (End)

			The sum of the first five cubes is 225, which ends in 5, so 5 is in the sequence.

Vaclav Kotesovec, Table of n, a(n) for n = 1..61
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
Wikipedia, Hensel's lemma

Cf. A000537, A199685.

seq = {}; Do[If[StringTake[ToString[k^2*(k+1)^2/4], -StringLength[ToString[k]]] == ToString[k], seq = Join[seq, {k}]], {k, 0, 1000000}]; seq (* Vaclav Kotesovec, Mar 29 2018 *)

A301912_list, k, n = [], 1, 1
while len(A301912_list) < 100:
    if n % 10**(len(str(k))) == k:
        A301912_list.append(k)
    k += 1
    n += k**3 # Chai Wah Wu, Mar 30 2018

Corrected and extended by Vaclav Kotesovec, Mar 29 2018

A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 4, 4, 5, 3, 6, 5, 5, 1, 7, 5, 8, 4, 6, 6, 9, 3, 9, 7, 8, 5, 10, 6, 11, 1, 7, 8, 7, 5, 12, 9, 8, 4, 13, 7, 14, 6, 7, 10, 15, 3, 16, 10, 9, 7, 16, 9, 8, 5, 10, 11, 17, 6, 18, 12, 8, 1, 9, 8, 19, 8, 11, 8, 20, 5, 21, 13, 11, 9, 9, 9, 22, 4, 16, 14, 23, 7, 10, 15, 12, 6

Offset: 1

Ilya Gutkovskiy, May 05 2018

nonn

			a(72) = 5 because 72 = 2^3*3^2 = prime(1)^3*prime(2)^2 and 1^3 + 2^2 = 5.

Ilya Gutkovskiy, Extended graphical example
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A002110, A003963, A008475, A056239, A066328, A156061, A222416.

a[n_] := Plus @@ (PrimePi[#[[1]]]^#[[2]]& /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 88}]

If gcd(u,v) = 1 then a(u*v) = a(u) + a(v).
a(p^k) = A000720(p)^k where p is a prime.
a(A002110(m)^k) = 1^k + 2^k + ... + m^k.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A038580(k)) = A006450(k).
a(A001248(k)) = a(A011757(k)) = A000290(k).
a(A030078(k)) = a(A055875(k)) = A000578(k).
a(A002110(k)) = a(A011756(k)) = A000217(k).
a(A061742(k)) = A000330(k).
a(A115964(k)) = A000537(k).
a(A080696(k)) = A007504(k).
a(A076954(k)) = A001923(k).

A327448 Number of ways the first n cubes can be partitioned into three sets with equal sums.

Original entry on oeis.org

1, 0, 0, 691, 3416, 0, 233, 1168, 0, 8857, 18157, 0, 2176512, 3628118, 0, 3204865, 8031495, 0, 79514209, 205927212, 0, 5152732369, 13493840291, 0

Offset: 23

N. J. A. Sloane, Sep 19 2019

Note the offset.

			The unique smallest solution (for n = 23) is
27 + 216 + 1000 + 2197 + 5832 + 6859 + 9261 =
1 + 64 + 343 + 512 + 1728 + 4096 + 8000 + 10648 =
8 + 125 + 729 + 1331 + 2744 + 3375 + 4913 + 12167.

Keith F. Lynch, Posting to Math Fun Mailing List, Sep 17 2019.

Cf. A000537, A000578, A007494, A112972, A275714, A113263, A327449, A327450.

s:= proc(n) option remember; `if`(n<2, 0, n^3+s(n-1)) end:
b:= proc(n, x, y) option remember; `if`(n=1, 1, (p-> (l->
      add(`if`(p>l[i], 0, b(n-1, sort(subsop(i=l[i]-p, l))
            [1..2][])), i=1..3))([x, y, s(n)-x-y]))(n^3))
    end:
a:= n-> `if`(irem(1+s(n), 3, 'q')=0, b(n, q-1, q)/2, 0):
seq(a(n), n=23..27);  # Alois P. Heinz, Sep 30 2019

s[n_] := s[n] = If[n < 2, 0, n^3 + s[n - 1]];
b[n_, x_, y_] := b[n, x, y] = If[n == 1, 1, With[{p = n^3}, Sum[If[p > #[[i]], 0, b[n - 1, Sequence @@ Sort[ReplacePart[#, i -> #[[i]] - p]][[1 ;; 2]]]], {i, 1, 3}]]&[{x, y, s[n] - x - y}]];
a[n_] := a[n] = If[q = Quotient[1 + s[n], 3]; Mod[1 + s[n], 3] == 0, b[n, q - 1, q]/2, 0];
Table[Print[n, " ", a[n]]; a[n], {n, 23, 34}] (* Jean-François Alcover, Nov 08 2020, after Alois P. Heinz *)

a(n) > 0 => n in { A007494 }. - Alois P. Heinz, Sep 30 2019

a(32), a(33), a(35) recomputed and a(36)-a(38) added by Alois P. Heinz, Sep 30 2019

a(39)-a(46) from Bert Dobbelaere, May 15 2021

A329708 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+...+(n+1)x^n)^2.

Original entry on oeis.org

1, 1, 4, 4, 1, 4, 10, 12, 9, 1, 4, 10, 20, 25, 24, 16, 1, 4, 10, 20, 35, 44, 46, 40, 25, 1, 4, 10, 20, 35, 56, 70, 76, 73, 60, 36, 1, 4, 10, 20, 35, 56, 84, 104, 115, 116, 106, 84, 49, 1, 4, 10, 20, 35, 56, 84, 120, 147, 164, 170, 164, 145, 112, 64

Offset: 0

Seiichi Manyama, Feb 29 2020

			Triangle begins:
  1;
  1, 4,  4;
  1, 4, 10, 12,  9;
  1, 4, 10, 20, 25, 24, 16;
  1, 4, 10, 20, 35, 44, 46, 40, 25;
  ...

Seiichi Manyama, Rows n = 0..99, flattened

Row sums give A000537(n+1).
T(n,2n) gives A000290(n+1).
Cf. A000292, A002260, A084608, A008794, A329709.

row[n_]:=CoefficientList[Series[(Sum[(i+1)x^i,{i,0,n}])^2,{x,0,2n}],x]; Array[row,8,0]//Flatten (* Stefano Spezia, Feb 15 2025 *)

for(n=0, 10, print(Vecrev(sum(k=0, n, (k+1)*x^k)^2), ", "))

T(n,k) = A000292(k+1) for k=0..n.

Sum_{k=0..2n} (-1)^k * T(n,k) = A008794(n+2). - Alois P. Heinz, Feb 14 2025

A364269 a(n) = Sum_{k=1..n} k^3*sigma_2(k), where sigma_2 is A001157.

Original entry on oeis.org

1, 41, 311, 1655, 4905, 15705, 32855, 76375, 142714, 272714, 435096, 797976, 1171466, 1857466, 2734966, 4131702, 5556472, 8210032, 10692990, 15060990, 19691490, 26186770, 32635280, 44385680, 54557555, 69497155, 85637215, 108686815, 129222353, 164322353

Offset: 1

Seiichi Manyama, Oct 20 2023

Wikipedia, Faulhaber's formula.

Cf. A356125, A364268.

Cf. A000537, A001157.

Accumulate[Table[n^3*DivisorSigma[2, n], {n, 1, 30}]] (* Amiram Eldar, Oct 20 2023 *)

f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);
a(n, s=3, t=2) = sum(k=1, n, k^(s+t)*f(n\k, s));

def A364269(n): return sum(k*(k**2*(m:=n//k)*(m+1)>>1)**2 for k in range(1,n+1)) # Chai Wah Wu, Oct 20 2023

from math import isqrt
def A364269(n): return ((((s:=isqrt(n))*(s+1))**4*(1-s*(s+1<<1))>>2) + sum(((q:=n//k)*(q+1))**2*k**3*(3*k**2+(q*(q+1<<1)-1)) for k in range(1,s+1)))//12 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} k^5 * A000537(floor(n/k)).

a(n) ~ (zeta(3)/6) * n^6. - Amiram Eldar, Oct 20 2023

A071910 a(n) = t(n)t(n+1)t(n+2), where t() are the triangular numbers.

Original entry on oeis.org

0, 18, 180, 900, 3150, 8820, 21168, 45360, 89100, 163350, 283140, 468468, 745290, 1146600, 1713600, 2496960, 3558168, 4970970, 6822900, 9216900, 12273030, 16130268, 20948400, 26910000, 34222500, 43120350, 53867268, 66758580, 82123650, 100328400, 121777920

Offset: 0

N. J. A. Sloane, Jun 13 2002

nonn

a(n) is also the number of three-dimensional cage assemblies such that the assembly is not a cube. See also A052149 for the two-dimensional version and to A059827 for the non-exclusive version. - Alejandro Rodriguez, Oct 20 2020

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A006542, (first differences of a(n) /18) A006414, (second differences of a(n) /18) A006322, (third differences of a(n) /18) A004068, (fourth differences of a(n) /18) A005891, (fifth differences of a(n) /18) A008706.

Join[{0},Times@@@Partition[Accumulate[Range[40]],3,1]] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,18,180,900,3150,8820,21168},40] (* Harvey P. Dale, Aug 08 2025 *)

t(n) = n*(n+1)/2;
a(n) = t(n)*t(n+1)*t(n+2); \\ Michel Marcus, Oct 21 2015

a(n) = 18*A006542(n+3). - Vladeta Jovovic, Jun 14 2002
G.f.: 18*x*(1+3*x+x^2)/(1-x)^7. - Vladeta Jovovic, Jun 14 2002
a(n) = ((n+1)*(n+2))^3/8 - Sum_{i=1..n+1} i^3. - Jon Perry, Feb 13 2004
a(n) = C(2+n, n)*C(3+n, 1+n)*C(4+n, 2+n). - Zerinvary Lajos, Jul 29 2005
a(n) = A059827(n+1) - A000537(n+1). - Michel Marcus, Oct 21 2015

A081175 Numbers of the form Sum_{i=1..k} i^j, j >= 1, k >= 1.

Original entry on oeis.org

1, 3, 5, 6, 9, 10, 14, 15, 17, 21, 28, 30, 33, 36, 45, 55, 65, 66, 78, 91, 98, 100, 105, 120, 129, 136, 140, 153, 171, 190, 204, 210, 225, 231, 253, 257, 276, 285, 300, 325, 351, 354, 378, 385, 406, 435, 441, 465, 496, 506, 513, 528, 561, 595, 630, 650, 666, 703

Offset: 1

N. J. A. Sloane, Apr 18 2003

Union of sums of k-th powers, for k >= 1.

			30 is in the set because 30 = 1^2 + 2^2 + 3^2 + 4^2 (j=2, k=4).

Robert Israel, Table of n, a(n) for n = 1..10000
Michael Penn, an excruciatingly deep dive into the power sum., YouTube video, 2022.

Cf. A000217, A000330, A000537, A000538, A000539, A000540, A000541, A000542, A007487, A023002.

For primes in this sequence see A164307.

N:= 1000: # to get all terms <= N
A:=select(`<=`,{1, seq(seq(sum(i^k,i=1..m), m=2..floor((N*(k+1))^(1/(k+1)))),k = 1 ..ilog2(N-1))},N):
sort(convert(A,list)); # Robert Israel, Jan 26 2015

Take[ Union[ Flatten[ Table[ Sum[ i^j, {i, 1, n}], {j, 1, 9}, {n, 1, 40}]]], 60]

Corrected and extended by Robert G. Wilson v, May 08 2003

cp's OEIS Frontend

A269947 Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A269948 Triangle read by rows, Stirling set numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+k^3*T(n-1, k), for n>=0 and 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A293550 a(n) = Sum_{k=0..n} k^3*binomial(2*n-k,n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A301912 Numbers k such that the decimal representation of k ends that of the sum of the first k cubes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A327448 Number of ways the first n cubes can be partitioned into three sets with equal sums.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A329708 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2*x+...+(n+1)*x^n)^2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A364269 a(n) = Sum_{k=1..n} k^3*sigma_2(k), where sigma_2 is A001157.

A293550 a(n) = Sum_{k=0..n} k^3binomial(2n-k,n).

A329708 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+...+(n+1)x^n)^2.

A071910 a(n) = t(n)t(n+1)t(n+2), where t() are the triangular numbers.