A107444 -id:A107444 - cp's OEIS frontend

A359665 a(n) = Sum_{k=0..n} binomial(k^3, k).

1, 2, 30, 2955, 638331, 235169606, 131748994154, 104332124742623, 110963563379491743, 152605484049946645638, 263562165946020159478038, 558488792578762177358255808, 1424733420462958066911824023728, 4307309064570490624823548890385698, 15228800547242034570505949850130312826

Offset: 0

Vaclav Kotesovec, Jan 10 2023

nonn

Cf. A005809, A107444, A188675, A295773.

Table[Sum[Binomial[k^3, k], {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(k^3, k)); \\ Michel Marcus, Jan 10 2023

a(n) ~ exp(n) * n^(2*n - 1/2) / sqrt(2*Pi).

A108131 Array read by antidiagonals: A(k,n) = C(n^k, n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 28, 84, 1, 1, 1, 120, 2925, 1820, 1, 1, 1, 496, 85320, 635376, 53130, 1, 1, 1, 2016, 2362041, 174792640, 234531275, 1947792, 1, 1, 1, 8128, 64304604, 45545029376, 782083984500, 131513824548, 85900584, 1

Offset: 0

Jonathan Vos Post, Jun 11 2007

			Array begins:
k.C(n^k, n)
1.|.1.1....1........1..............1...................1..........................1...
2.|.1.1....6.......84...........1820...............53130....................1947792...
3.|.1.1...28.....2925.........635376...........234531275...............131513824548...
4.|.1.1..120....85320......174792640........782083984500...........6505247592703944...
5.|.1.1..496..2362041....45545029376....2475588476563125......306455244538856615280...
6.|.1.1.2016.64304604.11710951848960.7756055513916018750.14320984850603177651837856...

Cf. A014062 (row 2), A107444 (row 3), A107446 (row 4).

Cf. A006516 (col 2), A026809 (col 3).

C:= func< n | Binomial(n^6, n) >; [ C(n) : n in [0..7]]; /* Row 6 example */

A227053 a(n) = (n^3)! / (n^3-n)! = number of ways of placing n labeled balls into n^3 labeled boxes with at most one ball in each box.

Original entry on oeis.org

1, 1, 56, 17550, 15249024, 28143753000, 94689953674560, 525169893772283760, 4469844204191484518400, 55337211594165488805417600, 955860613004397508326213120000, 22282564877342299983672172489536000, 682182070950002359574696677978908672000, 26812831292465310201469047550286967518976000

Offset: 0

Alex Ratushnyak, Jun 29 2013

Cf. A107444, A158362.

import math
for n in range(20):
  print(math.factorial(n**3)//math.factorial(n**3-n), end=', ')

A350693 Number of b > 0 which permit n^3 to be written as a sum of powers of b in n parts. Each exponent c is an integer >= 0, n^3 = b^c_1 + b^c_2 + ... + b^c_n.

Original entry on oeis.org

3, 5, 8, 7, 10, 13, 17, 19, 12, 20, 16, 18, 18, 25, 25, 21, 14, 28, 31, 34, 19, 22, 29, 34, 28, 33, 29, 38, 19, 33, 30, 31, 34, 51, 44, 30, 20, 41, 38, 44, 18, 37, 42, 52, 27, 30, 37, 59, 39, 50, 28, 35, 37, 82, 64, 44, 19, 36, 27, 36, 27, 52, 85, 65, 35, 40, 29

Offset: 2

Thomas Scheuerle, Jan 12 2022

nonn

If n^3 is written in different number bases, a(n) is an upper limit for the count of number bases which allow n^3 to be written as a base-b number with a digit sum of n (generalized Dudeney numbers).

a(n) has an upper limit in the number of divisors of n^3-n. Let d be one of these divisors, then it appears that a lower limit can be found by excluding all divisors d where d+1 does not share all its prime divisors with binomial(n^3, n) (A107444).

			a(2) = 3 because 2^3 = 2^2 + 2^2 = 4^1 + 4^1 = 7^1 + 7^0.

Cf. A000005, A046459, A092517, A107444.

a(n) = sum(d=2, n^3, s=sumdigits(n^3, d); s<=n&&(n-s)%(d-1)==0); \\ Jinyuan Wang, Jan 15 2022

a(n) <= A000005(n^3-n). Conjectured to become a(n) = A000005(n^3-n), if the definition would permit negative values for b and only the absolute value of the sum needs to be equal to n^3.

More terms from Jinyuan Wang, Jan 15 2022

cp's OEIS Frontend

A359665 a(n) = Sum_{k=0..n} binomial(k^3, k).

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A108131 Array read by antidiagonals: A(k,n) = C(n^k, n).

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A227053 a(n) = (n^3)! / (n^3-n)! = number of ways of placing n labeled balls into n^3 labeled boxes with at most one ball in each box.

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A350693 Number of b > 0 which permit n^3 to be written as a sum of powers of b in n parts. Each exponent c is an integer >= 0, n^3 = b^c_1 + b^c_2 + ... + b^c_n.

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