A014062 -id:A014062 - cp's OEIS frontend

A371472 Least positive number k such that binomial(k^2,k) is divisible by n^2.

1, 3, 8, 6, 9, 8, 10, 23, 16, 16, 16, 14, 23, 10, 9, 23, 45, 16, 27, 16, 34, 16, 33, 23, 94, 23, 105, 20, 77, 16, 54, 91, 16, 45, 19, 16, 83, 27, 23, 30, 58, 34, 114, 16, 16, 40, 133, 23, 130, 94, 45, 23, 75, 105, 16, 38, 27, 77, 145, 16, 106, 54, 47, 91, 49, 16, 190, 45, 80, 19, 123, 47, 283, 83, 94, 27, 40, 23, 137

Offset: 1

Seiichi Manyama, Mar 24 2024

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A014062, A081370, A371306.

a(n) = my(k=1); while(binomial(k^2, k)%n^2>0, k++); k;

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 */

A227052 a(n) = (n^2)! / (n^2-n)! = number of ways of placing n labeled balls into n^2 labeled boxes with at most one ball in each box.

Original entry on oeis.org

1, 1, 12, 504, 43680, 6375600, 1402410240, 432938943360, 178462987637760, 94670977328928000, 62815650955529472000, 50963773003971232204800, 49633807532904958383820800, 57141374006987657125324185600, 76763145767753986733306290176000, 119005648371962652004288345681920000

Offset: 0

Alex Ratushnyak, Jun 29 2013

Product of the entries in row n of an n X n square array with elements 1..n^2 listed in increasing order by rows. - Wesley Ivan Hurt, Mar 31 2025

Cf. A014062.

Table[(n^2)!/(n^2-n)!,{n,0,20}] (* Harvey P. Dale, Aug 10 2017 *)

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

A306763 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n^2) * x^n = Sum_{n>=0} binomial(n^2,n) * x^n.

Original entry on oeis.org

1, 5, 63, 1372, 41814, 1605215, 73586824, 3906566501, 235444126392, 15881634865780, 1185873283860557, 97147220190772317, 8665813010430379775, 836342349269443514470, 86843462603384158258103, 9655074380695222955712860, 1144404915485406530977640253, 144066096386630152751096708253, 19197014710932516253131393942286, 2699479675453423906131984772100102

Offset: 0

Paul D. Hanna, May 04 2019

nonn

			G.f.: A(x) = 1 + 5*x + 63*x^2 + 1372*x^3 + 41814*x^4 + 1605215*x^5 + 73586824*x^6 + 3906566501*x^7 + 235444126392*x^8 + 15881634865780*x^9 + ...
such that the following series are equal:
B(x) = 1 + A(x)*x + A(x)^4*x^2 + A(x)^9*x^3 + A(x)^16*x^4 + A(x)^25*x^5 + A(x)^36*x^6 + A(x)^49*x^7 + A(x)^64*x^8 + ...
B(x) = 1 + x + 6*x^2 + 84*x^3 + 1820*x^4 + 53130*x^5 + 1947792*x^6 + 85900584*x^7 + 4426165368*x^8 + 260887834350*x^9 + ... + binomial(n^2,n) * x^n + ...

Cf. A014062 (binomial(n^2,n)).

a[n_] := Module[{A = {1}}, For[i = 1, i <= n, i++, AppendTo[A, 0]; A[[-1]] = -Coefficient[Sum[x^m*(A.x^Range[0, Length[A]-1])^(m^2) - x^m* Binomial[m^2, m], {m, 0, Length[A]}], x, Length[A]]]; A[[n+1]] ];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, May 07 2019, from PARI *)

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = -polcoeff( sum(m=0,#A, x^m*Ser(A)^(m^2) - x^m*binomial(m^2,m) ), #A) );A[n+1]}
for(n=0,20,print1(a(n),", "))

A365628 a(n) = binomial(primorial(n), n).

Original entry on oeis.org

1, 2, 15, 4060, 78738660, 545754554499462, 1018081517447240182211275, 1793004475784081302284255717158418120, 1943305407393725342965469143054357602760779899437185, 3772316402417100592416011698371929155605067111502494326520988270728160

Offset: 0

Darío Clavijo, Sep 13 2023

nonn

Cf. A002110, A007318, A014062, A060604, A086687.

a(n) = binomial(vecprod(primes(n)), n); \\ Michel Marcus, Sep 14 2023

from sympy import binomial, primorial
a = lambda n: binomial(primorial(n), n)
print([a(n) for n in range(1,10)])

a(n) = binomial(A002110(n), n).

A371474 Numbers k such that binomial(k^2,k) == 0 (mod k^3).

Original entry on oeis.org

1, 2184, 6552, 12870, 13860, 19530, 23100, 33660, 40755, 47880, 51051, 58995, 81396, 88920, 101010, 113553, 114114, 114855, 121800, 125970, 136136, 141372, 142290, 142324, 145860, 150535, 154583, 157080, 158928, 164424, 171080, 180180, 193732, 195104, 197340, 214890, 225680, 229908, 230230, 230724, 238602, 243542, 249964, 253080, 257712, 267960, 284867, 291720, 294525, 297414, 300696

Offset: 1

Seiichi Manyama, Mar 24 2024

nonn

Cf. A014062, A081370.

PARI
```
isok(n) = binomial(n^2, n)%n^3==0;
```

from itertools import count, islice
from math import comb
def A371474_gen(): # generator of terms
    return filter(lambda k:not comb(k**2,k)%(k**3),count(1))
A371474_list = list(islice(A371474_gen(),3)) # Chai Wah Wu, Mar 25 2024

More terms from Vaclav Kotesovec, Mar 26 2024

cp's OEIS Frontend

A371472 Least positive number k such that binomial(k^2,k) is divisible by n^2.

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

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A227052 a(n) = (n^2)! / (n^2-n)! = number of ways of placing n labeled balls into n^2 labeled boxes with at most one ball in each box.

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A306763 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n^2) * x^n = Sum_{n>=0} binomial(n^2,n) * x^n.

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A365628 a(n) = binomial(primorial(n), n).

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A371474 Numbers k such that binomial(k^2,k) == 0 (mod k^3).

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