cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A257462 Number A(n,k) of factorizations of m^n into n factors, where m is a product of exactly k distinct primes and each factor is a product of k primes (counted with multiplicity); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 10, 10, 3, 1, 1, 1, 1, 26, 70, 25, 3, 1, 1, 1, 1, 71, 566, 465, 49, 4, 1, 1, 1, 1, 197, 4781, 11131, 2505, 103, 4, 1, 1, 1, 1, 554, 41357, 297381, 190131, 12652, 184, 5, 1, 1, 1, 1, 1570, 364470, 8349223, 16669641, 2928876, 57232, 331, 5, 1, 1
Offset: 0

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Author

Alois P. Heinz, Apr 24 2015

Keywords

Comments

Also number of ways to partition the multiset consisting of n copies each of 1, 2, ..., k into n multisets of size k.

Examples

			A(4,2) = 3: (2*3)^4 = 1296 = 6*6*6*6 = 9*6*6*4 = 9*9*4*4.
A(3,3) = 10: (2*3*5)^3 = 2700 = 30*30*30 = 45*30*20 = 50*27*20 = 50*30*18 = 50*45*12 = 75*20*18 = 75*30*12 = 75*45*8 = 125*18*12 = 125*27*8.
A(2,4) = 10: (2*3*5*7)^2 = 44100 = 210*210 = 225*196 = 294*150 = 315*140 = 350*126 = 441*100 = 490*90 = 525*84 = 735*60 = 1225*36.
Square array A(n,k) begins:
  1, 1, 1,  1,    1,      1, ...
  1, 1, 1,  1,    1,      1, ...
  1, 1, 2,  4,   10,     26, ...
  1, 1, 2, 10,   70,    566, ...
  1, 1, 3, 25,  465,  11131, ...
  1, 1, 3, 49, 2505, 190131, ...

Links

Crossrefs

Columns k=0+1, 2-5 give: A000012, A008619, A254233, A257114, A257518.
Rows n=0+1, 2-3 give: A000012, A257520, A333902.
Main diagonal gives A334286.
Cf. A257463, A333901 (ordered factorizations).

Programs

  • Maple
    with(numtheory):
b:= proc(n, i, k) option remember; `if`(n=1, 1,
      add(`if`(d>i or bigomega(d)<>k, 0,
      b(n/d, d, k)), d=divisors(n) minus {1}))
    end:
A:= (n, k)-> b(mul(ithprime(i), i=1..k)^n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..8);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 1, 1, Sum[If[d > i || PrimeOmega[d] != k, 0, b[n/d, d, k]], {d, Divisors[n] // Rest}]]; A[n_, k_] := Module[ {p = Product[Prime[i], {i, 1, k}]^n}, b[p, p, k]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A377063 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero up to permutations of rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 10, 6, 3, 1, 1, 1, 1, 26, 30, 12, 3, 1, 1, 1, 1, 71, 166, 117, 18, 4, 1, 1, 1, 1, 197, 981, 1421, 345, 30, 4, 1, 1, 1, 1, 554, 5937, 20326, 9691, 1042, 42, 5, 1, 1, 1, 1, 1570, 36646, 307063, 336596, 63076, 2746, 63, 5, 1, 1
Offset: 0

Views

Author

Andrew Howroyd, Oct 14 2024

Keywords

Comments

Columns are not permutable.
Equivalently, the number of n X k 0..2 arrays with row sums k and column sums n up to permutations of rows.

Examples

			Array begins:
===================================================
n\k | 0 1 2  3    4      5        6           7 ...
----+----------------------------------------------
  0 | 1 1 1  1    1      1        1           1 ...
  1 | 1 1 1  1    1      1        1           1 ...
  2 | 1 1 2  4   10     26       71         197 ...
  3 | 1 1 2  6   30    166      981        5937 ...
  4 | 1 1 3 12  117   1421    20326      307063 ...
  5 | 1 1 3 18  345   9691   336596    12650093 ...
  6 | 1 1 4 30 1042  63076  5328136   506525279 ...
  7 | 1 1 4 42 2746 369036 76292516 18490880339 ...
  ...

Links

Crossrefs

Main diagonal is A377064.
Rows n=0..4 are A000012, A000012, A257520, A377065, A377066.
Columns k=0..4 are A000012, A000012, A008619, A377067, A377068.
Cf. A334549.

A383527 Partial sums of A005773.

Original entry on oeis.org

1, 2, 4, 9, 22, 57, 153, 420, 1170, 3293, 9339, 26642, 76363, 219728, 634312, 1836229, 5328346, 15494125, 45137995, 131712826, 384900937, 1126265986, 3299509114, 9676690939, 28407473191, 83470059532, 245465090758, 722406781935, 2127562036990, 6270020029353
Offset: 0

Views

Author

Mélika Tebni, Apr 29 2025

Keywords

Comments

For p prime of the form 4*k+3 (A002145), a(p) == 0 (mod p).
For p Pythagorean prime (A002144), a(p) - 2 == 0 (mod p).
a(n) (mod 2) = A010059(n).
a(A000069(n+1)) is even.
a(A001969(n+1)) is odd.

Links

Crossrefs

Cf. A000069, A001969, A002144, A002145, A005773, A005774, A005775, A010059, A097893, A122896, A167630, A210736, A211278, A257520.

Programs

  • Maple
    gf := (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)):
a := n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n = 0 .. 29);
# Recurrence:
a:= proc(n) option remember; `if`(n<=2, 2^n, 3*a(n-1) - (6/n-1)*a(n-2) + (6/n-3)*a(n-3)) end:
seq(a(n), n = 0 .. 29);
  • Mathematica
    Module[{a, n}, RecurrenceTable[{a[n] == 3*a[n-1] - (6-n)*a[n-2]/n + 3*(2-n)*a[n-3]/n, a[0] == 1, a[1] == 2, a[2] == 4}, a, {n, 0, 30}]] (* Paolo Xausa, May 05 2025 *)
  • Python
    from math import comb as C
def a(n):
  return sum(C(n, k)*abs(sum((-1)**j*C(k, j) for j in range(k//2 + 1))) for k in range(n + 1))
print([a(n) for n in range(30)])

Formula

First differences of A211278.
a(n) = Sum_{k=0..n} A167630(n, k).
Binomial transform of A210736 (see Python program).
G.f.: (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)).
E.g.f.: (Integral_{x=-oo..oo} BesselI(0,2*x) dx + (1 + BesselI(0,2*x)) / 2)*exp(x).
Recurrence: n*a(n) = 3*n*a(n-1) - (6-n)*a(n-2) + 3*(2-n)*a(n-3). If n <= 2, a(n) = 2^n.
a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, May 02 2025
From Mélika Tebni, May 09 2025: (Start)
a(n) = A257520(n) + A097893(n-1) for n > 0.
a(n) = Sum_{j=0..n}(Sum_{k=0..j} A122896(j, k)).
a(n+2) - 3*a(n+1) + 2*a(n) = A005774(n).
a(n+2) - 4*a(n+1) + 4*a(n) - a(n-1) = A005775(n) for n >= 3. (End)

A379893 Triangle read by rows: T(n,k) is the number of standard Young tableaux with shapes in {lambda = (lambda_1,lambda_2,...) | lambda_1-lambda_2=k, lambda_i<=1 for i>=3, |lambda| = n}, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 3, 3, 3, 0, 1, 6, 9, 6, 4, 0, 1, 15, 21, 19, 10, 5, 0, 1, 36, 55, 50, 34, 15, 6, 0, 1, 91, 141, 139, 99, 55, 21, 7, 0, 1, 232, 371, 379, 293, 175, 83, 28, 8, 0, 1, 603, 982, 1043, 847, 551, 286, 119, 36, 9, 0, 1, 1585, 2628, 2872, 2441, 1684, 956, 441, 164, 45, 10, 0, 1
Offset: 0

Views

Author

Xiaomei Chen, Jan 05 2025

Keywords

Examples

			Triangle begins:
  [0]   1;
  [1]   0,   1;
  [2]   1,   0,   1;
  [3]   1,   2,   0,  1;
  [4]   3,   3,   3,  0,  1;
  [5]   6,   9,   6,  4,  0,  1;
  [6]  15,  21,  19, 10,  5,  0, 1;
  [7]  36,  55,  50, 34, 15,  6, 0, 1;
  [8]  91, 141, 139, 99, 55, 21, 7, 0, 1;
  ...

Links

Crossrefs

Row sums give A257520.
Column 1 gives A005043.

Programs

  • Sage
    def A379893_triangel(dim):
    M = matrix(ZZ, dim, dim)
    for n in range(dim):
        for k in range(n+1):
            for i in range(math.floor((n-k-1)/2)+1):
                for j in range(n-k-1-2*i+1):
                    if ((n+k-1-j)%2)==0:
                        M[n,k]=M[n, k]+(2*k+2)/(n+k+1-2*i-j)*binomial(n-2*i-2,j)*binomial(n-2*i-j-1,(n+k-j-1)/2-i)
            M[n,k]=M[n,k]-pow(-1,n+k+1)
    return M

Formula

T(n,k) = (-1)^(n+k) + Sum_{i=0..(n-k-1)/2} Sum_{j=0..n-k-1-2*i, j==n+k-1 (mod 2)} (2*k+2) / (n+k+1-2*i-j) * binomial(n-2*i-2,j) * binomial(n-2*i-j-1,(n+k-j-1)/2-i).
T(n+1,2*k-1) + T(n,2*k-1) = A379838(n+1,k) - A379838(n,k).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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