A052928 -id:A052928 - cp's OEIS frontend

A333119 Triangle T read by rows: T(n, k) = (n - k)(1 - (-1)^k + 2k)/4, with 0 <= k < n.

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

Offset: 1

Stefano Spezia, Mar 08 2020

T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A332566(n).
The h-th subdiagonal of the triangle T gives 0 followed by the multiples of h+1 repeated.
For k > 0, the (2*k-1)-th and (2*k)-th columns of the triangle T give the multiples of k.

			n\k| 0 1 2 3 4 5
---+------------
1  | 0
2  | 0 1
3  | 0 2 1
4  | 0 3 2 2
5  | 0 4 3 4 2
6  | 0 5 4 6 4 3
...
For n = 4 the matrix M(4) is
      0 1 1 2
      1 0 1 1
      1 1 0 1
      2 1 1 0
and therefore T(4, 0) = 0, T(4, 1) = 3, T(4, 2) = 2 and T(4, 3) = 2.

Cf. A332566.

Cf. A000004: 1st column; A000027: 2nd and 3rd column; A004526: diagonal; A005843: 4th and 5th column; A052928: 1st subdiagonal; A168237: 2nd subdiagonal; A168273: 3rd subdiagonal; A173196: row sums.

T[n_,k_]:=(n-k)(1-(-1)^k+2k)/4; Flatten[Table[T[n,k],{n,1,12},{k,0,n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[y*(x*(2 + y + y^2) - (1 + y + 2*y^2))/((1 - x)^2 *(1 - y)^3 (1 + y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 12]]

O.g.f.: y*(x*(2 + y + y^2) - (1 + y + 2*y^2))/((1 - x)^2*(1 - y)^3*(1 + y)^2).
T(n, k) = k*(n - k)/2 for k even.
T(n, k) = (1 + k)*(n - k)/2 for k odd.

A372602 The maximal exponent in the prime factorization of the largest square dividing n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, 2, 0, 2, 2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0

Offset: 1

Amiram Eldar, May 07 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008833, A051903, A052928.

Similar sequences: A007424, A368781, A372601, A372603, A372604.

f[n_] := 2 * Floor[n/2]; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = n \ 2 * 2;
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A008833(n)).
a(n) = A052928(A051903(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * Sum_{i>=1} (1 - (1/zeta(2*i))) = 0.98112786070359477197... .

A128217 Nonnegative integers n such that the square-root of n differs from its nearest integer by less than 1/4.

Original entry on oeis.org

0, 1, 4, 5, 8, 9, 10, 15, 16, 17, 18, 23, 24, 25, 26, 27, 34, 35, 36, 37, 38, 39, 46, 47, 48, 49, 50, 51, 52, 61, 62, 63, 64, 65, 66, 67, 68, 77, 78, 79, 80, 81, 82, 83, 84, 85, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125

Offset: 1

John W. Layman, Feb 19 2007

nonn

The squares are a subsequence; apparently A052928(n-1) = number of terms between (n-1)^2 and n^2. - Reinhard Zumkeller, Jun 20 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A063656. See the first differences in A128218.

Cf. A052928, A000290, A063657.

a128217 n = a128217_list !! (n-1)
a128217_list = filter f [0..] where
   f x = 4 * abs (root - fromIntegral (round root)) < 1
         where root = sqrt $ fromIntegral x
-- Reinhard Zumkeller, Jun 20 2015

nsrQ[n_]:=Module[{sr=Sqrt[n]},Abs[First[sr-Nearest[{Floor[sr], Ceiling[sr]},sr]]]<1/4]; Select[Range[0,150],nsrQ] (* Harvey P. Dale, Aug 19 2011 *)

from itertools import count, islice
from math import isqrt
def A128217_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:(m:=n<<4)<(k:=(isqrt(n)<<2)+1)**2 or m>(k+2)**2, count(max(startvalue,0)))
A128217_list = list(islice(A128217_gen(),40)) # Chai Wah Wu, Jun 06 2025

Offset changed by Reinhard Zumkeller, Jun 20 2015

A142961 Irregular triangle read by rows: coefficients of polynomials related to a family of convolutions of certain central binomial sequences.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, -2, 1, 30, 35, -10, 5, 70, 63, 8, -2, -75, 35, 315, 231, 56, -14, -245, 105, 693, 429, -272, 36, 2268, -525, -5880, 2310, 12012, 6435, -2448, 324, 9660, -2037, -16632, 6006, 25740, 12155, 3968, -304, -31260, 3840, 73395, -14091, -90090, 30030, 109395, 46189, 43648, -3344

Offset: 0

Wolfdieter Lang, Sep 15 2008

The row length sequence {r(k)} of this irregular triangle a(k, p) is given by r(0) = 1 = r(1) and r(k) = 2*(floor(k/2)) = A052928(k), k >= 2. This is {1,1,2,2,4,4,6,6,8,8,10,10,...}.
The array of the k-family of convolutions Sigma(k, n) := Sum_{p=0..n} p^k * binomial(2*p, p) * binomial(2*(n-p), n-p) can be written as Sigma(k, n) = ((4^n)*c(k, n) / A046161(k)) * Sum_{p=0..r(k)-1} a(k, p)*n^p, where c(k, n) = n for even k >= 2 and c(k, n) = n^2 for odd k >= 3, with c(0, n) = 1, c(1, n).
The author was led to compute such sums by a question asked by M. Greiter, Jun 27 2008.

			The irregular triangle a(k, p) begins:
k\p  0  1   2   3   4   5 ...
0:   1
1:   1
2:   1  3
3:   3  5
4:   2 -1  30  35
5: -10  5  70  63
6:   8 -2 -75  35 315 231
...
k=3: Sigma(3, n) = Sum_{p=0..n} p^3 * binomial(2*p, p) * binomial(2*(n-p), n-p) = (4*n/16)*n^2*(3 + 5*n), for n >= 0. This is the sequence {0, 2, 52, 648, 5888, 44800, 304128, 1906688, 11272192, 63700992, ...}.

Wolfdieter Lang, First eleven rows and more

Cf. A001147, A046161, A048993, A052928, A001147.

a(k, p)= [n^p] P(k, n) where c(k, n)*P(k, n) = A046161(k)*Sigma(k, n)/(4^n), with c(k, n) and the array Sigma(k, n) given above. A046161(k) are the denominators of binomial(2*k, k)/4^k: [1, 2, 8, 16, 128, 256, 1024, 2048, 32768, 65536, 262144,...].

Sigma(k, n)/4^n = Sum_{p=0..min(n, k)} binomial(n, p)*(2*p -1)!!*S2(k, p)/2^p, with the double factorials (2*p -1)!!= A001147(p), with (-1)!! := 1, and the Stirling numbers of the second kind S2(k, p):=A048993(k, p). (Proof from the product of the o.g.f.s and the normal ordering (x^d_x)^k = Sum_{p=0..k} (S2(k, p)*x^p*d_x^p), with the derivative operator d_x.)

Name changed, edited and corrected by Wolfdieter Lang, Aug 23 2019

A262666 Irregular table read by rows: T(n,k) is the number of binary bisymmetric n X n matrices with exactly k 1's; n>=0, 0<=k<=n^2.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 0, 4, 0, 8, 0, 12, 0, 14, 0, 12, 0, 8, 0, 4, 0, 1, 1, 1, 4, 4, 10, 10, 20, 20, 31, 31, 40, 40, 44, 44, 40, 40, 31, 31, 20, 20, 10, 10, 4, 4, 1, 1, 1, 0, 6, 0, 21, 0, 56, 0, 120, 0, 216, 0, 336, 0, 456, 0

Offset: 0

Kival Ngaokrajang, Sep 26 2015

T(n,k) = 0 if n is even and k is odd.

T(n,k) = T(n,k+1) if n is odd and k is even.

			Irregular table begins:
n\k 0   1   2   3   4   5   6   7   8   9   ...
0:  1
1:  1   1
2:  1   0   2   0   1
3:  1   1   2   2   2   2   2   2   1   1
4:  1   0   4   0   8   0  12   0  14   0   ...
5:  1   1   4   4  10  10  20  20  31  31   ...
...

Alois P. Heinz, Rows n = 0..32, flattened
Kival Ngaokrajang, Illustration of initial terms
Dennis P. Walsh, Notes on binary bisymmetric matrices
Wikipedia, Bisymmetric Matrix

Row sums give A060656(n+1).
Columns k=0-3 give: A000012, A000035, A052928, A237420(n+1).
Cf. A184948, A261242.

T:= n-> seq(coeff((t->(1+x^2)^(n-t)*(1+x)^t*(1+x^4)^
      (((n-2)*n+t)/4))(irem(n, 2)), x, i), i=0..n^2):
seq(T(n), n=0..6);  # Alois P. Heinz, Sep 27 2015

G.f. for row n: (1+x)^t*(1+x^2)^(n-t)*(1+x^4)^(((n-2)*n+t)/4) where t = n mod 2. - Alois P. Heinz, Sep 27 2015

More terms from Alois P. Heinz, Sep 27 2015

A268819 Column 32769 of array A269158: a(n) = F(n,65537), function F as defined in A269158.

Original entry on oeis.org

0, 98305, 3, 0, 6, 98306, 2, 98305, 12, 98311, 14, 3, 1, 98307, 9, 0, 24, 98317, 24, 6, 16, 98319, 27, 98306, 0, 98304, 23, 2, 30, 98312, 2, 98305, 48, 98329, 0, 12, 52, 98329, 6, 98311, 3, 98321, 3, 14, 14, 98330, 3, 3, 41, 98305, 43, 1, 4, 98326, 45, 98307, 6, 98335, 43, 9, 27, 98307, 19, 0, 27, 98353, 2, 24, 100, 98305, 1

Offset: 1

Antti Karttunen, Feb 25 2016

Terms a(1) .. a(65536) occur as column 32769 in arrays A268728 and A269158.

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A010873, A052928, A165471, A165472.
Cf. arrays A268728 and A269158.
Cf. A269157 (indices of zeros).

;; Two variants, both give same results in range n=1..65536:
(define (A268819 n) (A268728auxbi n 65537))
(define (A268819 n) (A269158auxbi n 65537))

a(n) = F(n,65537) = A269158(n,32769), function F as defined in A269158.
Other identities. For n = 1..65536:
a(n) = A268728(n,32769).
A165471(n) = 1 - A010873(A052928(a(n))).

A275730 Square array A(n,d): overwrite with zero the digit at position d from right (indicating radix d+2) in the factorial base representation of n, then convert back to decimal, read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), etc.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 0, 1, 0, 2, 0, 1, 2, 1, 4, 0, 1, 2, 3, 0, 4, 0, 1, 2, 3, 4, 1, 6, 0, 1, 2, 3, 4, 5, 6, 6, 0, 1, 2, 3, 4, 5, 0, 7, 8, 0, 1, 2, 3, 4, 5, 6, 1, 6, 8, 0, 1, 2, 3, 4, 5, 6, 7, 2, 7, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 3, 6, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 4, 7, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 12, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 13, 14

Offset: 0

Antti Karttunen, Aug 08 2016

			Columns 0-4 of rows 0 - 24 of the array:
  0, 0, 0, 0, 0, ... [No matter which digit of zero we clear, it stays zero forever]
  0, 1, 1, 1, 1  ... [When clearing the least significant digit (pos. 0) of one, "1", we get zero, and clearing any other digit past the most significant digit keeps one as one]
2, 0, 2, 2, 2, ... [Clearing the least significant digit of 2, "10", doesn't affect it, but clearing the digit-1 zeros the whole number].
  2, 1, 3, 3, 3, ... [Clearing the least significant factorial base digit of 3 ("11") gives "10", 2, clearing the digit-1 gives "01" = 1, and clearing any digit past the most significant keeps "11" as it is, 3].
  4, 0, 4, 4, 4
  4, 1, 5, 5, 5
  6, 6, 0, 6, 6
  6, 7, 1, 7, 7
  8, 6, 2, 8, 8
  8, 7, 3, 9, 9
  10, 6, 4, 10, 10
  10, 7, 5, 11, 11
  12, 12, 0, 12, 12
  12, 13, 1, 13, 13
  14, 12, 2, 14, 14
  14, 13, 3, 15, 15
  16, 12, 4, 16, 16
  16, 13, 5, 17, 17
  18, 18, 0, 18, 18
  18, 19, 1, 19, 19
  20, 18, 2, 20, 20
  20, 19, 3, 21, 21
  22, 18, 4, 22, 22
  22, 19, 5, 23, 23
  24, 24, 24, 0, 24
  ...

Transpose: A275731.
Column 0: A052928, Main diagonal: A001477.
Cf. A084558, A257687.
Can be used when computing A275732 and A275736.

(define (A275730 n) (A275730bi (A002262 n) (A025581 n)))
(define (A275730bi n c) (let loop ((z 0) (n n) (m 2) (f 1) (c c)) (let ((d (modulo n m))) (cond ((zero? n) z) ((zero? c) (loop z (/ (- n d) m) (+ 1 m) (* f m) (- c 1))) (else (loop (+ z (* f d)) (/ (- n d) m) (+ 1 m) (* f m) (- c 1)))))))

Other identities:
For all n >= 1, A(n,A084558(n)-1) = A257687(n).
For all n >= 0, A(n,A084558(n)) = n.

A316354 Triangle read by rows: T(1,1)=1, T(n,k) = T(n,k+1)+T(n-k,max(2*floor(k/2)-1,1)) and T(n,k) = 0 if k > A316355(n).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 7, 3, 1, 13, 6, 2, 24, 11, 4, 45, 21, 8, 1, 84, 39, 15, 2, 1, 156, 72, 27, 3, 1, 291, 135, 51, 6, 2, 543, 252, 96, 12, 4, 1013, 470, 179, 23, 8, 1889, 876, 333, 42, 15, 3524, 1635, 622, 79, 28, 1, 6575, 3051, 1162, 149, 53, 2, 1, 12266, 5691

Offset: 1

Seiichi Manyama, Jun 30 2018

			  n\k |   1   2   3  4  5
  ----+-------------------
   1  |   1;
   2  |   1;
   3  |   2,  1;
   4  |   4,  2,  1;
   5  |   7,  3,  1;
   6  |  13,  6,  2;
   7  |  24, 11,  4;
   8  |  45, 21,  8, 1;
   9  |  84, 39, 15, 2, 1;
  10  | 156, 72, 27, 3, 1;

Seiichi Manyama, Rows n = 1..402, flattened

Columns 1,2 give A224704, A316356 (for n>0).

Cf. A052928, A288267, A316355.

A346663 The number of nonreal roots of Sum_{k=0..n} prime(k+1)*x^k.

Original entry on oeis.org

0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 62, 62, 64, 64, 66, 66, 68, 68, 70, 70, 72, 72

Offset: 0

W. Edwin Clark, Jul 27 2021

nonn

This agrees with A052928 until n = 2436. a(2436) = 2434 and A052928(2436) = 2436. For n >= 2436 we have a(n) = n-2 for n even and a(n) = n - 1 for n odd, for an as yet undetermined number of n. A052928(n) = n - (n mod 2) for all n. See the paper by W. Clark and M. Shattuck linked to below.

W. Edwin Clark and Mark Shattuck, The Integer Sequence Transform a --> b where b_n is the Number of Real Roots of the Polynomial a_0 + a_1 x + a_2 x^2 + ... + a_n x^n, arXiv:2107.05572 [math.CO], 2021.
Index entries for sequences which agree for a long time but are different

Cf. A052928, A346442, A346443, A346444, A346445.

A360764 Number T(n,k) of sets of nonempty strict integer partitions with a total of k parts and total sum of n; triangle T(n,k), n>=0, 0<=k<=max(i:T(n,i)>0), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 4, 2, 0, 1, 4, 6, 1, 0, 1, 6, 8, 4, 0, 1, 6, 13, 9, 1, 0, 1, 8, 18, 16, 6, 0, 1, 8, 24, 29, 13, 2, 0, 1, 10, 30, 43, 29, 6, 0, 1, 10, 39, 64, 52, 19, 1, 0, 1, 12, 46, 89, 89, 42, 7, 0, 1, 12, 56, 122, 139, 85, 22, 1

Offset: 0

Alois P. Heinz, Feb 19 2023

T(n,k) is defined for all n >= 0 and k >= 0. Terms that are not in the triangle are zero.

			T(6,1) = 1: {[6]}.
T(6,2) = 4: {[1],[5]}, {[2],[4]}, {[1,5]}, {[2,4]}.
T(6,3) = 6: {[1,2,3]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[3],[1,2]}, {[1],[2],[3]}.
T(6,4) = 1: {[1],[2],[1,2]}.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1;
  0, 1,  2;
  0, 1,  2,  1;
  0, 1,  4,  2;
  0, 1,  4,  6,  1;
  0, 1,  6,  8,  4;
  0, 1,  6, 13,  9,  1;
  0, 1,  8, 18, 16,  6;
  0, 1,  8, 24, 29, 13,  2;
  0, 1, 10, 30, 43, 29,  6;
  0, 1, 10, 39, 64, 52, 19, 1;
  ...

Alois P. Heinz, Rows n = 0..250, flattened

Columns k=0-2 give: A000007, A057427, A052928(n-1) for n>=3.
Row sums give A050342.
Cf. A000009, A008289, A055884, A330462, A360742, A360763.

h:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i-1)))))
    end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i), k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);

h[n_, i_] := h[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, h[n, i - 1] + x*h[n - i, Min[n - i, i - 1]]]]];
g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i<0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Coefficient[h[n, n], x, i], k], {k, 0, j}]]]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]] ;
T[n_] := CoefficientList[b[n, n], x];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Nov 17 2023, after Alois P. Heinz *)

cp's OEIS Frontend

A333119 Triangle T read by rows: T(n, k) = (n - k)*(1 - (-1)^k + 2*k)/4, with 0 <= k < n.

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A372602 The maximal exponent in the prime factorization of the largest square dividing n.

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A128217 Nonnegative integers n such that the square-root of n differs from its nearest integer by less than 1/4.

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A142961 Irregular triangle read by rows: coefficients of polynomials related to a family of convolutions of certain central binomial sequences.

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A262666 Irregular table read by rows: T(n,k) is the number of binary bisymmetric n X n matrices with exactly k 1's; n>=0, 0<=k<=n^2.

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A268819 Column 32769 of array A269158: a(n) = F(n,65537), function F as defined in A269158.

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A275730 Square array A(n,d): overwrite with zero the digit at position d from right (indicating radix d+2) in the factorial base representation of n, then convert back to decimal, read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), etc.

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A316354 Triangle read by rows: T(1,1)=1, T(n,k) = T(n,k+1)+T(n-k,max(2*floor(k/2)-1,1)) and T(n,k) = 0 if k > A316355(n).

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A333119 Triangle T read by rows: T(n, k) = (n - k)(1 - (-1)^k + 2k)/4, with 0 <= k < n.