A006331 -id:A006331 - cp's OEIS frontend

A062862 Number of ways n can be written as the sum of squares of consecutive numbers (possibly including squares of negative numbers).

1, 4, 1, 0, 2, 4, 2, 0, 0, 2, 1, 0, 0, 2, 4, 2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 1, 2, 4, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 4, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 4, 2, 0, 0, 0, 2, 4, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 0

Henry Bottomley, Jun 25 2001

nonn

			a(25)=4 since 25 = 5^2 = 3^2+4^2 = (-4)^2+(-3)^2 = (-5)^2.

Index entries for sequences related to sums of squares

If a(n) is zero then n is in A062863, otherwise in A062861. If a(n) is odd then n is in A006331.

A108582 n appears n^3 times.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Jonathan Vos Post, Jul 25 2005

From Jonathan Vos Post, Mar 18 2006: (Start)
The key to this sequence is: 1^3 + 2^3 + 3^3 + ... + n^3 = (1+2+3+...+n)^2.
Since the last occurrence of n comes one before the first occurrence of n+1 and the former is at Sum_{i=0..n} i^3 = A000537(n) = (A000217(n))^2 = (n*(n+1)/2)^2 = (C(n+1,2))^2, have a(A000537(n)) = a((A000217(n))^2) = n and thus a(1+A000537(n)) = a(1+(A000217(n))^2) = n+1.
The current sequence is, loosely, the inverse function of the square of the triangular number sequence. (End)

Boris Putievskiy, Table of n, a(n) for n = 1..8281
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.

Cf. A000027, A000578, A002024, A072649, A074279, A000217.

Cf. A000330, A000537, A006331, A050446, A050447, A006003, A005900.

Flatten @ Table[ Table[k, {k^3}], {k, 5}] (* Giovanni Resta, Jun 17 2016 *)
a[n_]:=Ceiling[1/2 (Sqrt[8 Sqrt[n]+1]-1)]
Nmax=225; Table[a[n],{n,1,Nmax}] (* Boris Putievskiy, Jun 19 2024 *)

from sympy import integer_nthroot
def A108582(n): return (m:=integer_nthroot(k:=n<<2,4)[0])+(k>(m*(m+1))**2) # Chai Wah Wu, Nov 04 2024

a(n) = ceiling((1/2)*(sqrt(8*sqrt(n) + 1) - 1)). - Boris Putievskiy, Jun 19 2024
From Chai Wah Wu, Nov 04 2024: (Start)
a(n) = m+1 if n>(m(m+1))^2/4 and a(n) = m otherwise where m = floor((4n)^(1/4)).
More generally, for a sequence a_k(n) where n appears n^(k-1) times, a_k(n) = m+1 if n > Sum_{i=1..m} i^(k-1) and a_k(n) = m otherwise where m = floor((kn)^(1/k)).
Note that Sum_{i=1..m} i^(k-1) can be written as a k-th order polynomial of m using Faulhaber's formula. (End)

Two missing terms from Giovanni Resta, Jun 17 2016

A112524 a(n) = a(n-1) + 2*n^2 with a(1) = 1.

Original entry on oeis.org

1, 9, 27, 59, 109, 181, 279, 407, 569, 769, 1011, 1299, 1637, 2029, 2479, 2991, 3569, 4217, 4939, 5739, 6621, 7589, 8647, 9799, 11049, 12401, 13859, 15427, 17109, 18909, 20831, 22879, 25057, 27369, 29819, 32411, 35149, 38037, 41079, 44279, 47641

Offset: 1

Dennis Farr (dfarr(AT)comcast.net), Dec 13 2005

This is the total number of operations or total storage if a process first replaces a square array by an array one smaller, repeatedly down to 1 and then regrows the array to the original size.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Condensation
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006331.

[n*(n+1)*(2*n+1)/3 - 1: n in [1..40]]; // G. C. Greubel, Jan 12 2022

a[1]:=1: for n from 2 to 50 do a[n]:=a[n-1]+2*n^2 od: seq(a[n],n=1..50); # Emeric Deutsch, Feb 13 2006
a:=n->sum(k^2, k=1..n):seq(a(n)+sum(k^2, k=2..n), n=1...40); # Zerinvary Lajos, Jun 11 2008

Table[n*(n+1)*(2n+1)/3 - 1, {n, 50}] (* Stefan Steinerberger, Mar 11 2006 *)
2*Accumulate[Range[50]^2]-1 (* or *) LinearRecurrence[{4,-6,4,-1},{1,9,27,59},50] (* Harvey P. Dale, Dec 03 2012 *)

[n*(n+1)*(2*n+1)/3 - 1 for n in (1..40)] # G. C. Greubel, Jan 12 2022

Twice the sum of the first n square numbers - 1 = n*(n + 1)*(2n + 1)/3 - 1. - Stefan Steinerberger, Mar 11 2006
From R. J. Mathar, Sep 09 2008: (Start)
G.f.: x*(1 +5*x -3*x^2 +x^3)/(1-x)^4.
a(n) = A006331(n) - 1. (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(1)=1, a(2)=9, a(3)=27, a(4)=59. - Harvey P. Dale, Dec 03 2012
E.g.f.: ( 3 + (-3 + 6*x + 9*x^2 + 2*x^3)*exp(x) )/3. - G. C. Greubel, Jan 12 2022

Definition corrected by Alexandre Wajnberg, Jan 02 2006
More terms from Emeric Deutsch, Feb 13 2006
More terms from Stefan Steinerberger, Mar 11 2006

A183157 Triangle read by rows: T(n,k) is the number of partial isometries of an n-chain of height k (height of alpha = |Im(alpha)|).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 10, 2, 1, 16, 28, 12, 2, 1, 25, 60, 40, 14, 2, 1, 36, 110, 100, 54, 16, 2, 1, 49, 182, 210, 154, 70, 18, 2, 1, 64, 280, 392, 364, 224, 88, 20, 2, 1, 81, 408, 672, 756, 588, 312, 108, 22, 2, 1, 100, 570, 1080, 1428, 1344, 900, 420, 130, 24, 2

Offset: 0

Abdullahi Umar, Dec 28 2010

Rows also give the coefficients of the clique polynomial of the n X n bishop graph. - Eric W. Weisstein, Jun 04 2017

			T (3,2) = 10 because there are exactly 10 partial isometries (on a 3-chain) of height 2, namely: (1,2)-->(1,2); (1,2)-->(2,1); (1,2)-->(2,3); (1,2)-->(3,2); (2,3)-->(1,2); (2,3)-->(2,1); (2,3)-->(2,3); (2,3)-->(3,2); (1,3)-->(1,3); (1,3)-->(3,1) - the mappings are coordinate-wise.
The triangle starts
  1;
  1,    1;
  1,    4,    2;
  1,    9,   10,    2;
  1,   16,   28,   12,    2;
  1,   25,   60,   40,   14,    2;
  1,   36,  110,  100,   54,   16,    2;
  1,   49,  182,  210,  154,   70,   18,    2;
  1,   64,  280,  392,  364,  224,   88,   20,    2;
  1,   81,  408,  672,  756,  588,  312,  108,   22,    2;
  1,  100,  570, 1080, 1428, 1344,  900,  420,  130,   24,    2;

R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.2558 [math.GR], 2011.
Eric Weisstein's World of Mathematics, Bishop Graph
Eric Weisstein's World of Mathematics, Clique Polynomial

Cf. A183156 (row sums), A006331 (k=2), A008911 (k=3), A067056 (k=4).

A183157 := proc(n,k) if k =0 then 1; elif k = 1 then n^2 ; else 2*(2*n-k+1)*binomial(n,k)/(k+1) ; end if; end proc: # R. J. Mathar, Jan 06 2011

T[, 0] = 1; T[n, 1] := n^2; T[n_, k_] := 2*(2*n - k + 1)*Binomial[n, k] / (k + 1);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 25 2017 *)

T(n,0)=1, T(n,1) = n^2 and T(n,k)=2*(2*n-k+1)*binomial(n,k)/(k+1), k > 1.

A208532 Mirror image of triangle in A125185; unsigned version of A120058.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 9, 10, 4, 5, 16, 28, 24, 8, 6, 25, 60, 80, 56, 16, 7, 36, 110, 200, 216, 128, 32, 8, 49, 182, 420, 616, 560, 288, 64, 9, 64, 280, 784, 1456, 1792, 1408, 640, 128, 10, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256

Offset: 0

Philippe Deléham, Feb 27 2012

Subtriangle of the triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Equals A007318*A134309*A097806 as infinite lower triangular matrix.
Row sums are powers of 3 (A000244).
Diagonal sums are powers of 2 (A000079).

			Triangle begins :
1
2, 1
3, 4, 2
4, 9, 10, 4
5, 16, 28, 24, 8
6, 25, 60, 80, 56, 16
7, 36, 110, 200, 216, 128, 32
8, 49, 182, 420, 616, 560, 288, 64
9, 64, 280, 784, 1456, 1792, 1408, 640, 128
10, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256
Triangle (1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, ...) begins :
1
1, 0
2, 1, 0
3, 4, 2, 0
4, 9, 10, 4, 0
5, 16, 28, 24, 8, 0
6, 25, 60, 80, 56, 16, 0

Cf. Columns: A000027, A000290, A006331, A112742.

Cf. Diagonals: A011782, 2*A045623,

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-1,k-1), T(0,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1-y*x)/((1-x)*(1-(1+2*y)*x)).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A083085(n), A084567(n), A000012(n), A000027(n+1), A000244(n), A083065(n), A083076(n) for x = -3, -2, -1, 0, 1, 2, 3 respectively.

A259317 a(n) = 2(2n+1)A000538(n) - 4A000330(n)^2.

Original entry on oeis.org

0, 2, 70, 588, 2772, 9438, 26026, 61880, 131784, 257754, 471086, 814660, 1345500, 2137590, 3284946, 4904944, 7141904, 10170930, 14202006, 19484348, 26311012, 35023758, 46018170, 59749032, 76735960, 97569290, 122916222, 153527220, 190242668, 233999782

Offset: 0

N. J. A. Sloane, Jun 24 2015

			n=3: 588 = 2*7*92-4*14^2.

Colin Barker, Table of n, a(n) for n = 0..1000
J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000538, A000330, A006331, A259108, A259109, A259318.

LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,2,70,588,2772,9438,26026},30] (* Harvey P. Dale, Jul 12 2025 *)

concat(0, Vec(-2*x*(x^4+28*x^3+70*x^2+28*x+1)/(x-1)^7 + O(x^100))) \\ Colin Barker, Jun 28 2015

def A259317(n): return n*(n*(n**2*(n*(16*n + 48) + 40) - 11) - 3)//45 # Chai Wah Wu, Dec 07 2021

Also a(n) = (2*n+1)*A259108(n) - A006331(n)^2.
a(n) = (n*(1+2*n)^2*(-3+n+8*n^2+4*n^3))/45. - Colin Barker, Jun 28 2015
G.f.: -2*x*(x^4+28*x^3+70*x^2+28*x+1) / (x-1)^7. - Colin Barker, Jun 28 2015

A268434 Triangle read by rows, Lah numbers of order 2, 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)^2+k^2)*T(n-1,k), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 10, 10, 1, 0, 100, 140, 28, 1, 0, 1700, 2900, 840, 60, 1, 0, 44200, 85800, 31460, 3300, 110, 1, 0, 1635400, 3476200, 1501500, 203060, 10010, 182, 1, 0, 81770000, 185874000, 90563200, 14700400, 943800, 25480, 280, 1

Offset: 0

Peter Luschny, Mar 07 2016

0

			[1]
[0,        1]
[0,        2,         1]
[0,       10,        10,        1]
[0,      100,       140,       28,        1]
[0,     1700,      2900,      840,       60,      1]
[0,    44200,     85800,    31460,     3300,    110,     1]
[0,  1635400,   3476200,  1501500,   203060,  10010,   182,   1]

Peter Luschny, The P-transform.

Cf. A038207 (order 0), A111596 (order 1), A269946 (order 3).

Cf. A036969, A105278, A204579, A269938, A269941, A269944, A269945.

T := proc(n,k) option remember;
if n=k then return 1 fi; if k<0 or k>n then return 0 fi;
T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k) end:
seq(seq(T(n,k), k=0..n), n=0..8);
# Alternatively with the P-transform (cf. A269941):
A268434_row := n -> PTrans(n, n->`if`(n=1,1, ((n-1)^2+1)/(n*(4*n-2))),
(n,k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A268434_row(n)), n=0..8);

T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0 < k < n := T[n, k] = T[n-1, k-1] + ((n-1)^2 + k^2)*T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

#cached_function
def T(n, k):
    if n==k: return 1
    if k<0 or k>n: return 0
    return T(n-1, k-1)+((n-1)^2+k^2)*T(n-1, k)
for n in range(8): print([T(n, k) for k in (0..n)])
# Alternatively with the function PtransMatrix (cf. A269941):
PtransMatrix(8, lambda n: 1 if n==1 else ((n-1)^2+1)/(n*(4*n-2)), lambda n, k: (-1)^k*factorial(2*n)/factorial(2*k))

T(n,k) = (-1)^k*((2*n)!/(2*k)!)*P[n,k](s(n)) where P is the P-transform and s(n) = ((n-1)^2+1)/(n*(4*n-2)). The P-transform is defined in the link. Compare also the Sage and Maple implementations below.
T(n,k) = Sum_{j=k..n} A269944(n,j)*A269945(j,k).
T(n,1) = Product_{k=1..n} (k-1)^2+1 for n>=1 (cf. A101686).
T(n,n-1) = (n-1)*n*(2*n-1)/3 for n>=1 (cf. A006331).
Row sums: A269938.

A300758 a(n) = 2n(n+1)(2n+1).

Original entry on oeis.org

0, 12, 60, 168, 360, 660, 1092, 1680, 2448, 3420, 4620, 6072, 7800, 9828, 12180, 14880, 17952, 21420, 25308, 29640, 34440, 39732, 45540, 51888, 58800, 66300, 74412, 83160, 92568, 102660, 113460, 124992, 137280, 150348, 164220, 178920, 194472, 210900, 228228

Offset: 0

Christopher Purcell, Mar 12 2018

The altitude h(n) = a(n)/A001844(n) of the (A005408(n), A046092(n) and A001844(n)) rectangular triangle is an irreducible fraction. - Ralf Steiner, Feb 25 2020

In this case, area A = a(n)/2 = A055112(n). - Bernard Schott, Feb 27 2020

S. P. Borgatti and M. G. Everett, Notions of Position in Social Network Analysis, Sociological Methodology, 22 (1992), 1-35.
C. Purcell and P. Rombach, Role colouring graphs in hereditary classes, arXiv:1802.10180 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A002492, A006331, A055112, A059270, A005408, A046092, A001844, A000384, A016754, A060300.

Cf. A027480.

a(n) = 12*A000330(n).
G.f.: 12*x*(1+x)/(1-x)^4. - Colin Barker, Mar 12 2018
a(n) = 6*A006331(n) = 4*A059270(n) = 3*A002492(n) = 2*A055112(n). - Omar E. Pol, Apr 04 2018
From Ralf Steiner, Feb 27 2020: (Start)
a(n) = 2*n*A000384(n+1).
a(n) = sqrt(A016754(n)*A060300(n)).
(End)
a(n) = A005408(n) * A046092(n). - Bruce J. Nicholson, Apr 24 2020

Edited by N. J. A. Sloane, Aug 01 2019

A368045 Triangle read by rows. T(n, k) = (k(k + 1)(2k + 1) + n(n + 1)(2n + 1)) / 6.

Original entry on oeis.org

0, 1, 2, 5, 6, 10, 14, 15, 19, 28, 30, 31, 35, 44, 60, 55, 56, 60, 69, 85, 110, 91, 92, 96, 105, 121, 146, 182, 140, 141, 145, 154, 170, 195, 231, 280, 204, 205, 209, 218, 234, 259, 295, 344, 408, 285, 286, 290, 299, 315, 340, 376, 425, 489, 570

Offset: 0

Peter Luschny, Dec 09 2023

Consider a sequence-to-triangle transformation a -> T, where a is a 0-based sequence and T a regular (0, 0)-based triangular array. The transformation is recursively defined, starting with T(0, 0) = 0, and T(n, n) = a(n) + T(n, n - 1) for n > 0. For k <> n let T(n, k) = a(n) + T(n-1, k).
If a(n) = 1, then T = A051162; if a(n) = n, then T = A367964 (generalizing the triangular numbers); if a(n) = n^2, then T is this triangle.
In the multiplicative form of the transformation, T(0, 0) is set to 1, and the operation '+' is replaced by '*'. For instance, a(n) = 2 is then mapped to T = A368043 and a(n) = n to A143216.

			Triangle T(n, k) starts:
  [0] [  0]
  [1] [  1,   2]
  [2] [  5,   6,  10]
  [3] [ 14,  15,  19,  28]
  [4] [ 30,  31,  35,  44,  60]
  [5] [ 55,  56,  60,  69,  85, 110]
  [6] [ 91,  92,  96, 105, 121, 146, 182]
  [7] [140, 141, 145, 154, 170, 195, 231, 280]
  [8] [204, 205, 209, 218, 234, 259, 295, 344, 408]
  [9] [285, 286, 290, 299, 315, 340, 376, 425, 489, 570]

Cf. A000330 (T(n,0)), A056520 (T(n,1)), A005900 (T(n-1,n)), A006331 (T(n,n)), A094952 (T(2*n,n)), A368046 (row sums), A368047 (alternating row sums).
Cf. A051162 (transform of n^0), A367964 (transform of n^1), this sequence (transform of n^2).
Cf. A368043, A143216.

Module[{n=1},NestList[Append[#+n^2,Last[#]+2(n++^2)]&,{0},10]] (* or *)
Table[(k(k+1)(2k+1)+n(n+1)(2n+1))/6,{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 10 2023 *)

from functools import cache
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [0]
    row = Trow(n - 1) + [0]
    for k in range(n): row[k] += n * n
    row[n] = row[n - 1] + n * n
    return row
print([k for n in range(10) for k in Trow(n)])

T(n, k) = A000330(k) + A000330(n).

A181773 Molecular topological indices of the cocktail party graphs.

Original entry on oeis.org

0, 48, 240, 672, 1440, 2640, 4368, 6720, 9792, 13680, 18480, 24288, 31200, 39312, 48720, 59520, 71808, 85680, 101232, 118560, 137760, 158928, 182160, 207552, 235200, 265200, 297648, 332640, 370272, 410640

Offset: 1

Eric W. Weisstein, Jul 10 2011

a(n) is the number of 2 X 2 matrices (all four elements distinct) having entries in {-n,...,0,...,n} with determinant equal to the permanent. - Indranil Ghosh, Dec 25 2016

Eric Weisstein's World of Mathematics, Molecular Topological Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A280059 (2 X 2 matrices, elements can be repeated).

A181773:=n->8*(n-1)*n*(2*n-1): seq(A181773(n), n=1..50); # Wesley Ivan Hurt, Apr 11 2017

a(n) = 8*(n-1)*n*(2n-1).
a(n) = 16*A059270(n-1).
G.f.: 48*x^2*(x+1)/(x-1)^4. - Colin Barker, Oct 17 2012
a(n) = 48*A000330(n-1). - R. J. Mathar, Jan 04 2017
From Omar E. Pol, Jan 05 2017: (Start)
a(n) = 24*A006331(n-1) = 12*A002492(n-1) = 8*A055112(n-1).
a(n) = 2*A069074(n-2), n >= 2. (End)

cp's OEIS Frontend

A062862 Number of ways n can be written as the sum of squares of consecutive numbers (possibly including squares of negative numbers).

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A108582 n appears n^3 times.

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A112524 a(n) = a(n-1) + 2*n^2 with a(1) = 1.

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A183157 Triangle read by rows: T(n,k) is the number of partial isometries of an n-chain of height k (height of alpha = |Im(alpha)|).

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A208532 Mirror image of triangle in A125185; unsigned version of A120058.

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A259317 a(n) = 2*(2*n+1)*A000538(n) - 4*A000330(n)^2.

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A268434 Triangle read by rows, Lah numbers of order 2, 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)^2+k^2)*T(n-1,k), for n>=0 and 0<=k<=n.

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Sage

Formula

A300758 a(n) = 2n*(n+1)*(2n+1).

A259317 a(n) = 2(2n+1)A000538(n) - 4A000330(n)^2.

A300758 a(n) = 2n(n+1)(2n+1).

A368045 Triangle read by rows. T(n, k) = (k(k + 1)(2k + 1) + n(n + 1)(2n + 1)) / 6.