A027480 -id:A027480 - cp's OEIS frontend

A061579 Reverse one number (0), then two numbers (2,1), then three (5,4,3), then four (9,8,7,6), etc.

0, 2, 1, 5, 4, 3, 9, 8, 7, 6, 14, 13, 12, 11, 10, 20, 19, 18, 17, 16, 15, 27, 26, 25, 24, 23, 22, 21, 35, 34, 33, 32, 31, 30, 29, 28, 44, 43, 42, 41, 40, 39, 38, 37, 36, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66

Offset: 0

Henry Bottomley, May 21 2001

A self-inverse permutation of the nonnegative numbers.
a(n) is the smallest nonnegative integer not yet in the sequence such that n + a(n) is one less than a square. - Franklin T. Adams-Watters, Apr 06 2009
From Michel Marcus, Mar 01 2021: (Start)
Array T(n,k) = (n+k)^2/2 + (n+3*k)/2 for n,k >= 0 read by descending antidiagonals.
Array T(n,k) = (n+k)^2/2 + (3*n+k)/2 for n,k >= 0 read by ascending antidiagonals. (End)

			Read as a triangle, the sequence is:
    0
    2   1
    5   4   3
    9   8   7   6
   14  13  12  11  10
  (...)
As an infinite square matrix (cf. the "table" link, 2nd paragraph) it reads:
    0    2    5    9   14   20   ...
    1    4    8   13   19   22   ...
    3    7   12   18   23   30   ...
    6   11   17   24   31   39   ...
  (...)

Harry J. Smith, Table of n, a(n) for n = 0..1000
Madeline Brandt and Kåre Schou Gjaldbæk, Classification of Quadratic Packing Polynomials on Sectors of R^2, arXiv:2102.13578 [math.NT], 2021. See Figure 9 p. 17.
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024.
Index entries for sequences that are permutations of the natural numbers

Fixed points are A046092.
Row sums give A027480.
Each reversal involves the numbers from A000217 through to A000096.
Cf. A038722. Transpose of A001477.
Cf. A002419, A119771, A226725.

T:= (n,k)-> n*(n+3)/2-k:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 10 2023

Module[{nn=20},Reverse/@TakeList[Range[0,(nn(nn+1))/2],Range[nn]]]// Flatten (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jul 06 2018 *)

A061579_row(n)=vector(n+=1, j, n*(n+1)\2-j)
A061579_upto(n)=concat([A061579_row(r)|r<-[0..sqrtint(2*n)]]) \\ yields approximately n terms: actual number differs by less than +- sqrt(n). - M. F. Hasler, Nov 09 2021

from math import isqrt
def A061579(n): return (r:=isqrt((n<<3)+1)-1>>1)*(r+2)-n # Chai Wah Wu, Feb 10 2023

a(n) = floor(sqrt(2n+1)-1/2)*floor(sqrt(2n+1)+3/2) - n = A005563(A003056(n)) - n.
Row (or antidiagonal) n = 0, 1, 2, ... contains the integers from A000217(n) to A000217(n+1)-1 in reverse order (for diagonals, "reversed" with respect to the canonical "falling" order, cf. A001477/table). - M. F. Hasler, Nov 09 2021
From Alois P. Heinz, Feb 10 2023: (Start)
T(n,k) = n*(n+3)/2 - k.
Sum_{k=0..n} k * T(n,k) = A002419(n).
Sum_{k=0..n} k^2 * T(n,k) = A119771(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A226725(n). (End)

A004320 a(n) = n(n+1)(n+2)^2/6.

Original entry on oeis.org

0, 3, 16, 50, 120, 245, 448, 756, 1200, 1815, 2640, 3718, 5096, 6825, 8960, 11560, 14688, 18411, 22800, 27930, 33880, 40733, 48576, 57500, 67600, 78975, 91728, 105966, 121800, 139345, 158720, 180048, 203456, 229075, 257040, 287490, 320568, 356421, 395200

Offset: 0

N. J. A. Sloane

Consider the set B(n) = {1,2,3,...n}. Let a(0) = 0. Then a(n) = Sum [ b(i)^2 - b(j)^2] for all i, j = 1 to n, b(i) belongs to B(n). E.g., a(3) = (3^2-1^2) + (3^2-2^2) + (2^2-1^2) = 16. - Amarnath Murthy, Jun 01 2001
Partial sums of A016061. - J. M. Bergot, Jun 18 2013
For n >= 3, a(n-2) is the number of permutations of n symbols that 3-commute with an n-cycle (see A233440 for definition). - Luis Manuel Rivera Martínez, Feb 24 2014
a(n) is the sum of all pairs with repetitions allowed drawn from the set of triangular numbers from A000217(0) to A000217(n). This is similar to A027480 but uses triangular numbers instead of the integers. Example for n=2: 0+1, 0+3, 1+1, 1+3, 3+3 gives sum of 16 = a(2). - J. M. Bergot, Mar 23 2016
From Mircea Dan Rus, Jul 29 2020: (Start)
a(n) is the number of lattice rectangles (squares included) inside half of an Aztec diamond of order n. This shape is obtained by stacking n rows of consecutive unit lattice squares, with the centers of rows vertically aligned and consisting successively of 2n, 2n-2,..., 4, 2 squares. See below the representation for n=6.
              
           ||_|_
         ||_|||_
       ||_|||_||
     ||_|||_|||_|_
   ||_|||_|||_|||_
  |||_|||_|||_|||_|
(End)
a(n-1) = (n+1)*binomial(n+1, 3) is the number of certain rectangles (squares included) in an n X n square filled with 1 X 1 squares. Divide the n X n square, for n >= 2, into two complementary staircases by the boundary consisting of 2*n length 1 edges. For n = 1 there is no boundary. See a A000332 figure in the Mircea Dan Rus comment for the staircase with basis length n = 4. The complementary staircase is upside down with basis length n-1 = 3. Then a(n-1) is the number of rectangles in the n X n square which have at least one border link in their interior. This counting is based on the binomial identity given in the formula section, using A096948 (for n=m), A000332(n+3) and A000332(n+2). - Wolfdieter Lang, Sep 22 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Teofil Bogdan and Mircea Dan Rus, Counting the lattice rectangles inside Aztec diamonds and square biscuits, arXiv:2007.13472 [math.CO], 2020.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A016061, A047929, A233440, A000332, A096948, A000217.

[n*(n+1)*(n+2)^2/6: n in [0..40] ]; // Vincenzo Librandi, Aug 19 2011

[seq ((n+2)*(binomial(n+2,3)), n=0..45)]; # Zerinvary Lajos, May 26 2006

Table[n (n + 1) (n + 2)^2/6, {n, 0, 40}] (* Wesley Ivan Hurt, Oct 28 2014 *)
LinearRecurrence[{5,-10,10,-5,1},{0,3,16,50,120,245},40] (* Harvey P. Dale, Nov 09 2022 *)

a(n)=n*(n+1)*(n+2)^2/6 \\ Charles R Greathouse IV, Jun 18 2013

G.f.: x*(3+x)/(1-x)^5. - Paul Barry, Feb 27 2003
a(n) = (n+2)*A000292(n). - Zerinvary Lajos, May 26 2006
a(n) = A047929(n+2)/6. - Zerinvary Lajos, May 09 2007
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Oct 28 2014
a(n) = 3*A000332(n+3) + A000332(n+2). - Mircea Dan Rus, Jul 29 2020
Sum_{n>=1} 1/a(n) = Pi^2/2 - 9/2. - Jaume Oliver Lafont, Jul 13 2017
a(n-1) = T(n)^2 - (s(n) + s(n-1)), with T(n) = binomial(n+1, 2) = A000217(n) and s(n) = binomial(n+3, 4) = A000332(n+3), for n >= 1. See a comment above, and the formula by Mircea Dan Rus. - Wolfdieter Lang, Sep 22 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/4 + 12*log(2) - 21/2. - Amiram Eldar, Jan 28 2022
E.g.f.: exp(x)*x*(18 + 30*x + 11*x^2 + x^3)/6. - Stefano Spezia, Mar 04 2023
a(n) = Sum_{j=0..n+1} binomial(n+1,2) + binomial(n+1,3). - Detlef Meya, Jan 20 2024

A135503 a(n) = n*(n^2 - 1)/2.

Original entry on oeis.org

0, 0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, 1092, 1365, 1680, 2040, 2448, 2907, 3420, 3990, 4620, 5313, 6072, 6900, 7800, 8775, 9828, 10962, 12180, 13485, 14880, 16368, 17952, 19635, 21420, 23310, 25308, 27417, 29640, 31980, 34440

Offset: 0

Cino Hilliard, Feb 09 2008

Previous name was: Integer values of sqrt(b) solving sqrt(d) + sqrt(b) = sqrt(c) with d^2 + b = c.
Squaring the first equation and setting the result equal to the second, we need d + b + 2*sqrt(d*b) = d^2+b -> d + 2*sqrt(d*b) = d^2 -> d^2 - d = 2*sqrt(d*b)
-> d^2*(d-1)^2 = 4*d*b -> b = d*(d-1)^2/4 -> sqrt(b) = (d-1)*sqrt(d)/2. Setting d = (n+1)^2 yields sqrt(b) = A027480(n).
This is the case k = 2 for FLTR, Fermat's Last Theorem with rational exponents 1/k: Consider x + y = x + y. Then (x^k)^(1/k) + (y^k)^(1/k) = ((x+y)^k)^(1/k).
For k > 2, there are infinitely many solutions to d^(1/k) + b^(1/k) = c^(1/k). E.g., 8^(1/3) + 27^(1/3) = 125^(1/3) at k = 3. However, in conjunction with d^2 + b = c, I could not find any nontrivial solutions.
A shifted version of A027480. - R. J. Mathar, Apr 07 2009
For n > 2, a(n) is the maximum value of the magic constant in a perimeter-magic n-gon of order n (see A342758). - Stefano Spezia, Mar 21 2021
a(n) is equal to the total number of P_3 edge-disjoint subgraphs of the complete graph on n vertices. - Samuel J. Bevins, May 09 2023

			For d = 9, b = 144, c = 225, 9^(1/2) + 144^(1/2) = 225^(1/2) and 9^2 + 144 = 225. So b^(1/2) = 12 is the 4th entry in the sequence.

G. C. Greubel, Table of n, a(n) for n = 0..1000
C. D. Bennet, A. M. W. Glass and G. J. Székely, Fermat's Last Theorem for Rational Exponents, Am. Math. Monthly 111 (2004), 322-329. - _R. J. Mathar_, Apr 21 2009
Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20.

Cf. A000217, A000578, A002411, A004188, A006003, A007531, A007588, A027480, A342758.

Array[# (#^2 - 1)/2 &, 42, 0] (* Michael De Vlieger, Feb 20 2018 *)

flt2(n,p) = { local(a,b); for(a=0,n, b = (a^3-a)/2; print1(b", ") ) }

a(n) = 3*A000292(n-1).
From R. J. Mathar Feb 20 2008: (Start)
O.g.f.: 3*x^2/(-1+x)^4.
a(n) = n*(n^2 - 1)/2 = A007531(n+1)/2. (End)
G.f.: 3*x^2*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + (k+1)/(k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
a(n) = A006003(n+1) - A000326(n+1). - J. M. Bergot, Dec 04 2014
E.g.f.: (1/2)* x^2 *(3 + x)*exp(x). - G. C. Greubel, Oct 15 2016
From Miquel Cerda, Dec 25 2016: (Start)
a(n) = A000578(n) - A006003(n).
a(n) = A004188(n) - A000578(n).
a(n) = A007588(n) - A004188(n). (End)
a(n) = A002411(n) - A000217(n). - Justin Gaetano, Feb 20 2018
From Amiram Eldar, Jan 09 2021: (Start)
Sum_{n>=2} 1/a(n) = 1/2.
Sum_{n>=2} (-1)^n/a(n) = 4*log(2) - 5/2. (End)

Edited by R. J. Mathar, Apr 21 2009

New name using R. J. Mathar's formula, Joerg Arndt, Dec 05 2014

A253283 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 3, 12, 10, 0, 4, 30, 60, 35, 0, 5, 60, 210, 280, 126, 0, 6, 105, 560, 1260, 1260, 462, 0, 7, 168, 1260, 4200, 6930, 5544, 1716, 0, 8, 252, 2520, 11550, 27720, 36036, 24024, 6435, 0, 9, 360, 4620, 27720, 90090, 168168, 180180, 102960, 24310

Offset: 0

Peter Luschny, Mar 20 2015

The rows give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1) / (n!*(n+1)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 31 2022

This is related to the cluster fans of type B (see Fomin and Zelevinsky reference) - F. Chapoton, Nov 17 2022.

			[1]
[0, 1]
[0, 2,   3]
[0, 3,  12,   10]
[0, 4,  30,   60,   35]
[0, 5,  60,  210,  280,  126]
[0, 6, 105,  560, 1260, 1260,  462]
[0, 7, 168, 1260, 4200, 6930, 5544, 1716]
.
R_0(x) = 1/(x-1)^0.
R_1(x) = 0/(x-1)^1 + 1/(x-1)^2.
R_2(x) = 0/(x-1)^2 + 2/(x-1)^3 + 3/(x-1)^4.
R_3(x) = 0/(x-1)^3 + 3/(x-1)^4 + 12/(x-1)^5 + 10/(x-1)^6.
Then k!*[x^k] R_n(x) is A001286(k+2) and A001754(k+3) for n = 2, 3 respectively.
.
Seen as an array A(n, k) = binomial(n + k, k)*binomial(n + 2*k - 1, n + k):
[0] 1, 1,   3,   10,    35,    126,     462, ...
[1] 0, 2,  12,   60,   280,   1260,    5544, ...
[2] 0, 3,  30,  210,  1260,   6930,   36036, ...
[3] 0, 4,  60,  560,  4200,  27720,  168168, ...
[4] 0, 5, 105, 1260, 11550,  90090,  630630, ...
[5] 0, 6, 168, 2520, 27720, 252252, 2018016, ...
[6] 0, 7, 252, 4620, 60060, 630630, 5717712, ...

Seiichi Manyama, Rows n = 0..139, flattened
F. Chapoton Enumerative Properties of Generalized Associahedra, Sém. Loth. Comb. B51b (2004).
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.
S. Fomin and A. Zelevinsky Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3.
Yunlong Shen and Lixin Shen, Orthogonal Fourier-Mellin moments for invariant pattern recognition, J. Opt. Soc. Am. A 11 (6) (1994) p 1748-1757, eq. (6).

T(n, n) = C(2*n-1, n) = A001700(n-1).
T(n, n-1) = A005430(n-1) for n >= 1.
T(n, n-2) = A051133(n-2) for n >= 2.
T(n, 2) = A027480(n-1) for n >= 2.
T(2*n, n) = A208881(n) for n >= 0.
A002002 (row sums).
Cf. A000142, A001286, A001754, A001755, A001777, A182626, A266732 (row k=3), A008316, A033282, A063007, A345013, A269944.

T_row := proc(n) local egf, k, F, t;
if n=0 then RETURN(1) fi;
egf := (x/(1-x))^n/n!; t := diff(egf,[x$n]);
F := convert(t,parfrac,x);
# print(seq(k!*coeff(series(F,x,20),x,k),k=0..7));
# gives A000142, A001286, A001754, A001755, A001777, ...
seq(coeff(F,(x-1)^(-k)),k=n..2*n) end:
seq(print(T_row(n)),n=0..7);
# 2nd version by R. J. Mathar, Dec 18 2016:
A253283 := proc(n,k)
    binomial(n,k)*binomial(n+k-1,k-1) ;
end proc:

Table[Binomial[n, k] Binomial[n + k - 1, k - 1], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 22 2017 *)

T(n,k) = binomial(n,k)*binomial(n+k-1,k-1);
tabl(nn) = for(n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Apr 29 2018

The exponential generating functions for the rows of the square array L(n,k) = ((n+k)!/n!)*C(n+k-1,n-1) (associated to the unsigned Lah numbers) are given by R_n(x) = Sum_{k=0..n} T(n,k)/(x-1)^(n+k).
T(n,k) = C(n,k)*C(n+k-1,k-1).
Sum_{k=0..n} T(n,k) = (-1)^n*hypergeom([-n,n],[1],2) = (-1)^n*A182626(n).
Row generating function: Sum_{k>=1} T(n,k)*z^k = z*n* 2F1(1-n,n+1 ; 2; -z). - R. J. Mathar, Dec 18 2016
From Peter Bala, Feb 22 2017: (Start)
G.f.: (1/2)*( 1 + (1 - t)/sqrt(1 - 2*(2*x + 1)*t + t^2) ) = 1 + x*t + (2*x + 3*x^2)*t^2 + (3*x + 12*x^2 + 10*x^3)*t^3 + ....
n-th row polynomial R(n,x) = (1/2)*(LegendreP(n, 2*x + 1) - LegendreP(n-1, 2*x + 1)) for n >= 1.
The row polynomials are the black diamond product of the polynomials x^n and x^(n+1) (see Dukes and White 2016 for the definition of this product).
exp(Sum_{n >= 1} R(n,x)*t^n/n) = 1 + x*t + x*(1 + 2*x)*t^2 + x*(1 + 5*x + 5*x^2)*t^3 + ... is a g.f. for A033282, but with a different offset.
The polynomials P(n,x) := (-1)^n/n!*x^(2*n)*(d/dx)^n(1 + 1/x)^n begin 1, 3 + 2*x , 10 + 12*x + 3*x^2, ... and are the row polynomials for the row reverse of this triangle. (End)
Let Q(n, x) = Sum_{j=0..n} (-1)^(n - j)*A269944(n, j)*x^(2*j - 1) and P(x, y) = (LegendreP(x, 2*y + 1) - LegendreP(x-1, 2*y + 1)) / 2 (see Peter Bala above). Then n!*(n - 1)!*[y^n] P(x, y) = Q(n, x) for n >= 1. - Peter Luschny, Oct 31 2022
From Peter Bala, Apr 18 2024: (Start)
G.f.: Sum_{n >= 0} binomial(2*n-1, n)*(x*t)^n/(1 - t)^(2*n) = 1 + x*t + (2*x + 3*x^2)*t^2 + (3*x + 12*x^2 + 10*x^3)*t^3 + ....
n-th row polynomial R(n, x) = [t^n] ( (1 - t)/(1 - (1 + x)*t) )^n.
It follows that for integer x, the sequence {R(n, x) : n >= 0} satisfies the Gauss congruences: R(n*p^r, x) == R(n*p^(r-1), x) (mod p^r) for all primes p and positive integers n and r.
R(n, -2) = (-1)^n * A002003(n) for n >= 1.
R(n, 3) = A299507(n). (End)

A109649 Entries in 3-dimensional version of Pascal triangle: trinomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 3, 1, 3, 6, 3, 3, 3, 1, 1, 4, 6, 4, 1, 4, 12, 12, 4, 6, 12, 6, 4, 4, 1, 1, 5, 10, 10, 5, 1, 5, 20, 30, 20, 5, 10, 30, 30, 10, 10, 20, 10, 5, 5, 1, 1, 6, 15, 20, 15, 6, 1, 6, 30, 60, 60, 30, 6, 15, 60, 90, 60, 15, 20, 60, 60, 20, 15, 30, 15, 6

Offset: 0

Philippe Deléham, Aug 03 2005

Greatest numbers in each 2D triangle form A022916 (multinomial coefficient n!/([n/3]![(n+1)/3]![(n+2)/3]!)).
2D triangle sums are powers of 3.
See A046816 for another version.

			.1 3 3 1 ... Here is the third slice of the pyramid
. 3 6 3
.. 3 3
... 1 .....

Cf. A022916, A046816.

Cf. A007318, A002378, A027480, A033487, A033488.

Coefficients of x, y, z in (x+y+z)^n.

A158842 a(n) = 1 + n(n+1)(n-1)/2.

Original entry on oeis.org

1, 1, 4, 13, 31, 61, 106, 169, 253, 361, 496, 661, 859, 1093, 1366, 1681, 2041, 2449, 2908, 3421, 3991, 4621, 5314, 6073, 6901, 7801, 8776, 9829, 10963, 12181, 13486, 14881, 16369, 17953, 19636, 21421, 23311, 25309, 27418, 29641, 31981, 34441, 37024, 39733, 42571, 45541, 48646

Offset: 0

Gary W. Adamson & Roger L. Bagula, Mar 28 2009

Binomial transform of the sequence 1, 0, 3, 3, 0, 0, 0, ... .

			a(4) = 31 = sum of row 4 terms of triangle A158841: (13 + 9 + 6 + 3).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Row sums of A158841.

[1+ n*(n+1)*(n-1)/2: n in [1..50]]; // Vincenzo Librandi, Nov 16 2011

A158842 := proc(n)
        1+n*(n+1)*(n-1)/2 ;
end proc:
seq(A158842(n),n=0..30) ; # R. J. Mathar, Nov 05 2011

Table[1 + n*(n + 1)*(n - 1)/2, {n, 40}] (* and *) LinearRecurrence[{4, -6, 4, -1}, {1, 4, 13, 31}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

a(n) = 1+A027480(n-1) for n>=1. - R. J. Mathar, Mar 28 2009
G.f.: 1-x*(-1-3*x^2+x^3) / (x-1)^4 . - R. J. Mathar, Nov 05 2011
E.g.f.: exp(x)*(1 + x^3/2 + 3*x^2/2). - Nikolaos Pantelidis, Feb 13 2023

a(0)=1 prepended by Andrew Howroyd, Feb 14 2023

A229183 a(n) = n*(n^2 + 3)/2.

Original entry on oeis.org

0, 2, 7, 18, 38, 70, 117, 182, 268, 378, 515, 682, 882, 1118, 1393, 1710, 2072, 2482, 2943, 3458, 4030, 4662, 5357, 6118, 6948, 7850, 8827, 9882, 11018, 12238, 13545, 14942, 16432, 18018, 19703, 21490, 23382, 25382, 27493, 29718, 32060, 34522, 37107, 39818

Offset: 0

Derek Orr, Sep 15 2013

Numbers a(n) such that (a(n) + B)^(1/3) + (a(n) - B)^(1/3) = n, where B = sqrt(a(n)^2 + 1).
4*a(n) is the sum of two cubes. In fact: 2*n*(n^2 + 3) = (n-1)^3 + (n+1)^3. - Bruno Berselli, Apr 11 2016
From Olivier Gérard, Aug 07 2016 (Start)
Row sums of n consecutive integers, starting at 2, seen as a triangle:
.
2   | 2
7   | 3  4
18  | 5  6  7
38  | 8  9  10 11
70  | 12 13 14 15 16
117 | 17 18 19 20 21 22
(End)
Take a long horizontal strip of paper 1 unit high and mark two points on the top edge, n/2 and n units from the top left corner. Then fold over the top left corner so that the fold line passes through the bottom left corner and the point n units along the top edge. Then draw a line from the bottom left corner of the strip through the new position of the n/2 point. The point at which that shallow diagonal line meets the top edge of the strip of paper will be a(n) from the top left corner. - Elliott Line, Jul 09 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006003 (row sums of integers, starting with 1).

Cf. A027480 (row sums of integers, starting with 0).

[n*(n^2 + 3) div 2: n in [0..50]]; // Vincenzo Librandi, Sep 23 2013

A229183 := proc(n)
    n*(n^2+3) /2;
end proc:
[seq(A229183(n),n=0..30)] ; # R. J. Mathar, Aug 16 2019

Table[(n^3 + 3n)/2, {n, 0, 100}] (* T. D. Noe, Sep 16 2013 *)
CoefficientList[Series[x (2 - x + 2 x^2)/(x - 1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Sep 23 2013 *)

vector(100,n,((n-1)^3+3*n-3)/2) \\ Derek Orr, Mar 12 2015

{print((n**3+3*n)/2,end=', ') for n in range(0,100)} # Simplified by Derek Orr, Mar 12 2015

G.f.: x*(2 - x + 2*x^2) / (x-1)^4. - R. J. Mathar, Sep 22 2013
a(n)^2 + 1 = (n^2 + 1)^2 * ((n/2)^2 + 1). - Joerg Arndt, Jan 22 2015
E.g.f.: exp(x)*x*(4 + 3*x + x^2)/2. - Stefano Spezia, Jul 04 2021

A254407 a(n) = n(n+1)(11*n +10)/6.

Original entry on oeis.org

0, 7, 32, 86, 180, 325, 532, 812, 1176, 1635, 2200, 2882, 3692, 4641, 5740, 7000, 8432, 10047, 11856, 13870, 16100, 18557, 21252, 24196, 27400, 30875, 34632, 38682, 43036, 47705, 52700, 58032, 63712, 69751, 76160, 82950, 90132, 97717, 105716, 114140, 123000

Offset: 0

Bruno Berselli, Jan 30 2015

Similar sequences of the type m*P(s,m) - Sum_{i=1..m} P(s-1,i), where P(s,m) is the m-th s-gonal number:
s=3: A027480(n) = (n+1)*A000217(n+1) - Sum_{i=1..n+1} i;
s=4: A162148(n) = (n+1)*A000290(n+1) - Sum_{i=1..n+1} A000217(i);
s=5: A245301(n) = (n+1)*A000326(n+1) - Sum_{i=1..n+1} A000290(i);
s=6: A085788(n) = (n+1)*A000384(n+1) - Sum_{i=1..n+1} A000326(i);
s=7:       a(n) = (n+1)*A000566(n+1) - Sum_{i=1..n+1} A000384(i).

			532 is the 7th term because A000566(7)=112 and Sum_{i=1..7} A000384(i)=252, therefore 7*112-252 = 532.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Polygonal numbers: Table of values.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A011875, A027480, A057145, A085788, A132112, A162148, A245301, A254963.

Magma
```
[n*(n+1)*(11*n+10)/6: n in [0..40]];
```

A254407:= n-> n*(n+1)*(11*n+10)/6; seq(A254407(n), n=0..50); # G. C. Greubel, Mar 31 2021

Table[n (n + 1) (11 n + 10)/6, {n, 0, 40}]
Column[CoefficientList[Series[x (7 + 4 x) / (1 - x)^4, {x, 0, 60}], x]] (* Vincenzo Librandi, Jan 31 2015 *)

makelist(n*(n+1)*(11*n+10)/6, n, 0, 40);

vector(40, n, n--; n*(n+1)*(11*n+10)/6)

Sage
```
[n*(n+1)*(11*n+10)/6 for n in (0..40)]
```

G.f.: x*(7 + 4*x)/(1 - x)^4.
a(-n) = -A132112(n-1).
a(n) = Sum_{k=0..n} A011875(11*k+2).
Equivalently, partial sums of A254963.
E.g.f.: x*(42 + 54*x + 11*x^2)*exp(x)/6. - G. C. Greubel, Mar 31 2021

A035328 a(n) = n(2n-1)(2n+1).

Original entry on oeis.org

0, 3, 30, 105, 252, 495, 858, 1365, 2040, 2907, 3990, 5313, 6900, 8775, 10962, 13485, 16368, 19635, 23310, 27417, 31980, 37023, 42570, 48645, 55272, 62475, 70278, 78705, 87780, 97527, 107970, 119133, 131040, 143715, 157182, 171465, 186588

Offset: 0

N. J. A. Sloane

Bisection of A027480. For n>1, gives area of triangle two of whose cevians bound three smaller triangles with areas n-1, n, n+1 contiguously. - Lekraj Beedassy, Dec 21 2006

Eric Harold Neville, Jacobian Elliptic Functions, 2nd ed., 1951, p. 38.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")

Cf. A000292, A000447, A027480, A046092, A204558, A244009.

[n*(2*n-1)*(2*n+1): n in [0..40]]; // Vincenzo Librandi, Jun 07 2011

Table[n(2n-1)(2n+1),{n,0,40}] (* Harvey P. Dale, Jan 11 2014 *)

vector(100,n,(n-1)*(2*n-1)*(2*n-3)) \\ Derek Orr, Jan 29 2015

a(n) = 3*A000447(n) = 3*A000292(2*n-1).
Sum_{n>=1} 1/a(n) = 2*log(2) - 1. - Benoit Cloitre, Apr 05 2002
a(n) = A204558(2*n) / (2*n). - Reinhard Zumkeller, Jan 18 2012
G.f.: 3*x*(1 + 6*x + x^2)/(1 - x)^4. - Colin Barker, Mar 27 2012
Product_{n>=1} 4*n^3/a(n) = Pi/2. - Daniel Suteu, Feb 05 2017
a(n) = Sum_{i=0..2*n} A046092(n-1)+i = Sum_{i=2*n+1..4*n-1} A046092(n-1)+i for n>0. Example: for n = 5, A046092(4) = 40 and a(5) = 40 + 41 + 42 + ... + 49 + 50 = 51 + 52 + 53 + ... + 58 + 59 = 495. - Bruno Berselli, Oct 26 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - log(2) (A244009). - Amiram Eldar, Jan 30 2021
E.g.f.: exp(x)*x*(3 + 12*x + 4*x^2). - Stefano Spezia, Sep 03 2023

More terms from Benoit Cloitre, Apr 05 2002

A061928 Array T(n,m) = 1/beta(n+1,m+1) read by antidiagonals.

Original entry on oeis.org

6, 12, 12, 20, 30, 20, 30, 60, 60, 30, 42, 105, 140, 105, 42, 56, 168, 280, 280, 168, 56, 72, 252, 504, 630, 504, 252, 72, 90, 360, 840, 1260, 1260, 840, 360, 90, 110, 495, 1320, 2310, 2772, 2310, 1320, 495, 110, 132, 660, 1980, 3960, 5544, 5544, 3960

Offset: 1

Frank Ellermann, May 22 2001

beta(n+1,m+1) = Integral_{x=0..1} x^n * (1-x)^m dx for real n, m.

			Antidiagonals:
   6,
  12, 12,
  20, 30, 20,
  30, 60, 60, 30,
  ...
Array:
   6  12  20   30   42
  12  30  60  105  168
  20  60 140  280  504
  30 105 280  630 1260
  42 168 504 1260 2772

G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 26.

Rows: 1/b(n, 2): A002378, 1/b(n, 3): A027480, 1/b(n, 4): A033488. Diagonals: 1/b(n, n): A002457, 1/b(n, n+1) A005430, 1/b(n, n+2): A000917.

T(i, j)=A003506(i+1, j+1).

t[n_, m_] := 1/Beta[n+1, m+1]; Take[ Flatten[ Table[ t[n+1-m, m], {n, 1, 10}, {m, 1, n}]], 52] (* Jean-François Alcover, Oct 11 2011 *)

A(i,j)=if(i<1||j<1,0,1/subst(intformal(x^i*(1-x)^j),x,1)) /* Michael Somos, Feb 05 2004 */

A(i,j)=if(i<1||j<1,0,1/sum(k=0,i,(-1)^k*binomial(i,k)/(j+1+k))) /* Michael Somos, Feb 05 2004 */

from sympy import factorial as f
def T(n, m): return f(n + m + 1)/(f(n)*f(m))
for n in range(1, 11): print([T(m, n - m + 1) for m in range(1, n + 1)]) # Indranil Ghosh, Apr 29 2017

beta(n+1, m+1) = gamma(n+1)*gamma(m+1)/gamma(n+m+2) = n!*m!/(n+m+1)!.

cp's OEIS Frontend

A061579 Reverse one number (0), then two numbers (2,1), then three (5,4,3), then four (9,8,7,6), etc.

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A004320 a(n) = n*(n+1)*(n+2)^2/6.

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A135503 a(n) = n*(n^2 - 1)/2.

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A253283 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.

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A109649 Entries in 3-dimensional version of Pascal triangle: trinomial coefficients.

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A158842 a(n) = 1 + n*(n+1)*(n-1)/2.

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A229183 a(n) = n*(n^2 + 3)/2.

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A004320 a(n) = n(n+1)(n+2)^2/6.

A158842 a(n) = 1 + n(n+1)(n-1)/2.

A254407 a(n) = n(n+1)(11*n +10)/6.

A035328 a(n) = n(2n-1)(2n+1).