A051925 -id:A051925 - cp's OEIS frontend

A007290 a(n) = 2*binomial(n,3).

0, 0, 0, 2, 8, 20, 40, 70, 112, 168, 240, 330, 440, 572, 728, 910, 1120, 1360, 1632, 1938, 2280, 2660, 3080, 3542, 4048, 4600, 5200, 5850, 6552, 7308, 8120, 8990, 9920, 10912, 11968, 13090, 14280, 15540, 16872, 18278, 19760, 21320, 22960, 24682, 26488, 28380, 30360, 32430, 34592, 36848, 39200

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of acute triangles made from the vertices of a regular n-polygon when n is even (cf. A000330). - Sen-Peng Eu, Apr 05 2001
a(n+2) is (-1)*coefficient of X in Zagier's polynomial (n,n-1). - Benoit Cloitre, Oct 12 2002
Definite integrals of certain products of 2 derivatives of (orthogonal) Chebyshev polynomials of the 2nd kind are pi-multiple of this sequence. For even (p+q): Integrate[ D[ChebyshevU[p, x], x] D[ChebyshevU[q, x], x] (1 - x^2)^(1/2), {x,-1,1}] / Pi = a(n), where n=Min[p,q]. Example: a(3)=20 because Integrate[ D[ChebyshevU[3, x], x] D[ChebyshevU[5, x], x] (1 - x^2)^(1/2), {x,-1,1}]/Pi = 20 since 3=Min[3,5] and 3+5 is even. - Christoph Pacher (Christoph.Pacher(AT)arcs.ac.at), Dec 16 2004
If Y is a 2-subset of an n-set X then, for n>=3, a(n-1) is the number of 3-subsets and 4-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
a(n) is also the number of proper colorings of the cycle graph Csub3 (also the complete graph Ksub3) when n colors are available. - Gary E. Stevens, Dec 28 2008
a(n) is the reverse Wiener index of the path graph with n vertices. See the Balaban et al. reference, p. 927.
For n > 1: a(n) = sum of (n-1)-th row of A141418. - Reinhard Zumkeller, Nov 18 2012
This is the sequence for nuclear magic numbers in an idealized spherical nucleus under the harmonic oscillator model. - Jess Tauber, May 20 2013
Shifted non-vanishing diagonal of A132440^3/3. Second subdiagonal of A238363 (without zeros). For n>0, a(n+2)=n*(n+1)*(n+2)/3. Cf. A130534 for relations to colored forests and disposition of flags on flagpoles. - Tom Copeland, Apr 05 2014
a(n) is the number of ordered rooted trees with n non-root nodes that have 2 leaves; see A108838. - Joerg Arndt, Aug 18 2014
Number of floating point multiplications in the factorization of an (n-1)X(n-1) real matrix by Gaussian elimination as e.g. implemented in LINPACK subroutines sgefa.f or dgefa.f. The number of additions is given by A000330. - Hugo Pfoertner, Mar 28 2018
a(n+1) = Max_{s in S_n} Sum_{k=1..n} (k - s(k))^2 where S_n is the symmetric group of permutations of [1..n]; this maximum is obtained with the permutation s = (1, n) (2, n-1) (3, n-2) ... (k, n-k+1). (see Protat reference). - Bernard Schott, Dec 26 2022

Luigi Berzolari, Allgemeine Theorie der Höheren Ebenen Algebraischen Kurven, Encyclopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Band III_2. Heft 3, Leipzig: B. G. Teubner, 1906, p. 352.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.
Maurice Protat, Des Olympiades à l'Agrégation, un problème de maximum, Problème 36, p. 83, Ellipses, Paris 1997.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alexandru T. Balaban, Denise Mills, Ovidiu Ivanciuc and Subhash C. Basak,, Reverse Wiener indices, Croatica Chemica Acta, Vol. 73, No. 4 (2000), pp. 923-941.
A. Burstein, S. Kitaev and T. Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A. Vol. 19, No. 2-3 (2008), pp. 27-38.
Otto Haxel, J. Hans D. Jensen and Hans E. Suess, On the "Magic Numbers" in Nuclear Structure, Phys. Rev., Vol. 75 (1949), p. 1766.
Xiangdong Ji, Chapter 8: Structure of Finite Nuclei, Lecture notes for Phys 741 at Univ. of Maryland, p. 140 [From _Tom Copeland_, Apr 07 2014].
Sandi Klavžar, Balázs Patkós, Gregor Rus and Ismael G. Yero, On general position sets in Cartesian grids, arXiv:1907.04535 [math.CO], 2019.
Vladimir Ladma, Magic Numbers.
Cleve Moler, LINPACK subroutine sgefa.f, University of New Mexico, Argonne National Lab, 1978.
Hamzeh Mujahed and Benedek Nagy, Wiener Index on Lines of Unit Cells of the Body-Centered Cubic Grid, Mathematical Morphology and Its Applications to Signal and Image Processing, 12th International Symposium, ISMM 2015.
V. B. Priezzhev, Series expansion for rectilinear polymers on the square lattice, J. Phys. A, Vol. 12, No. 11 (1979), pp. 2131-2139.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Wikipedia, p-derivation.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A diagonal of A059419. Partial sums of A002378.
A diagonal of A008291. Row 3 of A074650.
Cf. A000292, A051925, A145066, A145067, A145068, A210569.

a007290 n = if n < 3 then 0 else 2 * a007318 n 3  -- Reinhard Zumkeller, Nov 18 2012

I:=[0, 0, 0, 2]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Jun 19 2012

A007290 := proc(n) 2*binomial(n,3) end proc:

Table[Integrate[ D[ChebyshevU[n, x], x] D[ChebyshevU[n, x], x] (1 - x^2)^(1/2), {x, -1, 1}]/Pi, {n, 1, 20}] (* Pacher *)
LinearRecurrence[{4,-6,4,-1},{0,0,0,2},50] (* Vincenzo Librandi, Jun 19 2012 *)

my(x='x+O('x^100)); concat([0, 0, 0], Vec(2*x^3/(1-x)^4)) \\ Altug Alkan, Nov 01 2015

apply( {A007290(n)=binomial(n,3)*2}, [0..55]) \\ M. F. Hasler, Jul 02 2021

G.f.: 2*x^3/(1-x)^4.
a(n) = a(n-1)*n/(n-3) = a(n-1) + A002378(n-2) = 2*A000292(n-2) = Sum_{i=0..n-2} i*(i+1) = n*(n-1)*(n-2)/3. - Henry Bottomley, Jun 02 2000 [Formula corrected by R. J. Mathar, Dec 13 2010]
a(n) = A000217(n-2) + A000330(n-2), n>1. - Reinhard Zumkeller, Mar 20 2008
a(n+1) = A000330(n) - A000217(n), n>=0. - Zak Seidov, Aug 07 2010
a(n) = A033487(n-2) - A052149(n-1) for n>1. - Bruno Berselli, Dec 10 2010
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 19 2012
a(n) = (2*n - 3*n^2 + n^3)/3. - T. D. Noe, May 20 2013
a(n+1) = A002412(n) - A000330(n) or "Hex Pyramidal" - "Square Pyramidal" (as can also be seen via above formula). - Richard R. Forberg, Aug 07 2013
Sum_{n>=3} 1/a(n) = 3/4. - Enrique Pérez Herrero, Nov 10 2013
E.g.f.: exp(x)*x^3/3. - Geoffrey Critzer, Nov 22 2015
a(n+2) = delta(-n) = -delta(n) for n >= 0, where delta is the p-derivation over the integers with respect to prime p = 3. - Danny Rorabaugh, Nov 10 2017
(a(n) + a(n+1))/2 = A000330(n-1). - Ezhilarasu Velayutham, Apr 05 2019
Sum_{n>=3} (-1)^(n+1)/a(n) = 6*log(2) - 15/4. - Amiram Eldar, Jan 09 2022
a(n) = Sum_{m=0..n-2} Sum_{k=0..n-2} abs(m-k). - Nicolas Bělohoubek, Nov 06 2022
From Bernard Schott, Jan 04 2023: (Start)
a(n) = 2 * A000292(n-2), for n >= 2.
a(n+1) = 2 *Sum_{k=1..floor(n/2)} (n-(2k-1))^2, for n >= 2. (End)

A211790 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k

Original entry on oeis.org

1, 7, 1, 23, 7, 1, 54, 22, 7, 1, 105, 51, 22, 7, 1, 181, 97, 50, 22, 7, 1, 287, 166, 96, 50, 22, 7, 1, 428, 263, 163, 95, 50, 22, 7, 1, 609, 391, 255, 161, 95, 50, 22, 7, 1, 835, 554, 378, 253, 161, 95, 50, 22, 7, 1, 1111, 756, 534, 374, 252, 161, 95, 50, 22, 7

Offset: 1

Clark Kimberling, Apr 21 2012

...
Let R be the array in A211790 and let R' be the array in A211793.  Then R(k,n) + R'(k,n) = 3^(n-1).  Moreover, (row k of R) =(row k of A211796) for k>2, by Fermat's last theorem; likewise, (row k of R')=(row k of A211799) for k>2.
...
Generalizations:  Suppose that b,c,d are nonzero integers, and let U(k,n) be the number of ordered triples (w,x,y) with all terms in {1,...,n} and b*w*k  c*x^k+d*y^k, where the relation  is one of these: <, >=, <=, >.  What additional assumptions force the limiting row sequence to be essentially one of these: A002412, A000330, A016061, A174723, A051925?
In the following guide to related arrays and sequences, U(k,n) denotes the number of (w,x,y) as described in the preceding paragraph:
                                 first 3 rows          limiting row sequence
A211790:  w^k < x^k+y^k    A004068, A211635, A211650       A002412
A211793:  w^k >= x^k+y^k   A000292, A211636, A211651       A000330
A211796:  w^k <= x^k+y^k   A002413, A211634, A211650       A002412
A211799:  w^k > x^k+y^k    A000292, A211637, A211651       A000330
A211802:  2w^k < x^k+y^k   A182260, A211800, A211801       A016061
A211805:  2w^k >= x^k+y^k  A055232, A211803, A211804       A000330
A211808:  2w^k <= x^k+y^k  A055232, A211806, A211807       A174723
A182259:  2w^k > x^k+y^k   A182260, A211810, A211811       A051925

			Northwest corner:
  1, 7, 23, 54, 105, 181, 287, 428, 609
  1, 7, 22, 51,  97, 166, 263, 391, 554
  1, 7, 22, 50,  96, 163, 255, 378, 534
  1, 7, 22, 50,  95, 161, 253, 374, 528
  1, 7, 22, 50,  95, 161, 252, 373, 527
For n=2 and k>=1, the 7 triples (w,x,y) are (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2).

Cf. A004068 (row1), A211635 (row2), A211650 (row3), A002412.

Cf. A211793, A211796, A211799, A211802, A211805, A211808, A182259

z = 48;
t[k_, n_] := Module[{s = 0},
   (Do[If[w^k < x^k + y^k, s = s + 1],
       {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
Table[t[1, n], {n, 1, z}]  (* A004068 *)
Table[t[2, n], {n, 1, z}]  (* A211635 *)
Table[t[3, n], {n, 1, z}]  (* A211650 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* A211790 *)
Table[n (n + 1) (4 n - 1)/6,
  {n, 1, z}] (* row-limit sequence, A002412 *)
(* Peter J. C. Moses, Apr 13 2012 *)

R(k,n) = n(n-1)(4n+1)/6 for 1<=k<=n, and

R(k,n) = Sum{Sum{floor[(x^k+y^k)^(1/k)] : 1<=x<=n, 1<=y<=n}} for 1<=k<=n.

A115262 Correlation triangle for n+1.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 8, 8, 4, 5, 11, 14, 11, 5, 6, 14, 20, 20, 14, 6, 7, 17, 26, 30, 26, 17, 7, 8, 20, 32, 40, 40, 32, 20, 8, 9, 23, 38, 50, 55, 50, 38, 23, 9, 10, 26, 44, 60, 70, 70, 60, 44, 26, 10, 11, 29, 50, 70, 85, 91, 85, 70, 50, 29, 11

Offset: 0

Paul Barry, Jan 18 2006

This sequence (formatted as a square array) gives the counts of all possible squares in an m X n rectangle. For example, 11 = 8 (1 X 1 squares) + 3 (2 X 2 square) in 4 X 2 rectangle. - Philippe Deléham, Nov 26 2009
From Clark Kimberling, Feb 07 2011: (Start)
Also the accumulation array of min{n,k}, when formatted as a rectangle.
This is the accumulation array of the array M=A003783 given by M(n,k)=min{n,k}; see A144112 for the definition of accumulation array.
The accumulation array of A115262 is A185957. (End)
From Clark Kimberling, Dec 22 2011: (Start)
As a square matrix, A115262 is the self-fusion matrix of A000027 (1,2,3,4,...).  See A193722 for the definition of fusion and A202673 for characteristic polynomials associated with A115622. (End)

			Triangle begins
  1;
  2,  2;
  3,  5,  3;
  4,  8,  8,  4;
  5, 11, 14, 11,  5;
  6, 14, 20, 20, 14,  6;
  ...
When formatted as a square matrix:
  1,  2,  3,  4,  5, ...
  2,  5,  8, 11, 14, ...
  3,  8, 14, 20, 26, ...
  4, 11, 20, 30, 40, ...
  5, 14, 26, 40, 55, ...
  ...

Cf. A000027, A202673, A271916.
For the triangular version: row sums are A001752. Diagonal sums are A097701. T(2n,n) is A000330(n+1).
Diagonals (1,5,...): A000330 (square pyramidal numbers),
diagonals (2,8,...): A007290,
diagonals (3,11,...): A051925,
diagonals (4,14,...): A159920,
antidiagonal sums: A001752.

U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k, {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
(* Clark Kimberling, Dec 22 2011 *)

Let f(m,n) = m*(m-1)*(3*n-m-1)/6. This array is (with a different offset) the infinite square array read by antidiagonals U(m,n) = f(n,m) if m < n, U(m,n) = f(m,n) if m <= n. See A271916. - N. J. A. Sloane, Apr 26 2016
G.f.: 1/((1-x)^2*(1-x*y)^2*(1-x^2*y)).
Number triangle T(n, k) = Sum_{j=0..n} [j<=k]*(k-j+1)[j<=n-k]*(n-k-j+1).
T(2n,n) - T(2n,n+1) = n+1.

A145066 Partial sums of A002522, starting at n=1.

Original entry on oeis.org

2, 7, 17, 34, 60, 97, 147, 212, 294, 395, 517, 662, 832, 1029, 1255, 1512, 1802, 2127, 2489, 2890, 3332, 3817, 4347, 4924, 5550, 6227, 6957, 7742, 8584, 9485, 10447, 11472, 12562, 13719, 14945, 16242, 17612, 19057, 20579, 22180, 23862, 25627

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 30 2008

			a(2) = a(1) + 2^2 + 1 = 2 + 4 + 1 = 7; a(3) = a(2) + 3^2 + 1 = 7 + 9 + 1 = 17.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A002522 (n^2 + 1), A005563 ((n+1)^2 - 1), A051925 (zero followed by partial sums of A005563), A081489 (partial sums of A002522 starting at n=0).

Accumulate[Range[50]^2+1] (* Harvey P. Dale, Dec 07 2016 *)

{a=0; for(n=1, 42, print1(a=a+n^2+1, ","))}

def A145066(n): return (n*(n*(2*n + 3) + 1))//6 + n # Chai Wah Wu, Oct 30 2024

a(1) = 2; a(n) = a(n-1) + n^2 + 1 for n > 1.
From Christoph Pacher (christoph.pacher(AT)ait.ac.at), Jul 23 2010: (Start)
a(n) = Sum_{k=1..n} (k^2 + 1).
a(n) = A000330(n) + n.
a(n) = n*(n+1)*(2*n+1)/6 + n. (End)
G.f.: x*(2-x+x^2)/(1-x)^4. - Colin Barker, Apr 04 2012
E.g.f.: (1/6)*x*(12 + 9*x + 2*x^2)*exp(x). - G. C. Greubel, Jul 22 2017
a(n) = A081489(n+1) - 1. - Jianing Song, Oct 10 2021

Edited by Klaus Brockhaus, Oct 17 2008

A024305 a(n) = 2(n+1) + 3n + ... + (k+1)*(n+2-k), where k = floor((n+1)/2).

Original entry on oeis.org

4, 6, 17, 22, 43, 52, 86, 100, 150, 170, 239, 266, 357, 392, 508, 552, 696, 750, 925, 990, 1199, 1276, 1522, 1612, 1898, 2002, 2331, 2450, 2825, 2960, 3384, 3536, 4012, 4182, 4713, 4902, 5491, 5700, 6350, 6580, 7294, 7546, 8327, 8602, 9453, 9752, 10676, 11000, 12000

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Bisection: 2*A051925(n).

Cf. A024854, A023855, A023856, A023857, A024868.

b:= func< n | (1-(-1)^n)/2 >;
[(2*n^3 + 3*(6 +b(n))*n^2 + 2*(14 +9*b(n))*n + 27*b(n))/24 : n in [1..50]] // G. C. Greubel, Jul 12 2022

seq(sum((i+1)*(k-i+2), i=1..ceil(k/2)), k=1..70); # Wesley Ivan Hurt, Sep 20 2013

Table[Ceiling[n/2]*(-2*Ceiling[n/2]^2+3n*Ceiling[n/2]+9n+14)/6,{n,100}] (* Wesley Ivan Hurt, Sep 20 2013 *)

def b(n): return (1-(-1)^n)/2
[(2*n^3 + 3*(6 +b(n))*n^2 + 2*(14 +9*b(n))*n + 27*b(n))/24 for n in (1..50)] # G. C. Greubel, Jul 12 2022

From Vladeta Jovovic, Jan 01 2003: (Start)
a(n) = (1/48)*(4*n^3 + (3*(-1)^(n+1) + 39)*n^2 + (18*(-1)^(n+1) + 74)*n + 27*(-1)^(n+1) + 27).
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).
G.f.: x*(4 + 2*x - x^2 - x^3)/((1+x)^3*(1-x)^4). (End)
a(n) = Sum_{i=1..ceiling(n/2)} (i+1)*(n-i+2) = ceiling(n/2)*(-2*ceiling(n/2)^2 + 3n*ceiling(n/2) + 9*n + 14)/6. - Wesley Ivan Hurt, Sep 20 2013
E.g.f.: (1/24)*( x*(69 + 24*x + 2*x^2)*cosh(x) + (27 + 48*x + 27*x^2 + 2*x^3)*sinh(x) ). - G. C. Greubel, Jul 12 2022

Name simplified by Jon E. Schoenfield, Jun 12 2019

A182259 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w^k<=x^k+y

Original entry on oeis.org

0, 3, 0, 11, 3, 0, 28, 11, 3, 0, 56, 28, 11, 3, 0, 99, 56, 26, 11, 3, 0, 159, 97, 52, 26, 11, 3, 0, 240, 153, 93, 50, 26, 11, 3, 0, 344, 230, 149, 85, 50, 26, 11, 3, 0, 475, 330, 222, 139, 85, 50, 26, 11, 3, 0, 635, 453, 314, 212, 133, 85, 50, 26, 11, 3, 0, 828

Offset: 1

Clark Kimberling, Apr 22 2012

Row 1:  A182260
Row 2:  A211810
Row 3:  A211811
Limiting row sequence: A051925
Let R be the array in A211808 and let R' be the array in A182259.  Then R(k,n)+R'(k,n)=3^(n-1).
See the Comments at A211790.

			Northwest corner (with antidiagonals read from northeast to southwest):
0...3...11...28...56...99...159
0...3...11...28...56...97...153
0...3...11...26...52...93...149
0...3...11...26...50...85...139
0...3...11...26...50...85...133

Cf. A211790.

z = 48;
t[k_, n_] := Module[{s = 0},
   (Do[If[2 w^k > x^k + y^k, s = s + 1],
       {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
Table[t[1, n], {n, 1, z}]  (* A182260 *)
Table[t[2, n], {n, 1, z}]  (* A211810 *)
Table[t[3, n], {n, 1, z}]  (* A211811 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1],
    {n, 1, 12}, {k, 1, n}]] (* A182259 *)
Table[k (k - 1) (2 k + 5)/6,
    {k, 1, z}] (* row-limit sequence, A051925 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A185874 Second accumulation array of A051340, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 6, 11, 10, 10, 21, 26, 20, 15, 34, 48, 50, 35, 21, 50, 76, 90, 85, 56, 28, 69, 110, 140, 150, 133, 84, 36, 91, 150, 200, 230, 231, 196, 120, 45, 116, 196, 270, 325, 350, 336, 276, 165, 55, 144, 248, 350, 435, 490, 504, 468, 375, 220, 66, 175, 306, 440, 560, 651, 700, 696, 630, 495, 286, 78, 209, 370, 540, 700, 833, 924, 960, 930, 825, 638, 364, 91, 246, 440, 650, 855, 1036, 1176, 1260, 1275, 1210, 1056, 806, 455, 105, 286, 516, 770, 1025, 1260, 1456, 1596, 1665, 1650, 1540, 1326, 1001, 560

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain: A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
.   1,   3,   6,   10,   15,   21,   28,   36,   45,   55, ...
.   4,  11,  21,   34,   50,   69,   91,  116,  144,  175, ...
.  10,  26,  48,   76,  110,  150,  196,  248,  306,  370, ...
.  20,  50,  90,  140,  200,  270,  350,  440,  540,  650, ...
.  35,  85, 150,  230,  325,  435,  560,  700,  855, 1025, ...
.  56, 133, 231,  350,  490,  651,  833, 1036, 1260, 1505, ...
.  84, 196, 336,  504,  700,  924, 1176, 1456, 1764, 2100, ...
. 120, 276, 468,  696,  960, 1260, 1596, 1968, 2376, 2820, ...
. 165, 375, 630,  930, 1275, 1665, 2100, 2580, 3105, 3675, ...
. 220, 495, 825, 1210, 1650, 2145, 2695, 3300, 3960, 4675, ...
...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

Cf. A144112, A051340, A141419, A185875, A185876.
Row 1 to 5: A000217, A115056, 2*A140096, 10*A000096, 5*A059845.
Column 1 to 3: A000292, A051925, A267370 and 3*A005581.
Main diagonal: A117066.

f[n_, k_] := (1/12)*k*n*(1 + n)*(1 + 3*k + 2*n);
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
Table[f[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten

T(n,k) = k*n*(n+1)*(2*n+3*k+1)/12 for k>=1, n>=1.

Edited by Bruno Berselli, Jan 14 2016

A185506 Accumulation array, T, of the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 7, 11, 10, 14, 23, 26, 20, 25, 42, 51, 50, 35, 41, 70, 88, 94, 85, 56, 63, 109, 140, 156, 155, 133, 84, 92, 161, 210, 240, 250, 237, 196, 120, 129, 228, 301, 350, 375, 374, 343, 276, 165, 175, 312, 416, 490, 535, 550, 532, 476, 375, 220, 231, 415, 558, 664, 735, 771, 770, 728, 639, 495, 286

Offset: 1

Clark Kimberling, Jan 29 2011

Suppose that R={R(n,k) : n>=1, k>=1} is a rectangular array. The accumulation array of R is given by T(n,k) = Sum_{R(i,j): 1<=i<=n, 1<=j<=k}. (See A144112.)

The formula for the integer T(n,k) has denominator 12. The 2nd, 3rd, and 4th accumulation arrays of A000027 have formulas in which the denominators are 144, 2880, and 86400, respectively; see A185507, A185508, and A185509.

			The natural number array A000027 starts with
  1, 2,  4,  7, ...
  3, 5,  8, 12, ...
  6, 9, 13, 18, ...
  ...
T(n,k) is the sum of numbers in the rectangle with corners at (1,1) and (n,k) of A000027, so that a corner of T is as follows:
   1,  3,   7,  14,  25,  41
   4, 11,  23,  42,  70, 109
  10, 26,  51,  88, 140, 210
  20, 50,  94, 156, 240, 350
  35, 85, 155, 250, 375, 535

G. C. Greubel, Table of n, a(n) for the first 50 antidiagonals, flattened

Cf. A000027, A185507, A185508, A185509.

Cf. A004006 (row 1), A000292 (col 1), A051925 (col 2), A185505 (1st diagonal).

f[n_,k_]:=k*n*(2n^2+3(k+1)*n+2k^2-3k+5)/12;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = k*n*(2*n^2 + 3*(k+1)*n + 2*k^2 - 3*k + 5)/12.

A213750 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 2*(n-1+h)-1, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 5, 3, 14, 11, 5, 30, 26, 17, 7, 55, 50, 38, 23, 9, 91, 85, 70, 50, 29, 11, 140, 133, 115, 90, 62, 35, 13, 204, 196, 175, 145, 110, 74, 41, 15, 285, 276, 252, 217, 175, 130, 86, 47, 17, 385, 375, 348, 308, 259, 205, 150, 98, 53, 19, 506, 495, 465, 420

Offset: 1

Clark Kimberling, Jun 20 2012

Principal diagonal:  A007585
Antidiagonal sums:  A002417
row 1,  (1,2,3,4,5,...)**(1,3,5,7,9,...): A000330
row 2,  (1,2,3,4,5,...)**(3,5,7,9,...): A051925
row 3,  (1,2,3,4,5,...)**(5,7,9,11,...): (2*k^3 + 15*k^2 + 13*k)/6
row 4,  (1,2,3,4,5,...)**(7,9,11,13,...): (2*k^3 + 21*k^2 + 19*k)/6
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....5....14...30....55....91
3....11...26...50....85....133
5....17...38...70....115...175
7....23...50...90....145...217
9....29...62...110...175...259
11...35...74...130...205...301

Cf. A213500.

b[n_] := n; c[n_] := 2 n - 1;
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213750 *)
d = Table[t[n, n], {n, 1, 40}] (* A007585 *)
s1 = Table[s[n], {n, 1, 50}] (* A002417 *)
FindLinearRecurrence[s1]
FindGeneratingFunction[s1, x]

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = (2*n - 1) - (2*n - 3)*x and g(x) = (1 - x )^4.

A145067 Zero followed by partial sums of A008865.

Original entry on oeis.org

0, -1, 1, 8, 22, 45, 79, 126, 188, 267, 365, 484, 626, 793, 987, 1210, 1464, 1751, 2073, 2432, 2830, 3269, 3751, 4278, 4852, 5475, 6149, 6876, 7658, 8497, 9395, 10354, 11376, 12463, 13617, 14840, 16134, 17501, 18943, 20462, 22060, 23739, 25501

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 30 2008

			a(2) = a(1) + 1^2 - 2 = 0 + 1 - 2 = -1; a(3) = a(2) + 2^2 - 2 = -1 + 4 - 2 = 1.

Cf. A008865 (n^2 - 2), A002522 (n^2 + 1), A145066 (partial sums of A002522, starting at n=1), A005563 ((n+1)^2 - 1), A051925 (zero followed by partial sums of A005563), A000330.

lst={0}; s=0; Do[s+=n^2 - 2; AppendTo[lst, s], {n, 5!}]; lst
Table[Sum[(i^2 + n - 1), {i, 0, n}], {n, -1, 41}] (* Zerinvary Lajos, Jul 11 2009 *)
Join[{0},Accumulate[Range[50]^2-2]] (* Harvey P. Dale, Jul 23 2018 *)

{a=2; for(n=0, 42, print1(a=a+n^2-2, ","))}

a(1) = 0; a(n) = a(n-1) + (n-1)^2 - 2 for n > 0.
a(n) = Sum_{k=1...n-1} (k^2-2) = A000330(n-1)-2*A000027(n-1) = (n-1)*(2*n^2-n-12)/6. - Christoph Pacher (christoph.pacher(AT)ait.ac.at), Jul 23 2010
G.f.: -x^2*(1-5*x+2*x^2)/(1-x)^4. - Colin Barker, Apr 04 2012

Edited by Klaus Brockhaus, Oct 17 2008

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