A052149 -id:A052149 - 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)

A000914 Stirling numbers of the first kind: s(n+2, n).

Original entry on oeis.org

0, 2, 11, 35, 85, 175, 322, 546, 870, 1320, 1925, 2717, 3731, 5005, 6580, 8500, 10812, 13566, 16815, 20615, 25025, 30107, 35926, 42550, 50050, 58500, 67977, 78561, 90335, 103385, 117800, 133672, 151096, 170170, 190995, 213675, 238317, 265031

Offset: 0

N. J. A. Sloane

Sum of product of unordered pairs of numbers from {1..n+1}.
Number of edges of a complete k-partite graph of order k*(k+1)/2 (A000217), K_1,2,3,...,k. - Roberto E. Martinez II, Oct 18 2001
This sequence holds the x^(n-2) coefficient of the characteristic polynomial of the N X N matrix A formed by MAX(i,j), where i is the row index and j is the column index of element A[i][j], 1 <= i,j <= N. Here N >= 2. - Paul Max Payton, Sep 06 2005
The sequence contains the partial sums of A006002, which represent the areas beneath lines created by the triangular numbers plotted (t(1),t(2)) connected to (t(2),t(3)) then (t(3),t(4))...(t(n-1),t(n)) and the x-axis. - J. M. Bergot, May 05 2012
Number of functions f from [n+2] to [n+2] with f(x)=x for exactly n elements x of [n+2] and f(x)>x for exactly two elements x of [n+2]. To prove this, let the two elements of [n+2] with a larger image be labeled i and j. Note both i and j must be less than n+2. Then there are (n+2-i) choices for f(i) and (n+2-j) choices for f(j). Summing the product of the number of choices over all sets {i,j} gives us "Sum of product of unordered pairs of numbers from {1..n+1}" in the first line of the Comments Section. See the example in the Example Section below. - Dennis P. Walsh, Sep 06 2017
Zhu Shijie gives in his Magnus Opus "Jade Mirror of the Four Unknowns" the problem: "Apples are piled in the form of a triangular pyramid. The top apple is worth 2 and the price of the whole is 1320. Each apple in one layer costs 1 less than an apple in the next layer below." We find the solution 9 to this problem in this sequence 1320 = a(9). Zhu Shijie gave the solution polynomial: "Let the element tian be the number of apples in a side of the base. From the statement we have 31680 for the negative shi, 10 for the positive fang, 21 for the positive first lian, 14 for the positive last lian, and 3 for the positive yu."  This translates into the polynomial equation: 3*x^4 + 14*x^3 + 21*x^2 + 10*x - 31680 = 0. - Thomas Scheuerle, Feb 10 2025

			Examples include E(K_1,2,3) = s(2+2,2) = 11 and E(K_1,2,3,4,5) = s(4+2,4) = 85, where E is the function that counts edges of graphs.
For n=2 the a(2)=11 functions f:[4]->[4] with exactly two f(x)=x and two f(x)>x are given by the 11 image vectors of form <f(1),f(2),f(3),f(4)> that follow: <1,3,4,4>, <1,4,4,4>, <2,2,4,4>, <3,2,4,4>, <4,2,4,4>, <2,3,3,4>, <2,4,3,4>, <3,3,3,4>, <3,4,3,4>, <4,3,3,4>, and <4,4,3,4>. - _Dennis P. Walsh_, Sep 06 2017

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
George E. Andrews, Number Theory, Dover Publications, New York, 1971, p. 4.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
H. S. Hall and S. R. Knight, Higher Algebra, Fourth Edition, Macmillan, 1891, p. 518.
Zhu Shijie, Jade Mirror of the Four Unknowns (Siyuan yujian), Book III Guo Duo Die Gang (Piles of Fruit), Problem number 1, 1303.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., Vol. 1, No. 3 (1926), pp. 44-49. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Zhu Shijie, Jade Mirror of the Four Unknowns 2, Translation by Library of Chinese classics, original from 1303.
Wikipedia, Jade Mirror of the Four Unknowns.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000290, A033428, A033581, A033583, A008275, A052149.
Cf. similar sequences listed in A241765.
Cf. A001296.
Cf. A006325(n+1) (Zhu Shijie's problem number 2 uses a pyramid with square base).

a000914 n = a000914_list !! n
a000914_list = scanl1 (+) a006002_list
-- Reinhard Zumkeller, Mar 25 2014

[StirlingFirst(n+2, n): n in [0..40]]; // Vincenzo Librandi, May 28 2019

A000914 := n -> 1/24*(n+1)*n*(n+2)*(3*n+5);
A000914 := proc(n)
    combinat[stirling1](n+2,n) ;
end proc: # R. J. Mathar, May 19 2016

Table[StirlingS1[n+2,n],{n,0,40}] (* Harvey P. Dale, Aug 24 2011 *)
a[ n_] := n (n + 1) (n + 2) (3 n + 5) / 24; (* Michael Somos, Sep 04 2017 *)

a(n)=sum(i=1,n+1,sum(j=1,n+1,i*j*(i

a(n)=sum(i=1,n+1,sum(j=1,i-1,i*j)) \\ Charles R Greathouse IV, Apr 07 2015

a(n) = binomial(n+2, 3)*(3*n+5)/4 \\ Charles R Greathouse IV, Apr 07 2015

[stirling_number1(n+2, n) for n in range(41)] # Zerinvary Lajos, Mar 14 2009

a(n) = binomial(n+2, 3)*(3*n+5)/4 = (n+1)*n*(n+2)*(3*n+5)/24.
E.g.f.: exp(x)*x*(48 + 84*x + 32*x^2 + 3*x^3)/24.
G.f.: (2*x+x^2)/(1-x)^5. - Simon Plouffe in his 1992 dissertation.
a(n) = Sum_{i=1..n} i*(i+1)^2/2. - Jon Perry, Jul 31 2003
a(n) = A052149(n+1)/2. - J. M. Bergot, Jun 02 2012
-(3*n+2)*(n-1)*a(n) + (n+2)*(3*n+5)*a(n-1) = 0. - R. J. Mathar, Apr 30 2015
a(n) = a(n-1) + (n+1)*binomial(n+1,2) for n >= 1. - Dennis P. Walsh, Sep 21 2015
a(n) = A001296(-2-n) for all n in Z. - Michael Somos, Sep 04 2017
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 162*log(3)/5 - 18*sqrt(3)*Pi/5 - 384/25.
Sum_{n>=1} (-1)^(n+1)/a(n) = 36*sqrt(3)*Pi/5 - 96*log(2)/5 - 636/25. (End)
a(n) = 3*A000332(n+3) - A000292(n). - Yasser Arath Chavez Reyes, Apr 03 2024

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000
Comments from Michael Somos, Jan 29 2000
Erroneous duplicate of the polynomial formula removed by R. J. Mathar, Sep 15 2009

A153978 a(n) = n(n-1)(n+1)(3n-2)/12.

Original entry on oeis.org

0, 2, 14, 50, 130, 280, 532, 924, 1500, 2310, 3410, 4862, 6734, 9100, 12040, 15640, 19992, 25194, 31350, 38570, 46970, 56672, 67804, 80500, 94900, 111150, 129402, 149814, 172550, 197780, 225680, 256432, 290224, 327250, 367710, 411810, 459762

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Partial sums of A011379.

Antidiagonal sums of the convolution array A213819. - Clark Kimberling, Jul 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A003215, A000537, A000578, A005898, A027602, A006007, A153976, A153977, A011379, A052149, A213819.

With[{r=Range[0,50]},Accumulate[r^2+r^3]] (* Harvey P. Dale, Jan 16 2011 *)
Rest[CoefficientList[Series[-2 x^2 * (2 x + 1)/(x - 1)^5, {x, 0, 40}], x]] (* Vincenzo Librandi, Jun 30 2014 *)
LinearRecurrence[{5,-10,10,-5,1}, {0,2,14,50,130}, 25] (* G. C. Greubel, Sep 01 2016 *)

concat(0, Vec(-2*x^2*(2*x+1)/(x-1)^5 + O(x^100))) \\ Colin Barker, Jun 28 2014

a(n) = n*(n-1)*(n+1)*(3*n-2)/12 \\ Charles R Greathouse IV, Sep 01 2016

a(n) = 2 * A001296(n-1) = (n-1)*n*(n+1)*(3*n-2)/12 (n>0). - Bruno Berselli, Apr 21 2010
a(n) = Sum_{i=1..n-1} binomial(i+1,i)*i^2. - Enrique Pérez Herrero, Jun 28 2014
G.f.: 2*x^2*(2*x+1) / (1 - x)^5. - Colin Barker, Jun 28 2014
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4. - Vincenzo Librandi, Jun 30 2014
a(n) = Sum_{k=1..n-1}k*((n-1)*n/2 + k) for n > 1. - J. M. Bergot, Feb 16 2018
From Amiram Eldar, Aug 23 2022: (Start)
Sum_{n>=2} 1/a(n) = 141/5 - 9*sqrt(3)*Pi/5 - 81*log(3)/5.
Sum_{n>=2} (-1)^n/a(n) = 18*sqrt(3)*Pi/5 + 48*log(2)/5 - 129/5. (End)

Edited by Bruno Berselli, Jun 15 2010

Simpler definition as suggested by Wesley Ivan Hurt, Jun 29 2014

A173020 Triangle of Generalized Runyon numbers R_{n,k}^(3) read by rows.

Original entry on oeis.org

1, 1, 3, 1, 9, 12, 1, 18, 66, 55, 1, 30, 210, 455, 273, 1, 45, 510, 2040, 3060, 1428, 1, 63, 1050, 6650, 17955, 20349, 7752, 1, 84, 1932, 17710, 74382, 148764, 134596, 43263, 1, 108, 3276, 40950, 245700, 753480, 1184040, 888030, 246675, 1, 135, 5220, 85260, 690606, 2992626, 7125300, 9161100, 5852925, 1430715

Offset: 1

R. J. Mathar, Nov 08 2010

The Runyon numbers R_{n,k}^(1) are A001263, R_{n,k}^(2) are A108767.

Row sums are in A002293.

			The triangle starts in row n=1 as
  1;
  1,  3;
  1,  9,   12;
  1, 18,   66,    55;
  1, 30,  210,   455,   273;
  1, 45,  510,  2040,  3060,   1428;
  1, 63, 1050,  6650, 17955,  20349,   7752;
  1, 84, 1932, 17710, 74382, 148764, 134596, 43263;

Chunwei Song, The Generalized Schroeder Theory, El. J. Combin. 12 (2005) #R53

G. C. Greubel, Rows n = 1..100 of the triangle, flattened
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020. See Fig. 6.
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.

Cf. A010054 (m=0), A001263 (m=1), A108767 (m=2), this sequence (m=3).

Cf. A001764, A002293, A003408, A045943, A052149.

A173020:= func< n,k,m | Binomial(n,k)*Binomial(m*n,k-1)/n >;
[A173020(n,k,3): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 20 2021

T[n_, k_, m_]:= Binomial[n, k]*Binomial[m*n, k-1]/n;
Table[T[n, k, 3], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)

def A173020(n,k,m): return binomial(n,k)*binomial(m*n,k-1)/n
flatten([[A173020(n,k,3) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 20 2021

T(n, k) = R(n,k,3) with R(n,k,m)= binomial(n,k)*binomial(m*n,k-1)/n, 1<=k<=n.
T(n, n) = A001764(n).
T(n, n-1) = A003408(n-2).
T(n, 2) = A045943(n-1).
T(n, 3) = n*(n-1)*(n-2)*(3*n-1)/4 = 3*A052149(n-1).
O.g.f. is series reversion with respect to x of x/((1+x)*(1+x*u)^3). - Peter Bala, Sep 12 2012
Sum_{k=1..n} T(n, k, 3) = binomial(4*n, n)/(3*n+1) = A002293(n). - G. C. Greubel, Feb 20 2021
n-th row polynomial = x * hypergeom([1 - n, -3*n], [2], x). - Peter Bala, Aug 30 2023

A087645 Third column of A071223.

Original entry on oeis.org

2, 6, 24, 72, 172, 352, 646, 1094, 1742, 2642, 3852, 5436, 7464, 10012, 13162, 17002, 21626, 27134, 33632, 41232, 50052, 60216, 71854, 85102, 100102, 117002, 135956, 157124, 180672, 206772, 235602, 267346, 302194, 340342, 381992, 427352, 476636

Offset: 2

N. J. A. Sloane, Oct 26 2003

Colin Barker, Table of n, a(n) for n = 2..1000
Alvaro Carbonero, Beth Anne Castellano, Gary Gordon, Charles Kulick, Karie Schmitz, and Brittany Shelton, Permutations of point sets in R_d, arXiv:2106.14140 [math.CO], 2021.
T. M. Cover, The number of linearly inducible orderings of points in d-space, SIAM J. Applied Math., 15 (1967), 434-439.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A052149, A071223.

CoefficientList[Series[-2 x^2*(x^4 - 4 x^3 + 7 x^2 - 2 x + 1)/(x - 1)^5, {x, 0, 38}], x][[3 ;; -1]] (* Michael De Vlieger, Oct 19 2021 *)

Vec(-2*x^2*(x^4-4*x^3+7*x^2-2*x+1)/(x-1)^5 + O(x^100)) \\ Colin Barker, Dec 06 2014

a(n) = A052149(n+1) + 2.
a(n) = (3*n^4-10*n^3+9*n^2-2*n+24)/12. - Vladeta Jovovic, Oct 26 2003
G.f.: -2*x^2*(x^4-4*x^3+7*x^2-2*x+1) / (x-1)^5. - Colin Barker, Dec 06 2014

More terms from Vladeta Jovovic, Oct 26 2003

A035291 Number of ways to place a non-attacking white and black queen on n X n chessboard.

Original entry on oeis.org

0, 0, 16, 88, 280, 680, 1400, 2576, 4368, 6960, 10560, 15400, 21736, 29848, 40040, 52640, 68000, 86496, 108528, 134520, 164920, 200200, 240856, 287408, 340400, 400400, 468000, 543816, 628488, 722680, 827080, 942400, 1069376, 1208768

Offset: 1

Erich Friedman

			There are 16 ways of putting distinct queens on 3 X 3 so that neither can capture the other.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

[(3*n^4-10*n^3+9*n^2-2*n)/3: n in [1..40]]; // Vincenzo Librandi, Apr 22 2012

I:=[0, 0, 16, 88,280]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Apr 22 2012

CoefficientList[Series[8*x^3*(2+x)/(1-x)^5,{x,0,40}],x] (* Vincenzo Librandi, Apr 22 2012 *)

a(n) = (3 n^4 - 10 n^3 + 9 n^2 - 2 n)/3.
Equals 4 * A052149(n-1). [N. J. A. Sloane, Feb 20 2005]
G.f.: 8*x^3*(2+x)/(1-x)^5. [Colin Barker, Apr 17 2012]

A071910 a(n) = t(n)t(n+1)t(n+2), where t() are the triangular numbers.

Original entry on oeis.org

0, 18, 180, 900, 3150, 8820, 21168, 45360, 89100, 163350, 283140, 468468, 745290, 1146600, 1713600, 2496960, 3558168, 4970970, 6822900, 9216900, 12273030, 16130268, 20948400, 26910000, 34222500, 43120350, 53867268, 66758580, 82123650, 100328400, 121777920

Offset: 0

N. J. A. Sloane, Jun 13 2002

nonn

a(n) is also the number of three-dimensional cage assemblies such that the assembly is not a cube. See also A052149 for the two-dimensional version and to A059827 for the non-exclusive version. - Alejandro Rodriguez, Oct 20 2020

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A006542, (first differences of a(n) /18) A006414, (second differences of a(n) /18) A006322, (third differences of a(n) /18) A004068, (fourth differences of a(n) /18) A005891, (fifth differences of a(n) /18) A008706.

Join[{0},Times@@@Partition[Accumulate[Range[40]],3,1]] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,18,180,900,3150,8820,21168},40] (* Harvey P. Dale, Aug 08 2025 *)

t(n) = n*(n+1)/2;
a(n) = t(n)*t(n+1)*t(n+2); \\ Michel Marcus, Oct 21 2015

a(n) = 18*A006542(n+3). - Vladeta Jovovic, Jun 14 2002
G.f.: 18*x*(1+3*x+x^2)/(1-x)^7. - Vladeta Jovovic, Jun 14 2002
a(n) = ((n+1)*(n+2))^3/8 - Sum_{i=1..n+1} i^3. - Jon Perry, Feb 13 2004
a(n) = C(2+n, n)*C(3+n, 1+n)*C(4+n, 2+n). - Zerinvary Lajos, Jul 29 2005
a(n) = A059827(n+1) - A000537(n+1). - Michel Marcus, Oct 21 2015

cp's OEIS Frontend

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

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A000914 Stirling numbers of the first kind: s(n+2, n).

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A153978 a(n) = n*(n-1)*(n+1)*(3*n-2)/12.

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A173020 Triangle of Generalized Runyon numbers R_{n,k}^(3) read by rows.

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A087645 Third column of A071223.

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A035291 Number of ways to place a non-attacking white and black queen on n X n chessboard.

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A153978 a(n) = n(n-1)(n+1)(3n-2)/12.

A071910 a(n) = t(n)t(n+1)t(n+2), where t() are the triangular numbers.