A000537 -id:A000537 - cp's OEIS frontend

A005898 Centered cube numbers: n^3 + (n+1)^3.

1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, 10745, 12691, 14859, 17261, 19909, 22815, 25991, 29449, 33201, 37259, 41635, 46341, 51389, 56791, 62559, 68705, 75241, 82179, 89531, 97309, 105525, 114191, 123319, 132921

Offset: 0

N. J. A. Sloane

Write the natural numbers in groups: 1; 2,3,4; 5,6,7,8,9; 10,11,12,13,14,15,16; ..... and add the groups, i.e., a(n) = Sum_{j=n^2-2(n-1)..n^2} j. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Sep 05 2001
The numbers 1, 9, 35, 91, etc. are divisible by 1, 3, 5, 7, etc. Therefore there are no prime numbers in this list. 9 is divisible by 3 and every third number after 9 is also divisible by 3. 35 is divisible by 5 and 7 and every fifth number after 35 is also divisible by 5 and every seventh number after 35 is also divisible by 7. This pattern continues indefinitely. - Howard Berman (howard_berman(AT)hotmail.com), Nov 07 2008
n^3 + (n+1)^3 = (2n+1)*(n^2+n+1), hence all terms are composite. - Zak Seidov, Feb 08 2011
This is the order of an n-ball centered at a node in the Kronecker product (or direct product) of three cycles, each of whose lengths is at least 2n+2. - Pranava K. Jha, Oct 10 2011
Positive y values of 4*x^3 - 3*x^2 = y^2. - Bruno Berselli, Apr 28 2018

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 52.
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
Pranava K. Jha, Perfect r-domination in the Kronecker product of three cycles, IEEE Trans. Circuits and Systems-I: Fundamental Theory and Applications, vol. 49, no. 1, pp. 89 - 92, Jan. 2002.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
Michael Penn, what's the pattern, Kenneth?, YouTube video, 2021.
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
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Eric Weisstein's World of Mathematics, Centered Cube Number
D. Zeitlin, A family of Galileo sequences, Amer. Math. Monthly 82 (1975), 819-822.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

(1/12)*t*(2*n^3 - 3*n^2 + n) + 2*n - 1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
Cf. A003215, A000537, A000578. - Vladimir Joseph Stephan Orlovsky, Jan 03 2009
Partial sums of A005897.
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e:  A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

[n^3+(n+1)^3: n in [0..40]]; // Vincenzo Librandi, Dec 16 2015

A005898:=(z+1)*(z**2+4*z+1)/(z-1)**4; # Simon Plouffe in his 1992 dissertation

a[n_]:=n^3; Table[a[n]+a[n+1], {n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Jan 03 2009 *)
CoefficientList[Series[(1 + 5 x + 5 x^2 + x^3)/(1 - x)^4,{x, 0, 40}], x] (* Vincenzo Librandi, Dec 16 2015 *)

a(n)=n^3 + (n+1)^3 \\ Anders Hellström, Dec 16 2015

A005898_list, m = [], [12, -6, 2, 1]
for _ in range(10**2):
    A005898_list.append(m[-1])
    for i in range(3):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

[i^3+(i+1)^3 for i in range(0,39)] # Zerinvary Lajos, Jul 03 2008

a(n) = Sum_{i=0..n} A005897(i), partial sums. - Jonathan Vos Post, Feb 06 2011
G.f.: (x^2+4*x+1)*(1+x)/(1-x)^3. - Simon Plouffe (see MAPLE section) and Colin Barker, Jan 02 2012; edited by N. J. A. Sloane, Feb 07 2018
a(n) = A037270(n+1) - A037270(n). - Ivan N. Ianakiev, May 13 2012
a(n) = A000217(n+1)^2 - A000217(n-1)^2. - Bob Selcoe, Mar 25 2016
a(n) = A005408(n) * A002061(n+1). - Miquel Cerda, Oct 05 2016
From Ilya Gutkovskiy, Oct 06 2016: (Start)
E.g.f.: (1 + 8*x + 9*x^2 + 2*x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
a(n) = (A081435(n))^2 - (A081435(n) - 1)^2. - Sergey Pavlov, Mar 01 2017

A000538 Sum of fourth powers: 0^4 + 1^4 + ... + n^4.

Original entry on oeis.org

0, 1, 17, 98, 354, 979, 2275, 4676, 8772, 15333, 25333, 39974, 60710, 89271, 127687, 178312, 243848, 327369, 432345, 562666, 722666, 917147, 1151403, 1431244, 1763020, 2153645, 2610621, 3142062, 3756718, 4463999, 5273999, 6197520, 7246096, 8432017, 9768353

Offset: 0

N. J. A. Sloane

This sequence is related to A000537 by the transform a(n) = n*A000537(n) - Sum_{i=0..n-1} A000537(i). - Bruno Berselli, Apr 26 2010
A formula for the r-th successive summation of k^4, for k = 1 to n, is ((12*n^2+(12*n-5)*r+r^2)*(2*n+r)*(n+r)!)/((r+4)!*(n-1)!), (H. W. Gould). - Gary Detlefs, Jan 02 2014
The number of four dimensional hypercubes in a 4D grid with side lengths n. This applies in general to k dimensions. That is, the number of k-dimensional hypercubes in a k-dimensional grid with side lengths n is equal to the sum of 1^k + 2^k + ... + n^k. - Alejandro Rodriguez, Oct 20 2020

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 222.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991, p. 275.
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].
J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359.
J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359. [Annotated scanned copy]
Bruno Berselli, A description of the transform in Comments lines: website Matem@ticamente (in Italian).
Stefano Capparelli, Notes on Discrete Math, Società Editrice Esculapio SRL (2019) 3-4.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
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
Eric Weisstein, MathWorld: Faulhaber's Formula
Wikipedia, Faulhaber's formula
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000217, A000330, A000537, A000539, A000540, A000541, A000542, A007487, A023002, A064538, A101089.
Row 4 of array A103438.
Cf. A000583.
Cf. A254640.

a000538 n = (3 * n * (n + 1) - 1) * (2 * n + 1) * (n + 1) * n `div` 30
-- Reinhard Zumkeller, Nov 11 2012

[n*(1+n)*(1+2*n)*(-1+3*n+3*n^2)/30: n in [0..35]]; // Vincenzo Librandi, Apr 04 2015

A000538 := n-> n*(n+1)*(2*n+1)*(3*n^2+3*n-1)/30;

Accumulate[Range[0,40]^4] (* Harvey P. Dale, Jan 13 2011 *)
CoefficientList[Series[x (1 + 11 x + 11 x^2 + x^3)/(1 - x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 07 2015 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 17, 98, 354, 979}, 35] (* Jean-François Alcover, Feb 09 2016 *)
Table[x^5/5+x^4/2+x^3/3-x/30,{x,40}] (* Harvey P. Dale, Jun 06 2021 *)

A000538(n):=n*(n+1)*(2*n+1)*(3*n^2+3*n-1)/30$
makelist(A000538(n),n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n) = n*(1+n)*(1+2*n)*(-1+3*n+3*n^2)/30 \\ Charles R Greathouse IV, Nov 20 2012

concat(0, Vec(x*(1+11*x+11*x^2+x^3)/(1-x)^6 + O(x^100))) \\ Altug Alkan, Dec 07 2015

A000538_list, m = [0], [24, -36, 14, -1, 0, 0]
for _ in range(10**2):
    for i in range(5):
        m[i+1] += m[i]
    A000538_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014

def A000538(n): return n*(n**2*(n*(6*n+15)+10)-1)//30 # Chai Wah Wu, Oct 03 2024

[bernoulli_polynomial(n,5)/5 for n in range(1, 35)] # Zerinvary Lajos, May 17 2009

a(n) = n*(1+n)*(1+2*n)*(-1+3*n+3*n^2)/30.
The preceding formula is due to al-Kachi (1394-1437). - Juri-Stepan Gerasimov, Jul 12 2009
G.f.: x*(x+1)*(1+10*x+x^2)/(1-x)^6. Simon Plouffe in his 1992 dissertation. More generally, the o.g.f. for Sum_{k=0..n} k^m is x*E(m, x)/(1-x)^(m+2), where E(m, x) is the Eulerian polynomial of degree m (cf. A008292). The e.g.f. for these o.g.f.s is: x/(1-x)^2*(exp(y/(1-x))-exp(x*y/(1-x)))/(exp(x*y/(1-x))-x*exp(y/(1-x))). - Vladeta Jovovic, May 08 2002
a(n) = Sum_{i = 1..n} J_4(i)*floor(n/i), where J_4 is A059377. - Enrique Pérez Herrero, Feb 26 2012
a(n) = 5*a(n-1) - 10* a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + 24. - Ant King, Sep 23 2013
a(n) = -Sum_{j=1..4} j*Stirling1(n+1,n+1-j)*Stirling2(n+4-j,n). - Mircea Merca, Jan 25 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = -30*(4 + 3/cos(sqrt(7/3)*Pi/2))*Pi/7. - Vaclav Kotesovec, Feb 13 2015
a(n) = (n + 1)*(n + 1/2)*n*(n + 1/2 + sqrt(7/12))*(n + 1/2  - sqrt(7/12))/5, see the Graham et al. reference, p. 275. - Wolfdieter Lang, Apr 02 2015

The general V. Jovovic formula has been slightly changed after his approval by Wolfdieter Lang, Nov 03 2011

A008459 Square the entries of Pascal's triangle.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 36, 16, 1, 1, 25, 100, 100, 25, 1, 1, 36, 225, 400, 225, 36, 1, 1, 49, 441, 1225, 1225, 441, 49, 1, 1, 64, 784, 3136, 4900, 3136, 784, 64, 1, 1, 81, 1296, 7056, 15876, 15876, 7056, 1296, 81, 1, 1, 100, 2025, 14400, 44100, 63504, 44100, 14400, 2025, 100, 1

Offset: 0

N. J. A. Sloane

Number of lattice paths from (0, 0) to (n, n) with steps (1, 0) and (0, 1), having k right turns. - Emeric Deutsch, Nov 23 2003
Product of A007318 and A105868. - Paul Barry, Nov 15 2005
Number of partitions that fit in an n X n box with Durfee square k. - Franklin T. Adams-Watters, Feb 20 2006
From Peter Bala, Oct 23 2008: (Start)
Narayana numbers of type B. Row n of this triangle is the h-vector of the simplicial complex dual to an associahedron of type B_n (a cyclohedron) [Fomin & Reading, p. 60]. See A063007 for the corresponding f-vectors for associahedra of type B_n. See A001263 for the h-vectors for associahedra of type A_n. The Hilbert transform of this triangular array is A108625 (see A145905 for the definition of this term).
Let A_n be the root lattice generated as a monoid by {e_i - e_j: 0 <= i, j <= n + 1}. Let P(A_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the h-vectors of a unimodular triangulation of P(A_n) [Ardila et al.]. A063007 is the corresponding array of f-vectors for these type A_n polytopes. See A086645 for the array of h-vectors for type C_n polytopes and A108558 for the array of h-vectors associated with type D_n polytopes.
(End)
The n-th row consists of the coefficients of the polynomial P_n(t) = Integral_{s = 0..2*Pi} (1 + t^2 - 2*t*cos(s))^n/Pi/2 ds. For example, when n = 3, we get P_3(t) = t^6 + 9*t^4 + 9*t^2 + 1; the coefficients are 1, 9, 9, 1. - Theodore Kolokolnikov, Oct 26 2010
Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0, 2*sqrt(-y)). Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence n!^2 as defined in Wang and Wang. - Peter Bala, Jul 24 2013
From Colin Defant, Sep 16 2018: (Start)
Let s denote West's stack-sorting map. T(n,k) is the number of permutations pi of [n+1] with k descents such that s(pi) avoids the patterns 132, 231, and 321. T(n,k) is also the number of permutations pi of [n+1] with k descents such that s(pi) avoids the patterns 132, 312, and 321.
T(n,k) is the number of permutations of [n+1] with k descents that avoid the patterns 1342, 3142, 3412, and 3421. (End)
The number of convex polyominoes whose smallest bounding rectangle has size (k+1)*(n+1-k) and which contain the lower left corner of the bounding rectangle (directed convex polyominoes). - Günter Rote, Feb 27 2019
Let P be the poset [n] X [n] ordered by the product order.  T(n,k) is the number of antichains in P containing exactly k elements. Cf. A063746. - Geoffrey Critzer, Mar 28 2020

			Pascal's triangle begins
  1
  1  1
  1  2   1
  1  3   3   1
  1  4   6   4   1
  1  5  10  10   5   1
  1  6  15  20  15   6   1
  1  7  21  35  35  21   7   1
...
so the present triangle begins
  1
  1   1
  1   4    1
  1   9    9     1
  1  16   36    16     1
  1  25  100   100    25    1
  1  36  225   400   225   36   1
  1  49  441  1225  1225  441  49   1
...

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 12.
J. Riordan, An introduction to combinatorial analysis, Dover Publications, Mineola, NY, 2002, page 191, Problem 15. MR1949650
P. G. Tait, On the Linear Differential Equation of the Second Order, Proceedings of the Royal Society of Edinburgh, 9 (1876), 93-98 (see p. 97) [From Tom Copeland, Sep 09 2010, vol number corrected Sep 10 2010]

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
N. Alexeev and A. Tikhomirov, Singular Values Distribution of Squares of Elliptic Random Matrices and type-B Narayana Polynomials, arXiv preprint arXiv:1501.04615 [math.PR], 2015.
F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices, arXiv:0809.5123 [math.CO], 2008.
Peter Bala, A commutative diagram of triangular arrays
E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005), 62-78.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
Carl M. Bender and Gerald V. Dunne, Polynomials and operator orderings, J. Math. Phys. 29 (1988), 1727-1731.
Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, and André Schulz, The Number of Convex Polyominoes with Given Height and Width, arXiv:1903.01095 [math.CO], 2019.
John H. Conway and N. J. A. Sloane, Low-dimensional lattices. VII Coordination sequences, Proc. R. Soc. Lond. A (1997) 453, 2369-2389.
R. Cori and G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
R. Cori and G. Hetyei, How to count genus one partitions, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Sergey Fomin and Nathan Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005, 2008. [From _Peter Bala_, Oct 23 2008]
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 10, 13.
Abdelkader Necer, Séries formelles et produit de Hadamard, Journal de théorie des nombres de Bordeaux, 9:2 (1997), pp. 319-335.
Weiping Wang and Tianming Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
Yi Wang and Arthur L.B. Yang, Total positivity of Narayana matrices, arXiv:1702.07822 [math.CO], 2017.
Harold R. L. Yang and Philip B. Zhang, Stable multivariate Narayana polynomials and labeled plane trees, arXiv:2403.15058 [math.CO], 2024. See p. 2.

Row sums are in A000984. Columns 0-3 are A000012, A000290, A000537, A001249.
Family of polynomials (see A062145): this sequence (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
Cf. A007318, A055133, A116647, A001263, A086645, A063007, A108558, A108625 (Hilbert transform), A145903, A181543, A086645 (logarithmic derivative), A105868 (inverse binomial transform), A093118.

Flat(List([0..10],n->List([0..n],k->Binomial(n,k)^2))); # Muniru A Asiru, Mar 30 2018

/* As triangle */ [[Binomial(n, k)^2: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Dec 15 2016

seq(seq(binomial(n, k)^2, k=0..n), n=0..10);

Table[Binomial[n, k]^2, {n, 0, 11}, {k, 0, n}]//Flatten (* Alonso del Arte, Dec 08 2013 *)

create_list(binomial(n,k)^2,n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

T(n,k):=if n=k then 1 else if k=0 then 1 else T(n-1,k)*(n+k)/(n-k)+T(n-1,k-1); /* Vladimir Kruchinin, Oct 18 2014 */

A(x,y):=1/sqrt(1-2*x-2*x*y+x^2-2*x^2*y+x^2*y^2);
taylor(x*A(x,y)+x*y*A(x,y)+sqrt(1+4*x^2*y*A(x,y)^2),x,0,7,y,0,7); /* Vladimir Kruchinin, Oct 23 2020 */

{T(n, k) = if( k<0 || k>n, 0, binomial(n, k)^2)}; /* Michael Somos, May 03 2004 */

{T(n,k)=polcoeff(polcoeff(sum(m=0,n,(2*m)!/m!^2*x^(2*m)*y^m/(1-x-x*y+x*O(x^n))^(2*m+1)),n,x),k,y)} \\ Paul D. Hanna, Oct 31 2010

def A008459(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),n-comb(r+1,2))**2 # Chai Wah Wu, Nov 12 2024

T(n,k) = A007318(n,k)^2. - Sean A. Irvine, Mar 29 2018
E.g.f.: exp((1+y)*x)*BesselI(0, 2*sqrt(y)*x). - Vladeta Jovovic, Nov 17 2003
G.f.: 1/sqrt(1-2*x-2*x*y+x^2-2*x^2*y+x^2*y^2); g.f. for row n: (1-t)^n P_n[(1+t)/(1-t)] where the P_n's are the Legendre polynomials. - Emeric Deutsch, Nov 23 2003 [The original version of the bivariate g.f. has been modified with the roles of x and y interchanged so that now x corresponds to n and y to k. - Petros Hadjicostas, Oct 22 2017]
G.f. for column k is Sum_{j = 0..k} C(k, j)^2*x^(k+j)/(1 - x)^(2*k+1). - Paul Barry, Nov 15 2005
Column k has g.f. (x^k)*Legendre_P(k, (1+x)/(1-x))/(1 - x)^(k+1) = (x^k)*Sum_{j = 0..k} C(k, j)^2*x^j/(1 - x)^(2*k+1). - Paul Barry, Nov 19 2005
Let E be the operator D*x*D, where D denotes the derivative operator d/dx. Then (1/n!^2) * E^n(1/(1 - x)) = (row n generating polynomial)/(1 - x)^(2*n+1) = Sum_{k >= 0} binomial(n+k, k)^2*x^k. For example, when n = 3 we have (1/3!)^2*E^3(1/(1 - x)) = (1 + 9*x + 9*x^2 + x^3)/(1 - x)^7 = (1/3!)^2 * Sum_{k >= 0} ((k+1)*(k+2)*(k+3))^2*x^k. - Peter Bala, Oct 23 2008
G.f.: A(x, y) = Sum_{n >= 0} (2*n)!/n!^2 * x^(2*n)*y^n/(1 - x - x*y)^(2*n+1). - Paul D. Hanna, Oct 31 2010
From Peter Bala, Jul 24 2013: (Start)
Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0, 2*sqrt(-y)). Generating function: E(y)*E(x*y) = 1 + (1 + x)*y + (1 + 4*x + x^2)*y^2/2!^2 + (1 + 9*x + 9*x^2 + x^3)*y^3/3!^2 + .... Cf. the unsigned version of A021009 with generating function exp(y)*E(x*y).
The n-th power of this array has the generating function E(y)^n*E(x*y). In particular, the matrix inverse A055133 has the generating function E(x*y)/E(y). (End)
T(n,k) = T(n-1,k)*(n+k)/(n-k) + T(n-1,k-1), T(n,0) = T(n,n) = 1. - Vladimir Kruchinin, Oct 18 2014
Observe that the recurrence T(n,k) = T(n-1,k)*(n+k)/(n-k) - T(n-1,k-1), for n >= 2 and 1 <= k < n, with boundary conditions T(n,0) = T(n,n) = 1 gives Pascal's triangle A007318. - Peter Bala, Dec 21 2014
n-th row polynomial R(n, x) = [z^n] (1 + (1 + x)*z + x*z^2)^n. Note that 1/n*[z^(n-1)] (1 + (1 + x)*z + x*z^2)^n gives the row polynomials of A001263. - Peter Bala, Jun 24 2015
Binomial transform of A105868. If G(x,t) = 1/sqrt(1 - 2*(1 + t)*x + (1 - t)^2*x^2) denotes the o.g.f. of this array then 1 + x*d/dx log(G(x,t)) = 1 + (1 + t)*x + (1 + 6*t + t^2)*x^2 + ... is the o.g.f. for A086645. - Peter Bala, Sep 06 2015
T(n,k) = Sum_{i=0..n} C(n-i,k)*C(n,i)*C(n+i,i)*(-1)^(n-i-k). - Vladimir Kruchinin, Jan 14 2018
G.f. satisfies A(x,y) = x*A(x,y)+x*y*A(x,y)+sqrt(1+4*x^2*y*A(x,y)^2). - Vladimir Kruchinin, Oct 23 2020
G.f. satisfies the differential equation y * d^2(A(x,y))/dy^2 - x^2 * d^2(x*A(x,y))/dx^2 + 2*x^2* A(x,y)^3 = 0. - Sergii Voloshyn, Mar 07 2025
T(n,k) = Sum_{i=0..n} C(2*n+1,i)*C(n+k-i,n)^2*(-1)^i. - Natalia L. Skirrow, Apr 14 2025

A000539 Sum of 5th powers: 0^5 + 1^5 + 2^5 + ... + n^5.

Original entry on oeis.org

0, 1, 33, 276, 1300, 4425, 12201, 29008, 61776, 120825, 220825, 381876, 630708, 1002001, 1539825, 2299200, 3347776, 4767633, 6657201, 9133300, 12333300, 16417401, 21571033, 28007376, 35970000, 45735625, 57617001, 71965908, 89176276, 109687425, 133987425, 162616576

Offset: 0

N. J. A. Sloane

This sequence is related to A000538 by a(n) = n*A000538(n) - Sum_{i=0..n-1} A000538(i). - Bruno Berselli, Apr 26 2010

See comment in A008292 for a formula for r-th successive summation of Sum_{k=1..n} k^j. - Gary Detlefs, Jan 02 2014

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991, p. 275.
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].
Bruno Berselli, A description of the transform in Comments lines: website Matem@ticamente (in Italian).
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
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
Eric Weisstein's World of Mathematics, Faulhaber's Formula
Wikipedia, Faulhaber's formula
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Partial sums of A000584. Row 5 of array A103438.

Cf. A000217, A000537, A000538, A006542, A008292, A059378, A101092.

[n^2*(n+1)^2*(2*n^2+2*n-1)/12: n in [0..30]]; // Vincenzo Librandi, Apr 04 2015

A000539:=-(1+26*z+66*z**2+26*z**3+z**4)/(z-1)**7; # Simon Plouffe in his 1992 dissertation
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^5 od: seq(a[n], n=0..30); # Zerinvary Lajos, Feb 22 2008
a:=n->sum(j^5,j=0..n): seq(a(n), n=0..30); # Zerinvary Lajos, Jun 05 2008

Accumulate[Range[0, 40]^5]
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 1, 33, 276, 1300, 4425, 12201}, 41] (* Jean-François Alcover, Feb 09 2016 *)

A000539(n):=n^2*(n+1)^2*(2*n^2+2*n-1)/12$ makelist(A000539(n),n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n)=n^2*(n+1)^2*(2*n^2+2*n-1)/12 \\ Charles R Greathouse IV, Jul 15 2011

concat(0, Vec(x*(1+26*x+66*x^2+26*x^3+x^4)/(1-x)^7 + O(x^100))) \\ Altug Alkan, Dec 07 2015

A000539_list, m = [0], [120, -240, 150, -30, 1, 0, 0]
for _ in range(10**2):
    for i in range(6):
        m[i+1] += m[i]
    A000539_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014

def A000539(n): return n**2*(n**2*(n*(n+3<<1)+5)-1)//12 # Chai Wah Wu, Oct 03 2024

a(n) = n^2*(n+1)^2*(2*n^2+2*n-1)/12.
a(n) = sqrt(Sum_{j=1..n}Sum_{i=1..n}(i*j)^5). - Alexander Adamchuk, Oct 26 2004
a(n) = Sum_{i = 1..n} J_5(i)*floor(n/i), where J_5 is A059378. - Enrique Pérez Herrero, Feb 26 2012
a(n) = 6*a(n-1) - 15* a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + 120. - Ant King, Sep 23 2013
a(n) = 120*C(n+3,6) + 30*C(n+2,4) + C(n+1,2) (Knuth). - Gary Detlefs, Jan 02 2014
a(n) = -Sum_{j=1..5} j*Stirling1(n+1,n+1-j)*Stirling2(n+5-j,n). - Mircea Merca, Jan 25 2014
Sum_{n>=1} 1/a(n) = 60 - 4*Pi^2 + 8*sqrt(3)*Pi * tan(sqrt(3)*Pi/2). - Vaclav Kotesovec, Feb 13 2015
a(n) = (n + 1)^2*n^2*(n + 1/2 + sqrt(3/4))*(n + 1/2 - sqrt(3/4))/6. See the Graham et al. reference, p. 275. - Wolfdieter Lang, Apr 02 2015
G.f.: x*(1+26*x+66*x^2+26*x^3+x^4)/(1-x)^7. - Robert Israel, Dec 07 2015
a(n) = (4/3)*A000217(n)^3 - (1/3)*A000217(n)^2. - Michael Raney, Feb 19 2016
a(n) = (binomial(n+1,4) + 6*binomial(n+2,4) + binomial(n+3,4))*(binomial(n+2,3) - binomial(n+1,3)). - Tony Foster III, Oct 21 2018
a(n) = 24*A006542(n+2) + A000537(n). - Yasser Arath Chavez Reyes, May 04 2024
E.g.f.: exp(x)*x*(12 + 186*x + 360*x^2 + 195*x^3 + 36*x^4 + 2*x^5)/12. - Stefano Spezia, May 04 2024

A001715 a(n) = n!/6.

Original entry on oeis.org

1, 4, 20, 120, 840, 6720, 60480, 604800, 6652800, 79833600, 1037836800, 14529715200, 217945728000, 3487131648000, 59281238016000, 1067062284288000, 20274183401472000, 405483668029440000, 8515157028618240000, 187333454629601280000, 4308669456480829440000

Offset: 3

N. J. A. Sloane

The numbers (4, 20, 120, 840, 6720, ...) arise from the divisor values in the general formula a(n) = n*(n+1)*(n+2)*(n+3)* ... *(n+k)*(n*(n+k) + (k-1)*k/6)/((k+3)!/6) (which covers the following sequences: A000578, A000537, A024166, A101094, A101097, A101102). - Alexander R. Povolotsky, May 17 2008
a(n) is also the number of decreasing 3-cycles in the decomposition of permutations as product of disjoint cycles, a(3)=1, a(4)=4, a(5)=20. - Wenjin Woan, Dec 21 2008
Equals eigensequence of triangle A130128 reflected. - Gary W. Adamson, Dec 23 2008
a(n) is the number of n-permutations having 1, 2, and 3 in three distinct cycles. - Geoffrey Critzer, Apr 26 2009
From Johannes W. Meijer, Oct 20 2009: (Start)
The asymptotic expansion of the higher order exponential integral E(x,m=1,n=4) ~ exp(-x)/x*(1 - 4/x + 20/x^2 - 120/x^3 + 840/x^4 - 6720/x^5 + 60480/x^6 - 604800/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information.
(End)

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).

Vincenzo Librandi, Table of n, a(n) for n = 3..200
Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 263.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

Cf. A049352, A049458, A049460.
Cf. A034472, A130128.
Cf. A245334, A000142, A111530.
Cf. A001710, A001720, A007759, A094638.
Cf. A094587, A138533, A173333, A213936.

a001715 = (flip div 6) . a000142 -- Reinhard Zumkeller, Aug 31 2014

[Factorial(n)/6: n in [3..30]]; // Vincenzo Librandi, Jun 20 2011

f := proc(n) n!/6; end;
BB:= [S, {S = Prod(Z,Z,C), C = Union(B,Z,Z), B = Prod(Z,C)}, labelled]: seq(combstruct[count](BB, size=n)/12, n=3..20); # Zerinvary Lajos, Jun 19 2008
G(x):=1/(1-x)^4: f[0]:=G(x): for n from 1 to 18 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..16); # Zerinvary Lajos, Apr 01 2009

a[n_]:=n!/6; (*Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
Range[3,30]!/6 (* Harvey P. Dale, Aug 12 2012 *)

a(n)=n!/6 \\ Charles R Greathouse IV, Jan 12 2012

a(n) = A049352(n-2, 1) (first column of triangle).
E.g.f. if offset 0: 1/(1-x)^4.
a(n) = A173333(n,3). - Reinhard Zumkeller, Feb 19 2010
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
G.f.: W(0), where W(k) = 1 - x*(k+4)/( x*(k+4) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
a(n) = A245334(n,n-3) / 4. - Reinhard Zumkeller, Aug 31 2014
From Peter Bala, May 22 2017: (Start)
The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (4*x - 1)*A(x) + 1 = 0.
G.f. as an S-fraction: A(x) = 1/(1 - 4*x/(1 - x/(1 - 5*x/(1 - 2*x/(1 - 6*x/(1 - 3*x/(1 - ... - (n + 3)*x/(1 - n*x/(1 - ... ))))))))) (apply Stokes, 1982).
A(x) = 1/(1 - 3*x - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1 - 6*x/(1 - ... - n*x/(1 - (n+3)*x/(1 - ... ))))))))). (End)
H(x) = (1 - (1 + x)^(-3)) / 3 = x - 4 x^2/2! + 20 x^3/3! - ... is an e.g.f. of the signed sequence (n!/4!), which is the compositional inverse of G(x) = (1 - 3*x)^(-1/3) - 1, an e.g.f. for A007559. Cf. A094638, A001710 (for n!/2!), and A001720 (for n!/4!). Cf. columns of A094587, A173333, and A213936 and rows of A138533.- Tom Copeland, Dec 27 2019
E.g.f.: x^3 / (3! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=3} 1/a(n) = 6*e - 15.
Sum_{n>=3} (-1)^(n+1)/a(n) = 3 - 6/e. (End)

More terms from Harvey P. Dale, Aug 12 2012

A024166 a(n) = Sum_{1 <= i < j <= n} (j-i)^3.

Original entry on oeis.org

0, 1, 10, 46, 146, 371, 812, 1596, 2892, 4917, 7942, 12298, 18382, 26663, 37688, 52088, 70584, 93993, 123234, 159334, 203434, 256795, 320804, 396980, 486980, 592605, 715806, 858690, 1023526, 1212751, 1428976, 1674992, 1953776, 2268497, 2622522, 3019422

Offset: 0

Clark Kimberling

Convolution of the cubes (A000578) with the positive integers a(n)=n+1, where all sequences have offset zero. - Graeme McRae, Jun 06 2006
a(n) gives the n-th antidiagonal sum of the convolution array A212891. - Clark Kimberling, Jun 16 2012
In general, the r-th successive summation of the cubes from 1 to n is (6*n^2 + 6*n*r + r^2 - r)*(n+r)!/((r+3)!*(n-1)!), n>0. Here r = 2. - Gary Detlefs, Mar 01 2013
The inverse binomial transform is (essentially) row n=2 of A087127. - R. J. Mathar, Aug 31 2022

			4*a(7) = 6384 = (0*1)^2 + (1*2)^2 + (2*3)^2 + (3*4)^2 + (4*5)^2 + (5*6)^2 + (6*7)^2 + (7*8)^2. - _Bruno Berselli_, Feb 05 2014

Elisabeth Busser and Gilles Cohen, Neuro-Logies - "Chercher, jouer, trouver", La Recherche, April 1999, No. 319, page 97.

T. D. Noe, Table of n, a(n) for n = 0..1000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 1. - _N. J. A. Sloane_, Mar 23 2014
Claudio de J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7.
Alexander R. Povolotsky, Problem 1147, Pi Mu Epsilon Fall 2006 Problems.
Alexander R. Povolotsky, Problem, Pi Mu Epsilon Spring 2007 Problems.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Cf. A000292, A000332, A000389, A000579, A000580, A024166, A027555, A085438, A085439, A085440, A085441, A085442, A086020, A086021, A086022, A086023, A086024, A086025, A086026, A086027, A086028, A086029, A086030, A087127.

Cf. A000330, A000537 (first diffs), A001286, A003215, A100536, A101094 (partial sums), A101097, A101102, A222716.

a024166 n = sum $ zipWith (*) [n+1,n..0] a000578_list
-- Reinhard Zumkeller, Oct 14 2001

[n*(n+1)*(n+2)*(3*n^2 + 6*n + 1)/60: n in [0..30]]; // G. C. Greubel, Nov 21 2017

A024166:=n->n*(n+1)*(n+2)*(3*n^2 + 6*n + 1)/60: seq(A024166(n), n=0..50); # Wesley Ivan Hurt, Nov 21 2017

Nest[Accumulate,Range[0,40]^3,2] (* Harvey P. Dale, Jan 10 2016 *)
Table[n*(n+1)*(n+2)*(3*n^2 + 6*n + 1)/60, {n,0,30}] (* G. C. Greubel, Nov 21 2017 *)

a(n)=sum(j=1,n, sum(m=1, j, sum(i=m*(m+1)/2-m+1, m*(m+1)/2, (2*i-1)))) \\ Alexander R. Povolotsky, May 17 2008

for(n=0,30, print1(n*(n+1)*(n+2)*(3*n^2 + 6*n + 1)/60, ", ")) \\ G. C. Greubel, Nov 21 2017

From Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999: (Start)
a(n) = Sum_{i=0..n} A000537(i), partial sums of A000537.
a(n) = n*(n+1)*(n+2)*(3*n^2 + 6*n + 1)/60. (End)
a(A004772(n)) mod 2 = 0; a(A016813(n)) mod 2 = 1. - Reinhard Zumkeller, Oct 14 2001
a(n) = Sum_{k=0..n} k^3*(n+1-k). - Paul Barry, Sep 14 2003; edited by Jon E. Schoenfield, Dec 29 2014
a(n) = 2*n*(n+1)*(n+2)*((n+1)^2 + 2*n*(n+2))/5!. This sequence could be obtained from the general formula a(n) = n*(n+1)*(n+2)*(n+3)* ...* (n+k) *(n*(n+k) + (k-1)*k/6)/((k+3)!/6) at k=2. - Alexander R. Povolotsky, May 17 2008
O.g.f.: x*(1 + 4*x + x^2)/(-1 + x)^6. - R. J. Mathar, Jun 06 2008
a(n) = (6*n^2 + 12*n + 2)*(n+2)!/(120*(n-1)!), n > 0. - Gary Detlefs, Mar 01 2013
a(n) = A222716(n+1)/10 = A000292(n)*A100536(n+1)/10. - Jonathan Sondow, Mar 04 2013
4*a(n) = Sum_{i=0..n} A000290(i)*A000290(i+1). - Bruno Berselli, Feb 05 2014
a(n) = Sum_{i=1..n} Sum_{j=1..n} i*j*(n - max(i, j) + 1). - Melvin Peralta, May 12 2016
a(n) = n*binomial(n+3, 4) + binomial(n+2, 5). - Tony Foster III, Nov 14 2017
a(n) = Sum_{i=1..n} i*A143037(n,n-i+1). - J. M. Bergot, Aug 30 2022

A037270 a(n) = n^2*(n^2 + 1)/2.

Original entry on oeis.org

0, 1, 10, 45, 136, 325, 666, 1225, 2080, 3321, 5050, 7381, 10440, 14365, 19306, 25425, 32896, 41905, 52650, 65341, 80200, 97461, 117370, 140185, 166176, 195625, 228826, 266085, 307720, 354061, 405450, 462241, 524800, 593505, 668746, 750925, 840456, 937765

Offset: 0

Aaron Gulliver (gulliver(AT)elec.canterbury.ac.nz)

Sum of first n^2 positive integers.
Start from xanthene and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink below on chemistry. - Robert G. Wilson v, Aug 02 2002; Amarnath Murthy, Aug 01 2002
Sum of the next n multiples of n. - Amarnath Murthy, Aug 01 2002
The sum of the terms in an n X n spiral. These are also triangular numbers. - William A. Tedeschi, Feb 27 2008
Hypotenuse of Pythagorean triangles with smallest side a cube: A000578(n)^2 + A083374(n)^2 = a(n)^2. - Martin Renner, Nov 12 2011
For n>1, triangular numbers that can be represented as a sum of a square and a triangular number. For example, a(2)=10=4+6=9+1. - Ivan N. Ianakiev, Apr 24 2012
A037270 can be constructed in the following manner: Take A000217 and for every n not in A000290 delete the corresponding A000217(n). - Ivan N. Ianakiev, Apr 26 2012
Starting at a(1)=1 simply take 1*1=1, a(2)= 2*(2+3)=10, a(3)= 3*(4+5+6)=45, a(4)=4*(7+8+9+10) and so on. - J. M. Bergot, May 01 2015
Observation: The digital roots of the terms repeat in the sequence 1, 1, 9; e.g., the digital roots of 1, 10, 45, 136, 325, and 666 are 1, 1, 9, 1, 1, and 9. Verified for the first 10000 terms. - Rob Barton, Mar 28 2018
The above observation is easily explained and proved given that the digital root of a positive number equals the number modulo 9, and a(n + 9k) == a(n) (mod 9). - M. F. Hasler, Apr 05 2018
Number of unoriented rows of length 4 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=10, there are 4 achiral (AAAA, ABBA, BAAB, BBBB) and 6 chiral pairs (AAAB-BAAA, AABA-ABAA, AABB-BBAA, ABAB-BABA, ABBB-BBBA, BABB-BBAB). - Robert A. Russell, Nov 14 2018
For n > 0, a(2n+1) is the number of non-isomorphic 6C_m-snakes, where m = 2n+1 or m = 2n (for n>=2). A kC_n-snake is a connected graph in which the k>=2 blocks are isomorphic to the cycle C_n and the block-cutpoint graph is a path. - Christian Barrientos, May 15 2019
Number of achiral colorings of the edges of a tetrahedron with n available colors. - Robert A. Russell, Sep 07 2019

C. Alsina and R. B. Nelson, Charming Proofs: A Journey into Elegant Mathematics, MAA, 2010. See p. 5.
C. Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60(2001), 85-96.
Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 55.
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
R. A. Wilson, Cosmic Trigger, epilogue of S.-P. Sirag.

T. D. Noe, Table of n, a(n) for n = 0..1000
J. D. Bell, A translation of Leonhard Euler's "De Quadratis Magicis", E795, arXiv:math/0408230 [math.CO], 2004-2005.
N. G de Bruijn, Some classes of integer-valued functions, Nederl. Akad. Wetensch. Proc, Ser. A, 58 (1955), 363-367. See page 363.
Th. Gruner, A. Kerber, R. Laue, and M. Meringer, Mathematics for Combinatorial Chemistry, In: F. Keil, W. Mackens, H. Voß and J. Wether, Scientific Computing in Chemical Engineering II, Springer, 1999, 74-81.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A236770 (see crossrefs).
Row 4 of A277504.
Cf. A000583 (oriented), A083374 (chiral), A000290 (achiral).
Cf. A317617.
Row 3 of A327086 (achiral simplex edge colorings).

a:=List([0..30],n->n^2*(n^2+1)/2); # Muniru A Asiru, Mar 28 2018

[n^2*(n^2 + 1)/2: n in [0..30]] // Stefano Spezia, Jan 15 2019

seq(n^2*(n^2+1)/2,n=0..30); # Muniru A Asiru, Mar 28 2018

Table[ n^2*((n^2 + 1)/2), {n, 0, 30} ]
Table[(1/8) Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n, 0, 30}] (* Artur Jasinski, Feb 10 2010 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,10,45,136},30] (* Harvey P. Dale, Aug 03 2014 *)

a(n)=binomial(n^2+1,2) \\ Charles R Greathouse IV, Apr 25 2012

for n in range(0,30): print(n**2*(n**2+1)/2, end=', ') # Stefano Spezia, Jan 10 2019

a(n) = a(n-1) + n^3 + (n-1)^3.
a(n) = A000537(n)+A000537(n-1), i.e., square of sum of first n integers plus square of sum of first n-1 integers. - Henry Bottomley, Oct 15 2001
a(n) = Sum_{k=0..n^2} k. - William A. Tedeschi, Feb 27 2008
a(n) = (1/8)*sinh(2*arcsinh(n)). - Artur Jasinski, Feb 10 2010
G.f.: x*(1+x)*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Mar 22 2012
a(n) = a(n-1) + A005898(n-1). - Ivan N. Ianakiev, May 13 2012
a(n) = 2 * A000217(n-1) * A000217(n) + A000290(n). - Ivan N. Ianakiev, May 26 2012
a(n) = A000217(n^2). - J. M. Bergot, Jun 07 2012
a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5) n>4, a(0)=0, a(1)=1, a(2)=10, a(3)=45, a(4)=136. - Yosu Yurramendi, Sep 02 2013
For n>0, a(n) = A000217(n)^2 + A000217(n-1)^2. - Richard R. Forberg, Dec 25 2013
a(n) = T(T(n)) + T(T(n-1)) + T(T(n)-1) + T(T(n-1)-1), where T(n) = A000217(n). - Charlie Marion, Sep 10 2016
a(n) = t(n-3)*t(n)+t(n-1)*t(n+2), with t(n)=A000217(n). - J. M. Bergot, Apr 07 2018
From Robert A. Russell, Nov 14 2018: (Start)
a(n) = (A000583(n) + A000290(n)) / 2 = (n^4 + n^2) / 2.
a(n) = A000583(n) - A083374(n) = A083374(n) + A000290(n).
G.f.: (Sum_{j=1..4} S2(4,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..2} S2(2,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: Sum_{k=1..4} A145882(4,k) * x^k / (1-x)^5.
E.g.f.: (Sum_{k=1..4} S2(4,k)*x^k + Sum_{k=1..2} S2(2,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>4, a(n) = Sum_{j=1..5} -binomial(j-6,j) * a(n-j). (End)
a(n) = n*A006003(n). - Kritsada Moomuang, Dec 16 2018
For n > 0, a(n) = Sum_{k=1..n} A317617(n,k). - Stefano Spezia, Jan 10 2019
Sum_{n>=1} 1/a(n) = 1 + Pi^2/3 - Pi*coth(Pi) = 1.13652003875929052467672874379... - Vaclav Kotesovec, Jan 21 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi*csch(Pi) + Pi^2/6 - 1. - Amiram Eldar, Nov 02 2021

A103438 Square array T(m,n) read by antidiagonals: Sum_{k=1..n} k^m.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 5, 6, 4, 0, 1, 9, 14, 10, 5, 0, 1, 17, 36, 30, 15, 6, 0, 1, 33, 98, 100, 55, 21, 7, 0, 1, 65, 276, 354, 225, 91, 28, 8, 0, 1, 129, 794, 1300, 979, 441, 140, 36, 9, 0, 1, 257, 2316, 4890, 4425, 2275, 784, 204, 45, 10

Offset: 0

Ralf Stephan, Feb 11 2005

For the o.g.f.s of the column sequences for this array, see A196837 and the link given there. - Wolfdieter Lang, Oct 15 2011
T(m,n)/n is the m-th moment of the discrete uniform distribution on {1,2,...,n}. - Geoffrey Critzer, Dec 31 2018
T(1,n) divides T(m,n) for odd m. - Franz Vrabec, Dec 23 2020

			Square array begins:
  0, 1,  2,   3,    4,     5,     6,      7,      8,      9, ... A001477;
  0, 1,  3,   6,   10,    15,    21,     28,     36,     45, ... A000217;
  0, 1,  5,  14,   30,    55,    91,    140,    204,    285, ... A000330;
  0, 1,  9,  36,  100,   225,   441,    784,   1296,   2025, ... A000537;
  0, 1, 17,  98,  354,   979,  2275,   4676,   8772,  15333, ... A000538;
  0, 1, 33, 276, 1300,  4425, 12201,  29008,  61776, 120825, ... A000539;
  0, 1, 65, 794, 4890, 20515, 67171, 184820, 446964, 978405, ... A000540;
Antidiagonal triangle begins as:
  0;
  0, 1;
  0, 1,  2;
  0, 1,  3,  3;
  0, 1,  5,  6,  4;
  0, 1,  9, 14, 10,  5;
  0, 1, 17, 36, 30, 15, 6;

J. Faulhaber, Academia Algebrae, Darinnen die miraculosische inventiones zu den höchsten Cossen weiters continuirt und profitirt werden, Augspurg, bey Johann Ulrich Schönigs, 1631.

G. C. Greubel, Antidiagonals n = 0..50, flattened
José L. Cereceda, Sums of powers of integers and hyperharmonic numbers, arXiv:2005.03407 [math.NT], 2020.
T. A. Gulliver, Divisibility of sums of powers of odd integers, Int. Math. For. 5 (2010) 3059-3066.
T. A. Gulliver, Sums of Powers of Integers Divisible by Three, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 38, pp. 1895-1901. - From _N. J. A. Sloane_, Dec 22 2012
V. J. W. Guo and J. Zeng, A q-analogue of Faulhaber's formula for sums of powers, arXiv:math/0501441 [math.CO], 2005.
H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
T. Kim, q-analogues of the sums of powers of consecutive integers, arXiv:math/0502113 [math.NT], 2005.
D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 61 (1993), no. 203, 277-294.
Eric Weisstein's World of Mathematics, Discrete Uniform Distribution.
Wikipedia, Faulhaber's formula

Rows include A000027, A000217, A000330, A000537, A000538, A000539, A000540, A000541, A000542, A007487, A023002.
Columns include A000051, A001550, A001551, A001552, A001553, A001554, A001555, A001556, A001557.
Diagonals include A076015 and A031971.
Antidiagonal sums are in A103439.
Antidiagonals are the rows of triangle A192001.

T:= func< n,k | n eq 0 select k else (&+[j^n: j in [0..k]]) >;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 22 2021

seq(print(seq(Zeta(0,-k,1)-Zeta(0,-k,n+1),n=0..9)),k=0..6);
# (Produces the square array from the example.) Peter Luschny, Nov 16 2008
# alternative
A103438 := proc(m,n)
    (bernoulli(m+1,n+1)-bernoulli(m+1))/(m+1) ;
    if m = 0 then
        %-1 ;
    else
        % ;
    end if;
end proc: # R. J. Mathar, May 10 2013
# simpler:
A103438 := proc(m,n)
    (bernoulli(m+1,n+1)-bernoulli(m+1,1))/(m+1) ;
end proc: # Peter Luschny, Mar 20 2024

T[m_, n_]:= HarmonicNumber[m, -n]; Flatten[Table[T[m-n, n], {m, 0, 11}, {n, m, 0, -1}]] (* Jean-François Alcover, May 11 2012 *)

PARI
```
T(m,n)=sum(k=0,n,k^m)
```

from itertools import count, islice
from math import comb
from fractions import Fraction
from sympy import bernoulli
def A103438_T(m,n): return sum(k**m for k in range(1,n+1)) if n<=m else int(sum(comb(m+1,i)*(bernoulli(i) if i!=1 else Fraction(1,2))*n**(m-i+1) for i in range(m+1))/(m+1))
def A103438_gen(): # generator of terms
    for m in count(0):
        for n in range(m+1):
            yield A103438_T(m-n,n)
A103438_list = list(islice(A103438_gen(),100)) # Chai Wah Wu, Oct 23 2024

def T(n,k): return (bernoulli_polynomial(k+1, n+1) - bernoulli_polynomial(1, n+1)) /(n+1)
flatten([[T(n-k,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 22 2021

E.g.f.: e^x*(e^(x*y)-1)/(e^x-1).
T(m, n) = Zeta(-n, 1) - Zeta(-n, m + 1), for m >= 0 and n >= 0, where Zeta(z, v) is the Hurwitz zeta function. - Peter Luschny, Nov 16 2008
T(m, n) = HarmonicNumber(m, -n). - Jean-François Alcover, May 11 2012
T(m, n) = (Bernoulli(m + 1, n + 1) - Bernoulli(m + 1, 1)) / (m + 1). - Peter Luschny, Mar 20 2024
T(m, n) = Sum_{k=0...m-n} B(k)*(-1)^k*binomial(m-n,k)*n^(m-n-k+1)/(m-n-k+1), where B(k) = Bernoulli number A027641(k) / A027642(k). - Robert B Fowler, Aug 20 2024
T(m, n) = Sum_{i=1..n} J_m(i)*floor(n/i), where J_m is the m-th Jordan totient function. - Ridouane Oudra, Jul 19 2025

A331755 Number of vertices in a regular drawing of the complete bipartite graph K_{n,n}.

Original entry on oeis.org

2, 5, 13, 35, 75, 159, 275, 477, 755, 1163, 1659, 2373, 3243, 4429, 5799, 7489, 9467, 11981, 14791, 18275, 22215, 26815, 31847, 37861, 44499, 52213, 60543, 70011, 80347, 92263, 105003, 119557, 135327, 152773, 171275, 191721, 213547, 237953

Offset: 1

N. J. A. Sloane, Feb 02 2020

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..1000
Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5, Lemma 2.
S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.
Scott R. Shannon, Images of vertices for n=2.
Scott R. Shannon, Images of vertices for n=3.
Scott R. Shannon, Images of vertices for n=4.
Scott R. Shannon, Images of vertices for n=5.
Scott R. Shannon, Images of vertices for n=6
Scott R. Shannon, Images of vertices for n=7
Scott R. Shannon, Images of vertices for n=8
Scott R. Shannon, Images of vertices for n=9
Scott R. Shannon, Images of vertices for n=10.
Scott R. Shannon, Images of vertices for n=12.
Scott R. Shannon, Images of vertices for n=15.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Index entries for sequences related to stained glass windows

Cf. A290131 (regions), A290132 (edges), A333274 (polygons per vertex), A333276, A159065.

For K_n see A007569, A007678, A135563.

# Maple code from N. J. A. Sloane, Jul 16 2020
V106i := proc(n) local ans,a,b; ans:=0;
for a from 1 to n-1 do for b from 1 to n-1 do
if igcd(a,b)=1 then ans:=ans + (n-a)*(n-b); fi; od: od: ans; end; # A115004
V106ii := proc(n) local ans,a,b; ans:=0;
for a from 1 to floor(n/2) do for b from 1 to floor(n/2) do
if igcd(a,b)=1 then ans:=ans + (n-2*a)*(n-2*b); fi; od: od: ans; end; # A331761
A331755 := n -> 2*(n+1) + V106i(n+1) - V106ii(n+1);

a[n_]:=Module[{x,y,s1=0,s2=0}, For[x=1, x<=n-1, x++, For[y=1, y<=n-1, y++, If[GCD[x,y]==1,s1+=(n-x)*(n-y); If[2*x<=n-1&&2*y<=n-1,s2+=(n-2*x)*(n-2*y)]]]]; s1-s2]; Table[a[n]+ 2 n, {n, 1, 40}] (* Vincenzo Librandi, Feb 04 2020 *)

a(n) = A290132(n) - A290131(n) + 1.
a(n) = A159065(n) + 2*n.
This is column 1 of A331453.
a(n) = (9/(8*Pi^2))*n^4 + O(n^3 log(n)). Asymptotic to (9/(2*Pi^2))*A000537(n-1). [Stéphane Legendre, see A159065.]

A176271 The odd numbers as a triangle read by rows.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Reinhard Zumkeller, Apr 13 2010

A108309(n) = number of primes in n-th row.

			From _Philippe Deléham_, Oct 03 2011: (Start)
Triangle begins:
   1;
   3,  5;
   7,  9, 11;
  13, 15, 17, 19;
  21, 23, 25, 27, 29;
  31, 33, 35, 37, 39, 41;
  43, 45, 47, 49, 51, 53, 55;
  57, 59, 61, 63, 65, 67, 69, 71;
  73, 75, 77, 79, 81, 83, 85, 87, 89; (End)

G. C. Greubel, Rows n = 1..100 of the triangle, flattened
Eric Weisstein's World of Mathematics, Nicomachus's Theorem
Wikipedia, Nikomachos von Gerasa

Cf. A000466, A000537, A000578, A001105, A002061, A005408, A014209.
Cf. A016754, A027688, A027690, A027692, A027694, A028387, A048058.
Cf. A053755, A065599, A082111, A108195, A108309, A214604, A214661.

a176271 n k = a176271_tabl !! (n-1) !! (k-1)
a176271_row n = a176271_tabl !! (n-1)
a176271_tabl = f 1 a005408_list where
   f x ws = us : f (x + 1) vs where (us, vs) = splitAt x ws
-- Reinhard Zumkeller, May 24 2012

[n^2-n+2*k-1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024

A176271 := proc(n,k)
    n^2-n+2*k-1 ;
end proc: # R. J. Mathar, Jun 28 2013

Table[n^2-n+2*k-1, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 10 2024 *)

flatten([[n^2-n+2*k-1 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n, k) = n^2 - n + 2*k - 1 for 1 <= k <= n.
T(n, k) = A005408(n*(n-1)/2 + k - 1).
T(2*n-1, n) = A016754(n-1) (main diagonal).
T(2*n, n) = A000466(n).
T(2*n, n+1) = A053755(n).
T(n, k) + T(n, n-k+1) = A001105(n), 1 <= k <= n.
T(n, 1)   = A002061(n), central polygonal numbers.
T(n, 2)   = A027688(n-1) for n > 1.
T(n, 3)   = A027690(n-1) for n > 2.
T(n, 4)   = A027692(n-1) for n > 3.
T(n, 5)   = A027694(n-1) for n > 4.
T(n, 6)   = A048058(n-1) for n > 5.
T(n, n-3) = A108195(n-2) for n > 3.
T(n, n-2) = A082111(n-2) for n > 2.
T(n, n-1) = A014209(n-1) for n > 1.
T(n, n) = A028387(n-1).
Sum_{k=1..n} T(n, k) = A000578(n) (Nicomachus's theorem).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A065599(n) (alternating sign row sums).
Sum_{j=1..n} (Sum_{k=1..n} T(j, k)) = A000537(n) (sum of first n rows).

cp's OEIS Frontend

A005898 Centered cube numbers: n^3 + (n+1)^3.

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A000538 Sum of fourth powers: 0^4 + 1^4 + ... + n^4.

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A008459 Square the entries of Pascal's triangle.

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A000539 Sum of 5th powers: 0^5 + 1^5 + 2^5 + ... + n^5.

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A001715 a(n) = n!/6.

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