A024196 -id:A024196 - cp's OEIS frontend

A000447 a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2n-1)^2 = n(4*n^2 - 1)/3.

0, 1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, 2925, 3654, 4495, 5456, 6545, 7770, 9139, 10660, 12341, 14190, 16215, 18424, 20825, 23426, 26235, 29260, 32509, 35990, 39711, 43680, 47905, 52394, 57155, 62196, 67525, 73150, 79079, 85320, 91881, 98770, 105995, 113564, 121485

Offset: 0

N. J. A. Sloane

4 times the variance of the area under an n-step random walk: e.g., with three steps, the area can be 9/2, 7/2, 3/2, 1/2, -1/2, -3/2, -7/2, or -9/2 each with probability 1/8, giving a variance of 35/4 or a(3)/4. - Henry Bottomley, Jul 14 2003
Number of standard tableaux of shape (2n-1,1,1,1) (n>=1). - Emeric Deutsch, May 30 2004
Also a(n) = (1/6)*(8*n^3-2*n), n>0: structured octagonal diamond numbers (vertex structure 9). Cf. A059722 = alternate vertex; A000447 = structured diamonds; and structured tetragonal anti-diamond numbers (vertex structure 9). Cf. A096000 = alternate vertex; A100188 = structured anti-diamonds. Cf. A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
The n-th tetrahedral (or pyramidal) number is n(n+1)(n+2)/6. This sequence contains the tetrahedral numbers of A000292 obtained for n= 1,3,5,7,... (see A015219). - Valentin Bakoev, Mar 03 2009
Using three consecutive numbers u, v, w, (u+v+w)^3-(u^3+v^3+w^3) equals 18 times the numbers in this sequence. - J. M. Bergot, Aug 24 2011
This sequence is related to A070893 by A070893(2*n-1) = n*a(n)-sum(i=0..n-1, a(i)). - Bruno Berselli, Aug 26 2011
Number of integer solutions to 1-n <= x <= y <= z <= n-1. - Michael Somos, Dec 27 2011
Partial sums of A016754. - Reinhard Zumkeller, Apr 02 2012
Also the number of cubes in the n-th Haüy square pyramid. - Eric W. Weisstein, Sep 27 2017

			G.f. = x + 10*x^2 + 35*x^3 + 84*x^4 + 165*x^5 + 286*x^6 + 455*x^7 + 680*x^8 + ...
a(2) = 10 since (-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, 0), (-1, 0, 1), (-1, 1, 1), (0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1) are the 10 solutions (x, y, z) of -1 <= x <= y <= z <= 1.
a(0) = 0, which corresponds to the empty sum.

G. Chrystal, Textbook of Algebra, Vol. 1, A. & C. Black, 1886, Chap. XX, Sect. 10, Example 2.
F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
C. V. Durell, Advanced Algebra, Volume 1, G. Bell & Son, 1932, Exercise IIIe, No. 4.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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
J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359. [Annotated scanned copy]
Valentin Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, Vol. 275 (2004), pp. 17-41. - _Valentin Bakoev_, Mar 03 2009
F. E. Croxton and D. J. Cowden, Applied General Statistics, 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1955 [Annotated scans of just pages 742-743]
Milan Janjic, Two Enumerative Functions.
T. P. Martin, Shells of atoms, Phys. Reports, Vol. 273 (1996), pp. 199-241, eq. (11).
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, Haüy Construction.
Eric Weisstein's World of Mathematics, Square Pyramid.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Column 1 in triangles A008956 and A008958.
Cf. A035328, A069072, A190152.
A000447 is related to partitions of 2^n into powers of 2, as it is shown in the formula, example and cross-references of A002577. - Valentin Bakoev, Mar 03 2009
Cf. A002412, A016061.

a000447 n = a000447_list !! n
a000447_list = scanl1 (+) a016754_list
-- Reinhard Zumkeller, Apr 02 2012

[n*(4*n^2-1)/3: n in [0..50]]; // Vincenzo Librandi, Jan 12 2016

A000447:=z*(1+6*z+z**2)/(z-1)**4; # Simon Plouffe, 1992 dissertation.
A000447:=n->n*(4*n^2 - 1)/3; seq(A000447(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2014

Table[n (4 n^2 - 1)/3, {n, 0, 80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 10, 35}, 80] (* Harvey P. Dale, May 25 2012 *)
Join[{0}, Accumulate[Range[1, 81, 2]^2]] (* Harvey P. Dale, Jul 18 2013 *)
CoefficientList[Series[x (1 + 6 x + x^2)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)

A000447(n):=n*(4*n^2 - 1)/3$ makelist(A000447(n),n,0,20); /* Martin Ettl, Jan 07 2013 */

PARI
```
{a(n) = n * (4*n^2 - 1) / 3};
```

concat(0, Vec(x*(1+6*x+x^2)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 11 2016

def A000447(n): return n*((n**2<<2)-1)//3 # Chai Wah Wu, Feb 12 2023

a(n) = binomial(2*n+1, 3) = A000292(2*n-1).
G.f.: x*(1+6*x+x^2)/(1-x)^4.
a(n) = -a(-n) for all n in Z.
a(n) = A000330(2*n)-4*A000330(n) = A000466(n)*n/3 = A000578(n)+A007290(n-2) = A000583(n)-2*A024196(n-1) = A035328(n)/3. - Henry Bottomley, Jul 14 2003
a(n+1) = (2*n+1)*(2*n+2)(2*n+3)/6. - Valentin Bakoev, Mar 03 2009
a(0)=0, a(1)=1, a(2)=10, a(3)=35, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, May 25 2012
a(n) = v(n,n-1), where v(n,k) is the central factorial numbers of the first kind with odd indices. - Mircea Merca, Jan 25 2014
a(n) = A005917(n+1) - A100157(n+1), where A005917 are the rhombic dodecahedral numbers and A100157 are the structured rhombic dodecahedral numbers (vertex structure 9). - Peter M. Chema, Jan 09 2016
For any nonnegative integers m and n, 8*(n^3)*a(m) + 2*m*a(n) = a(2*m*n). - Ivan N. Ianakiev, Mar 04 2017
E.g.f.: exp(x)*x*(1 + 4*x  + (4/3)*x^2). - Wolfdieter Lang, Mar 11 2017
a(n) = A002412(n) + A016061(n-1), for n>0. - Bruce J. Nicholson, Nov 12 2017
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=1} 1/a(n) = 6*log(2) - 3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3 - 3*log(2). (End)

Chrystal and Durell references from R. K. Guy, Apr 02 2004

A028338 Triangle of coefficients in expansion of (x+1)(x+3)...*(x + 2n - 1) in rising powers of x.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 15, 23, 9, 1, 105, 176, 86, 16, 1, 945, 1689, 950, 230, 25, 1, 10395, 19524, 12139, 3480, 505, 36, 1, 135135, 264207, 177331, 57379, 10045, 973, 49, 1, 2027025, 4098240, 2924172, 1038016, 208054, 24640, 1708, 64, 1, 34459425, 71697105, 53809164, 20570444, 4574934, 626934, 53676, 2796, 81, 1

Offset: 0

Bill Gosper

Exponential Riordan array (1/sqrt(1-2*x), log(1/sqrt(1-2*x))). - Paul Barry, May 09 2011

The o.g.f.s D(d, x) of the column sequences, for d, d >= 0,(d=0 for the main diagonal) are P(d, x)/(1 - x)^(2*d+1), with the row polynomial P(d, x) = Sum_{m=0..d} A288875(d, m)*x^m. See A288875 for details. - Wolfdieter Lang, Jul 21 2017

			G.f. for n = 4: (x + 1)*(x + 3)*(x + 5)*(x + 7) = 105 + 176*x + 86*x^2 + 16*x^3 + x^4.
The triangle T(n, k) begins:
n\k       0        1        2        3       4      5     6    7  8  9
0:        1
1:        1        1
2:        3        4        1
3:       15       23        9        1
4:      105      176       86       16       1
5:      945     1689      950      230      25      1
6:    10395    19524    12139     3480     505     36     1
7:   135135   264207   177331    57379   10045    973    49    1
8:  2027025  4098240  2924172  1038016  208054  24640  1708   64  1
9: 34459425 71697105 53809164 20570444 4574934 626934 53676 2796 81  1
...
row n = 10: 654729075 1396704420 1094071221 444647600 107494190 16486680 1646778 106800 4335 100 1.
...  reformatted and extended. - _Wolfdieter Lang_, May 09 2017
O.g.f.s of diagonals d >= 0: D(2, x) = (3 + 8*x + x^2)/(1 - x)^5 generating [3, 23, 86, ...] = A024196(n+1), from the row d=2 entries of A288875 [3, 8, 1]. - _Wolfdieter Lang_, Jul 21 2017
Boas-Buck recurrence for column k=2 and n=4: T(4, 2) = (4!/2)*(2*(1+4*(5/12))*T(2,2)/2! + 1*(1 + 4*(1/2))*T(3,2)/3!) = (4!/2)*(8/3*1 + 3*9/3!) = 86. - _Wolfdieter Lang_, Aug 11 2017

T. D. Noe, Rows n=0..50 of triangle, flattened
Priyavrat Deshpande, Krishna Menon, and Anurag Singh, A combinatorial statistic for labeled threshold graphs, arXiv:2103.03865 [math.CO], 2021.
Thomas Godland and Zakhar Kabluchko, Projections and angle sums of permutohedra and other polytopes, arXiv:2009.04186 [math.MG], 2020.
Thomas Godland and Zakhar Kabluchko, Projections and Angle Sums of Belt Polytopes and Permutohedra, Res. Math. (2023) Vol. 78, Art. No. 140.
Z. Kabluchko, V. Vysotsky, and D. Zaporozhets, Convex hulls of random walks, hyperplane arrangements, and Weyl chambers, arXiv preprint arXiv:1510.04073 [math.PR], 2015.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Bruce E. Sagan and Joshua P. Swanson, q-Stirling numbers in type B, arXiv:2205.14078 [math.CO], 2022.

A039757 is signed version.
Row sums: A000165.
Columns: A001147, A004041, A028339, A028340, A028341; A000012, A000290, A024196, A024197, A024198.
Diagonals: A000012, A000290(n+1), A024196(n+1), A024197(n+1), A024198(n+1).
A161198 is a scaled triangle version and A109692 is a transposed triangle version.
Central terms: A293318.
Cf. A286718, A002208(n+1)/A002209(n+1).

nmax:=8; for n from 0 to nmax do a(n, 0) := doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, n) := 1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := (2*n-1)*a(n-1, m) + a(n-1, m-1) od; od: seq(seq(a(n, m), m=0..n), n=0..nmax); # Johannes W. Meijer, Jun 08 2009, revised Nov 25 2012

T[n_, k_] := Sum[(-2)^(n-i) Binomial[i, k] StirlingS1[n, i], {i, k, n}] (* Woodhouse *)
Join[{1},Flatten[Table[CoefficientList[Expand[Times@@Table[x+i,{i,1,2n+1,2}]],x],{n,0,10}]]] (* Harvey P. Dale, Jan 29 2013 *)

Triangle T(n, k), read by rows, given by [1, 2, 3, 4, 5, 6, 7, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 20 2005
T(n, k) = Sum_{i=k..n} (-2)^(n-i) * binomial(i, k) * s(n, i) where s(n, k) are signed Stirling numbers of the first kind. - Francis Woodhouse (fwoodhouse(AT)gmail.com), Nov 18 2005
G.f. of row polynomials in y: 1/(1-(x+x*y)/(1-2*x/(1-(3*x+x*y)/(1-4*x/(1-(5*x+x*y)/(1-6*x*y/(1-... (continued fraction). - Paul Barry, Feb 07 2009
T(n, m) = (2*n-1)*T(n-1,m) + T(n-1,m-1) with T(n, 0) = (2*n-1)!! and T(n, n) = 1. - Johannes W. Meijer, Jun 08 2009
From Wolfdieter Lang, May 09 2017: (Start)
E.g.f. of row polynomials in y: (1/sqrt(1-2*x))*exp(-y*log(sqrt(1-2*x))) = exp(-(1+y)*log(sqrt(1-2*x))) = 1/sqrt(1-2*x)^(1+y).
E.g.f. of column m sequence: (1/sqrt(1-2*x))* (-log(sqrt(1-2*x)))^m/m!. For the special Sheffer, also known as exponential Riordan array, see a comment above. (End)
Boas-Buck type recurrence for column sequence k: T(n, k) = (n!/(n - k)) * Sum_{p=k..n-1} 2^(n-1-p)*(1 + 2*k*beta(n-1-p))*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, and beta(k) = A002208(k+1)/A002209(k+1). See a comment and references in A286718. - Wolfdieter Lang, Aug 09 2017

A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).

Original entry on oeis.org

1, 1, 2, 3, 8, 4, 15, 46, 36, 8, 105, 352, 344, 128, 16, 945, 3378, 3800, 1840, 400, 32, 10395, 39048, 48556, 27840, 8080, 1152, 64, 135135, 528414, 709324, 459032, 160720, 31136, 3136, 128

Offset: 0

Johannes W. Meijer, Jun 08 2009, Jul 22 2011

The series expansion of (1-x)^((-1-2*n)/2) = sum(b(p)*x^p, p=0..infinity) for n = 0, 1, 2, .. can be described with b(p) = (F(p,n)/ (2*n-1)!!)*(binomial(2*p,p)/4^(p)) with F(x,n) = 2^n * product( x+(2*k-1)/2, k=1..n). The roots of the F(x,n) polynomials can be found at p = (1-2*k)/2 with k from 1 to n for n = 0, 1, 2, .. . The coefficients of the F(x,n) polynomials lead to the triangle given above. The triangle row sums lead to A001147.
Quite surprisingly we discovered that sum(b(p)*x^p, p=0..infinity) = (1-x)^(-1-2*n)/2, for n = -1, -2, .. . We assume that if m = n+1 then the value returned for product(f(k), k = m..n) is 1 and if m> n+1 then 1/product(f(k), k=n+1..m-1) is the value returned. Furthermore (1-2*n)!! = (-1)^(n+1)/(2*n-3)!! for n = 1, 2, 3 .. . This leads to b(p) = ((-1-2*n)!!/ G(p,n))*(binomial(2*p,p) /4^(p)) for n = -1, -2, .. . For the G(p,n) polynomials we found that G(p,n) = F(-p,-n). The roots of the G(p,n) polynomials can be found at p=(2*k-1)/2 with k from 1 to (-n) for n = -1, -2, .. . The coefficients of the G(p,n) polynomials lead to a second triangle that stands with its head on top of the first one. It is remarkable that the row sums lead once again to A001147.
These two triangles together look like an hourglass so we propose to call the F(p,n) and the G(p,n) polynomials the hourglass polynomials.
Triangle T(n,k), read by rows, given by (1, 2, 3, 4, 5, 6, 7, 8, 9, ...) DELTA (2, 0, 2, 0, 2, 0, 2, 0, 2, ...) where DELTA is the operator defined in A084938. Philippe Deléham, May 14 2015.

			From _Gary W. Adamson_, Jul 19 2011: (Start)
The first few rows of matrix M are:
  1, 2,  0,  0, 0, ...
  1, 3,  2,  0, 0, ...
  1, 4,  5,  2, 0, ...
  1, 5,  9,  7, 2, ...
  1, 6, 14, 16, 9, ... (End)
The first few G(p,n) polynomials are:
  G(p,-3) = 15 - 46*p + 36*p^2 - 8*p^3
  G(p,-2) = 3 - 8*p + 4*p^2
  G(p,-1) = 1 - 2*p
The first few F(p,n) polynomials are:
  F(p,0) = 1
  F(p,1) = 1 + 2*p
  F(p,2) = 3 + 8*p + 4*p^2
  F(p,3) = 15 + 46*p + 36*p^2 + 8*p^3
The first few rows of the upper and lower hourglass triangles are:
  [15, -46, 36, -8]
  [3, -8, 4]
  [1, -2]
  [1]
  [1, 2]
  [3, 8, 4]
  [15, 46, 36, 8]

Cf. A001790 [(1-x)^(-1/2)], A001803 [(1-x)^(-3/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/2)].
Cf. A002596 [(1-x)^(1/2)], A161200 [(1-x)^(3/2)] and A161202 [(1-x)^(5/2)].
A046161 gives the denominators of the series expansions of all (1-x)^((-1-2*n)/2).
A028338 is a scaled triangle version, A039757 is a scaled signed triangle version and A109692 is a transposed scaled triangle version.
A001147 is the first left hand column and equals the row sums.
A004041 is the second left hand column divided by 2, A028339 is the third left hand column divided by 4, A028340 is the fourth left hand column divided by 8, A028341 is the fifth left hand column divided by 16.
A000012, A000290, A024196, A024197 and A024198 are the first (n-m=0), second (n-m=1), third (n-m=2), fourth (n-m=3) and fifth (n-m=4) right hand columns divided by 2^m.
A074599 * A025549 is not always equals the second left hand column.
Cf. A029635. [Gary W. Adamson, Jul 19 2011]

nmax:=7; for n from 0 to nmax do a(n,n):=2^n: a(n,0):=doublefactorial(2*n-1) od: for n from 2 to nmax do for m from 1 to n-1 do a(n,m) := 2*a(n-1,m-1)+(2*n-1)*a(n-1,m) od: od: seq(seq(a(n,k), k=0..n), n=0..nmax);
nmax:=7: M := Matrix(1..nmax+1,1..nmax+1): A029635 := proc(n,k): binomial(n,k) + binomial(n-1,k-1) end: for i from 1 to nmax do for j from 1 to i+1 do M[i,j] := A029635(i,j-1) od: od: for n from 0 to nmax do B := M^n: for m from 0 to n do a(n,m):= B[1,m+1] od: od: seq(seq(a(n,m), m=0..n), n=0..nmax);
A161198 := proc(n,k) option remember; if k > n or k < 0 then 0 elif n = 0 and k = 0 then 1 else 2*A161198(n-1, k-1) + (2*n-1)*A161198(n-1, k) fi end:
seq(print(seq(A161198(n,k), k = 0..n)), n = 0..6);  # Peter Luschny, May 09 2013

nmax = 7; a[n_, 0] := (2*n-1)!!; a[n_, n_] := 2^n; a[n_, m_] := a[n, m] = 2*a[n-1, m-1]+(2*n-1)*a[n-1, m]; Table[a[n, m], {n, 0, nmax}, {m, 0, n}] // Flatten (* Jean-François Alcover, Feb 25 2014, after Maple *)

for(n=0,9, print(Vec(Ser( 2^n*prod( k=1,n, x+(2*k-1)/2 ),,n+1))))  \\ M. F. Hasler, Jul 23 2011

@CachedFunction
def A161198(n,k):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return 2*A161198(n-1,k-1)+(2*n-1)*A161198(n-1,k)
for n in (0..6): [A161198(n,k) for k in (0..n)]  # Peter Luschny, May 09 2013

a(n,m) := coeff(2^(n)*product((x+(2*k-1)/2),k=1..n), x, m) for n = 0, 1, .. ; m = 0, 1, .. .
a(n, m) = 2*a(n-1,m-1)+(2*n-1)*a(n-1,m) with a(n, n) = 2^n and a(n, 0) = (2*n-1)!!.
a(n,m) = the (m+1)-th term in the top row of M^n, where M is an infinite square production matrix; M[i,j] = A029635(i,j-1) = binomial(i, j-1) + binomial(i-1, j-2) with A029635 the (1.2)-Pascal triangle, see the examples and second Maple program. [Gary W. Adamson, Jul 19 2011]
T(n,k) = 2^k * A028338(n,k). - Philippe Deléham, May 14 2015

A103220 a(n) = n(n+1)(3*n^2+n-1)/6.

Original entry on oeis.org

0, 1, 13, 58, 170, 395, 791, 1428, 2388, 3765, 5665, 8206, 11518, 15743, 21035, 27560, 35496, 45033, 56373, 69730, 85330, 103411, 124223, 148028, 175100, 205725, 240201, 278838, 321958, 369895, 422995, 481616, 546128, 616913, 694365, 778890

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 25 2005

Row sums of A103219.
From Bruno Berselli, Dec 10 2010: (Start)
a(n) = n*A002412(n) - Sum_{i=0..n-1} A002412(i). More generally: n^2*(n+1)*(2*d*n-2*d+3)/6 - (Sum_{i=0..n-1} i*(i+1)*(2*d*i-2*d+3))/6 = n * (n+1) * (3*d*n^2-d*n+4*n-2*d+2)/12; in this sequence is d=2.
The inverse binomial transform yields 0, 1, 11, 22, 12, 0, 0 (0 continued). (End)
a(n-1) is also number of ways to place 2 nonattacking semi-queens (see A099152) on an n X n board. - Vaclav Kotesovec, Dec 22 2011
Also, one-half the even-indexed terms of the partial sums of A045947. - J. M. Bergot, Apr 12 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002412, A002418, A039755, A099152, A103219, A015237 (first diffs), A024196.

for(n=0,100,print1((3*n^4+4*n^3-n)/6,","))

CoefficientList[Series[- x (1 + 8 x + 3 x^2) / (x - 1)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 12 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,13,58,170},40] (* Harvey P. Dale, Jan 23 2016 *)

a(n)=n*(n+1)*(3*n^2+n-1)/6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(1+8*x+3*x^2)/(1-x)^5.
a(n) = Sum_{i=1..n} Sum_{j=1..n} max(i,j)^2. - Enrique Pérez Herrero, Jan 15 2013
a(n) = a(n-1) + (2*n-1)*n^2 with a(0)=0, see A015237. - J. M. Bergot, Jun 10 2017
From Wesley Ivan Hurt, Nov 20 2021: (Start)
a(n) = Sum_{k=1..n} k * C(2*k,2).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). (End)
From Peter Bala, Sep 03 2023: (Start)
a(n) = Sum_{1 <= i <= j <= n} (2*i - 1)*(2*j - 1).
Second subdiagonal of A039755. (End)

A099721 a(n) = n^2(2n+1).

Original entry on oeis.org

0, 3, 20, 63, 144, 275, 468, 735, 1088, 1539, 2100, 2783, 3600, 4563, 5684, 6975, 8448, 10115, 11988, 14079, 16400, 18963, 21780, 24863, 28224, 31875, 35828, 40095, 44688, 49619, 54900, 60543, 66560, 72963, 79764, 86975, 94608, 102675, 111188, 120159, 129600

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 07 2004

For a right triangle with sides of lengths 8*n^3 + 12*n^2 + 8*n + 2, 4*n^4 + 8*n^3 + 4*n^2, and 4*n^4 + 8*n^3 + 12*n^2 + 8*n + 2, dividing the area by the perimeter gives a(n). - J. M. Bergot, Jul 30 2013
This sequence is the difference between the centered icosahedral (or cuboctahedral) numbers (A005902(n)) and the centered octagonal pyramidal numbers (A000447(n+1)). - Peter M. Chema, Jan 09 2016
a(n) is the sum of the integers in the closed interval (n-1)*n to n*(n+1). - J. M. Bergot, Apr 19 2017

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

Row n=3 of A229079.
Cf. A000578, A001093, A011379, A015237, A027444, A033431, A033562, A034262, A053698, A061317, A066023, A071568, A098547.
Cf. A000290, A000447, A002378, A005408, A005902, A024196.

[n^2*(2*n+1): n in [0..50]]; // Vincenzo Librandi, May 01 2011

A099721 := proc(n) n^2*(2*n+1) ; end proc:
seq(A099721(n),n=0..10) ;

a[n_]:=2*n^3+n^2; (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{4,-6,4,-1},{0,3,20,63},40] (* Harvey P. Dale, Aug 19 2022 *)

a(n) = ceil(sum(i=n^2-(n-1), n^2+(n-1), if(!issquare(4*i+1), (2*i+1+sqrt(4*i+1))/2, 0))); \\ Michel Marcus, Nov 14 2014, after Richard R. Forberg

G.f.: x*(3 + 8*x + x^2)/(x-1)^4.
a(n) = A024196(n) - A024196(n-1). - Philippe Deléham, May 07 2012
a(n) = ceiling(Sum_{i=n^2-(n-1)..n^2+(n-1)} s(i)), for n > 0 and integer i, where s(i) are the real solutions to x = i + sqrt(x), and the summation range excludes the integer solutions which occur where i is an oblong number (A002378). The fractional portion of the summation converges to 2/3 for large n. If s(i) is replaced with i, then the summation equals n^2*(2*n-1) = A015237. - Richard R. Forberg, Oct 15 2014
a(n) = A005902(n) - A000447(n+1). - Peter M. Chema, Jan 09 2016
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 + 4*log(2) - 4.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - Pi - 2*log(2) + 4. (End)
From Elmo R. Oliveira, Aug 08 2025: (Start)
E.g.f.: x*(1 + 2*x)*(3 + x)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A000290(n)*A005408(n). (End)

A109692 Triangle of coefficients in expansion of (1+x)(1+3x)(1+5x)(1+7x)...*(1+(2n-1)x).

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 9, 23, 15, 1, 16, 86, 176, 105, 1, 25, 230, 950, 1689, 945, 1, 36, 505, 3480, 12139, 19524, 10395, 1, 49, 973, 10045, 57379, 177331, 264207, 135135, 1, 64, 1708, 24640, 208054, 1038016, 2924172, 4098240, 2027025

Offset: 0

Philippe Deléham, Aug 08 2005

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 2, 3, 4, 5, 6, 7, 8, 9, ...] where DELTA is the operator defined in A084938.

T(n,k), 0 <= k <= n, is the number of elements in the Coxeter group B_n with absolute length k. - Jose Bastidas, Jul 14 2023

			Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,   3;
  1,  9,  23,   15;
  1, 16,  86,  176,   105;
  1, 25, 230,  950,  1689,   945;
  1, 36, 505, 3480, 12139, 19524, 10395;
  ...

Seiichi Manyama, Rows n = 0..139, flattened

Cf. A039758 (signed version). A028338 transposed.
Diagonals: A000012, A000290, A024196, A024197, A024198; A001147, A004041, A028339, A028340, A028341.
Row sums: A000165.
Central terms: A293318.
Cf. A161198 (transposed scaled triangle version).

nmax:=8; mmax:=nmax: for n from 0 to nmax do a(n, n) := doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, 0):=1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := a(n-1,m) + (2*n-1)*a(n-1,m-1) od; od: seq(seq(a(n, m), m=0..n), n=0..nmax); # Johannes W. Meijer, Jun 08 2009, revised Nov 25 2012

T(n,m) = T(n-1,m) + (2*n-1)*T(n-1,m-1) with T(n,n) = (2*n-1)!! and T(n,0) = 1. - Johannes W. Meijer, Jun 08 2009

cp's OEIS Frontend

A000447 a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3.

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A028338 Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.

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A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).

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A103220 a(n) = n*(n+1)*(3*n^2+n-1)/6.

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A099721 a(n) = n^2*(2*n+1).

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A109692 Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).

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A000447 a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2n-1)^2 = n(4*n^2 - 1)/3.

A028338 Triangle of coefficients in expansion of (x+1)(x+3)...*(x + 2n - 1) in rising powers of x.

A103220 a(n) = n(n+1)(3*n^2+n-1)/6.

A099721 a(n) = n^2(2n+1).

A109692 Triangle of coefficients in expansion of (1+x)(1+3x)(1+5x)(1+7x)...*(1+(2n-1)x).