cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A000332 Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24.

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, 40920, 46376, 52360, 58905, 66045, 73815, 82251, 91390, 101270, 111930, 123410
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of intersection points of diagonals of convex n-gon where no more than two diagonals intersect at any point in the interior.
Also the number of equilateral triangles with vertices in an equilateral triangular array of points with n rows (offset 1), with any orientation. - Ignacio Larrosa Cañestro, Apr 09 2002. [See Les Reid link for proof. - N. J. A. Sloane, Apr 02 2016] [See Peter Kagey link for alternate proof. - Sameer Gauria, Jul 29 2025]
Start from cubane and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink on chemistry. - Robert G. Wilson v, Aug 02 2002
For n>0, a(n) = (-1/8)*(coefficient of x in Zagier's polynomial P_(2n,n)). (Zagier's polynomials are used by PARI/GP for acceleration of alternating or positive series.)
Figurate numbers based on the 4-dimensional regular convex polytope called the regular 4-simplex, pentachoron, 5-cell, pentatope or 4-hypertetrahedron with Schlaefli symbol {3,3,3}. a(n)=((n*(n-1)*(n-2)*(n-3))/4!). - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004, R. J. Mathar, Jul 07 2009
Maximal number of crossings that can be created by connecting n vertices with straight lines. - Cameron Redsell-Montgomerie (credsell(AT)uoguelph.ca), Jan 30 2007
If X is an n-set and Y a fixed (n-1)-subset of X then a(n) is equal to the number of 4-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Product of four consecutive numbers divided by 24. - Artur Jasinski, Dec 02 2007
The only prime in this sequence is 5. - Artur Jasinski, Dec 02 2007
For strings consisting entirely of 0's and 1's, the number of distinct arrangements of four 1's such that 1's are not adjacent. The shortest possible string is 7 characters, of which there is only one solution: 1010101, corresponding to a(5). An eight-character string has 5 solutions, nine has 15, ten has 35 and so on, congruent to A000332. - Gil Broussard, Mar 19 2008
For a(n)>0, a(n) is pentagonal if and only if 3 does not divide n. All terms belong to the generalized pentagonal sequence (A001318). Cf. A000326, A145919, A145920. - Matthew Vandermast, Oct 28 2008
Nonzero terms = row sums of triangle A158824. - Gary W. Adamson, Mar 28 2009
Except for the 4 initial 0's, is equivalent to the partial sums of the tetrahedral numbers A000292. - Jeremy Cahill (jcahill(AT)inbox.com), Apr 15 2009
If the first 3 zeros are disregarded, that is, if one looks at binomial(n+3, 4) with n>=0, then it becomes a 'Matryoshka doll' sequence with alpha=0: seq(add(add(add(i,i=alpha..k),k=alpha..n),n=alpha..m),m=alpha..50). - Peter Luschny, Jul 14 2009
For n>=1, a(n) is the number of n-digit numbers the binary expansion of which contains two runs of 0's. - Vladimir Shevelev, Jul 30 2010
For n>0, a(n) is the number of crossing set partitions of {1,2,..,n} into n-2 blocks. - Peter Luschny, Apr 29 2011
The Kn3, Ca3 and Gi3 triangle sums of A139600 are related to the sequence given above, e.g., Gi3(n) = 2*A000332(n+3) - A000332(n+2) + 7*A000332(n+1). For the definitions of these triangle sums, see A180662. - Johannes W. Meijer, Apr 29 2011
For n > 3, a(n) is the hyper-Wiener index of the path graph on n-2 vertices. - Emeric Deutsch, Feb 15 2012
Except for the four initial zeros, number of all possible tetrahedra of any size, having the same orientation as the original regular tetrahedron, formed when intersecting the latter by planes parallel to its sides and dividing its edges into n equal parts. - V.J. Pohjola, Aug 31 2012
a(n+3) is the number of different ways to color the faces (or the vertices) of a regular tetrahedron with n colors if we count mirror images as the same.
a(n) = fallfac(n,4)/4! is also the number of independent components of an antisymmetric tensor of rank 4 and dimension n >= 1. Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
Does not satisfy Benford's law [Ross, 2012] - N. J. A. Sloane, Feb 12 2017
Number of chiral pairs of colorings of the vertices (or faces) of a regular tetrahedron with n available colors. Chiral colorings come in pairs, each the reflection of the other. - Robert A. Russell, Jan 22 2020
From Mircea Dan Rus, Aug 26 2020: (Start)
a(n+3) is the number of lattice rectangles (squares included) in a staircase of order n; this is obtained by stacking n rows of consecutive unit lattice squares, aligned either to the left or to the right, which consist of 1, 2, 3, ..., n squares and which are stacked either in the increasing or in the decreasing order of their lengths. Below, there is a staircase or order 4 which contains a(7) = 35 rectangles. [See the Teofil Bogdan and Mircea Dan Rus link, problem 3, under A004320]
_
||
|||_
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(End)
a(n+4) is the number of strings of length n on an ordered alphabet of 5 letters where the characters in the word are in nondecreasing order. E.g., number of length-2 words is 15: aa,ab,ac,ad,ae,bb,bc,bd,be,cc,cd,ce,dd,de,ee. - Jim Nastos, Jan 18 2021
From Tom Copeland, Jun 07 2021: (Start)
Aside from the zeros, this is the fifth diagonal of the Pascal matrix A007318, the only nonvanishing diagonal (fifth) of the matrix representation IM = (A132440)^4/4! of the differential operator D^4/4!, when acting on the row vector of coefficients of an o.g.f., or power series.
M = e^{IM} is the matrix of coefficients of the Appell sequence p_n(x) = e^{D^4/4!} x^n = e^{b. D} x^n = (b. + x)^n = Sum_{k=0..n} binomial(n,k) b_n x^{n-k}, where the (b.)^n = b_n have the e.g.f. e^{b.t} = e^{t^4/4!}, which is that for A025036 aerated with triple zeros, the first column of M.
See A099174 and A000292 for analogous relationships for the third and fourth diagonals of the Pascal matrix. (End)
For integer m and positive integer r >= 3, the polynomial a(n) + a(n + m) + a(n + 2*m) + ... + a(n + r*m) in n has its zeros on the vertical line Re(n) = (3 - r*m)/2 in the complex plane. - Peter Bala, Jun 02 2024

Examples

			a(5) = 5 from the five independent components of an antisymmetric tensor A of rank 4 and dimension 5, namely A(1,2,3,4), A(1,2,3,5), A(1,2,4,5), A(1,3,4,5) and A(2,3,4,5). See the Dec 10 2015 comment. - _Wolfdieter Lang_, Dec 10 2015

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 70.
  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, p. 294.
  • J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
  • 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).
  • Charles W. Trigg, Mathematical Quickies, New York: Dover Publications, Inc., 1985, p. 53, #191.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 127.

Links

Crossrefs

binomial(n, k): A161680 (k = 2), A000389 (k = 5), A000579 (k = 6), A000580 (k = 7), A000581 (k = 8), A000582 (k = 9).
Cf. A053134, A053126, A075733, A006322, A006325.
Cf. also A000583, A014820, A092181, A092182, A092183.
Cf. A000217, A000292, A007318 (column k = 4).
Cf. A158824.
Cf. A006008 (Number of ways to color the faces (or vertices) of a regular tetrahedron with n colors when mirror images are counted as two).
Cf. A104712 (third column, k=4).
See A269747 for a 3-D analog.
Cf. A006008 (oriented), A006003 (achiral) tetrahedron colorings.
Row 3 of A325000, col. 4 of A007318.
Cf. A000292, A025036, A099174.

Programs

  • GAP
    A000332 := List([1..10^2], n -> Binomial(n, 4)); # Muniru A Asiru, Oct 16 2017
  • Magma
    [Binomial(n,4): n in [0..50]]; // Vincenzo Librandi, Nov 23 2014
  • Maple
    A000332 := n->binomial(n,4); [seq(binomial(n,4), n=0..100)];
  • Mathematica
    Table[ Binomial[n, 4], {n, 0, 45} ] (* corrected by Harvey P. Dale, Aug 22 2011 *)
Table[(n-4)(n-3)(n-2)(n-1)/24, {n, 100}] (* Artur Jasinski, Dec 02 2007 *)
LinearRecurrence[{5,-10,10,-5,1}, {0,0,0,0,1}, 45] (* Harvey P. Dale, Aug 22 2011 *)
CoefficientList[Series[x^4 / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 23 2014 *)
  • PARI
    a(n)=binomial(n,4);
  • Python
    # Starts at a(3), i.e. computes n*(n+1)*(n+2)*(n+3)/24
# which is more in line with A000217 and A000292.
def A000332():
    x, y, z, u = 1, 1, 1, 1
    yield 0
    while True:
        yield x
        x, y, z, u = x + y + z + u + 1, y + z + u + 1, z + u + 1, u + 1
a = A000332(); print([next(a) for i in range(41)]) # Peter Luschny, Aug 03 2019
  • Python
    print([n*(n-1)*(n-2)*(n-3)//24 for n in range(50)])
# Gennady Eremin, Feb 06 2022

Formula

a(n) = n*(n-1)*(n-2)*(n-3)/24.
G.f.: x^4/(1-x)^5. - Simon Plouffe in his 1992 dissertation
a(n) = n*a(n-1)/(n-4). - Benoit Cloitre, Apr 26 2003, R. J. Mathar, Jul 07 2009
a(n) = Sum_{k=1..n-3} Sum_{i=1..k} i*(i+1)/2. - Benoit Cloitre, Jun 15 2003
Convolution of natural numbers {1, 2, 3, 4, ...} and A000217, the triangular numbers {1, 3, 6, 10, ...}. - Jon Perry, Jun 25 2003
a(n) = A110555(n+1,4). - Reinhard Zumkeller, Jul 27 2005
a(n+1) = ((n^5-(n-1)^5) - (n^3-(n-1)^3))/24 - (n^5-(n-1)^5-1)/30; a(n) = A006322(n-2)-A006325(n-1). - Xavier Acloque, Oct 20 2003; R. J. Mathar, Jul 07 2009
a(4*n+2) = Pyr(n+4, 4*n+2) where the polygonal pyramidal numbers are defined for integers A>2 and B>=0 by Pyr(A, B) = B-th A-gonal pyramid number = ((A-2)*B^3 + 3*B^2 - (A-5)*B)/6; For all positive integers i and the pentagonal number function P(x) = x*(3*x-1)/2: a(3*i-2) = P(P(i)) and a(3*i-1) = P(P(i) + i); 1 + 24*a(n) = (n^2 + 3*n + 1)^2. - Jonathan Vos Post, Nov 15 2004
First differences of A000389(n). - Alexander Adamchuk, Dec 19 2004
For n > 3, the sum of the first n-2 tetrahedral numbers (A000292). - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005 [Corrected by Doug Bell, Jun 25 2017]
Starting (1, 5, 15, 35, ...), = binomial transform of [1, 4, 6, 4, 1, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
Sum_{n>=4} 1/a(n) = 4/3, from the Taylor expansion of (1-x)^3*log(1-x) in the limit x->1. - R. J. Mathar, Jan 27 2009
A034263(n) = (n+1)*a(n+4) - Sum_{i=0..n+3} a(i). Also A132458(n) = a(n)^2 - a(n-1)^2 for n>0. - Bruno Berselli, Dec 29 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=1. - Harvey P. Dale, Aug 22 2011
a(n) = (binomial(n-1,2)^2 - binomial(n-1,2))/6. - Gary Detlefs, Nov 20 2011
a(n) = Sum_{k=1..n-2} Sum_{i=1..k} i*(n-k-2). - Wesley Ivan Hurt, Sep 25 2013
a(n) = (A000217(A000217(n-2) - 1))/3 = ((((n-2)^2 + (n-2))/2)^2 - (((n-2)^2 + (n-2))/2))/(2*3). - Raphie Frank, Jan 16 2014
Sum_{n>=0} a(n)/n! = e/24. Sum_{n>=3} a(n)/(n-3)! = 73*e/24. See A067764 regarding the second ratio. - Richard R. Forberg, Dec 26 2013
Sum_{n>=4} (-1)^(n+1)/a(n) = 32*log(2) - 64/3 = A242023 = 0.847376444589... . - Richard R. Forberg, Aug 11 2014
4/(Sum_{n>=m} 1/a(n)) = A027480(m-3), for m>=4. - Richard R. Forberg, Aug 12 2014
E.g.f.: x^4*exp(x)/24. - Robert Israel, Nov 23 2014
a(n+3) = C(n,1) + 3*C(n,2) + 3*C(n,3) + C(n,4). Each term indicates the number of ways to use n colors to color a tetrahedron with exactly 1, 2, 3, or 4 colors.
a(n) = A080852(1,n-4). - R. J. Mathar, Jul 28 2016
From Gary W. Adamson, Feb 06 2017: (Start)
G.f.: Starting (1, 5, 14, ...), x/(1-x)^5 can be written
as (x * r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) = (1+x)^5;
as (x * r(x) * r(x^3) * r(x^9) * r(x^27) * ...) where r(x) = (1+x+x^2)^5;
as (x * r(x) * r(x^4) * r(x^16) * r(x^64) * ...) where r(x) = (1+x+x^2+x^3)^5;
... (as a conjectured infinite set). (End)
From Robert A. Russell, Jan 22 2020: (Start)
a(n) = A006008(n) - a(n+3) = (A006008(n) - A006003(n)) / 2 = a(n+3) - A006003(n).
a(n+3) = A006008(n) - a(n) = (A006008(n) + A006003(n)) / 2 = a(n) + A006003(n).
a(n) = A007318(n,4).
a(n+3) = A325000(3,n). (End)
Product_{n>=5} (1 - 1/a(n)) = cosh(sqrt(15)*Pi/2)/(100*Pi). - Amiram Eldar, Jan 21 2021

Extensions

Some formulas that referred to another offset corrected by R. J. Mathar, Jul 07 2009

A006561 Number of intersections of diagonals in the interior of a regular n-gon.

Original entry on oeis.org

0, 0, 0, 1, 5, 13, 35, 49, 126, 161, 330, 301, 715, 757, 1365, 1377, 2380, 1837, 3876, 3841, 5985, 5941, 8855, 7297, 12650, 12481, 17550, 17249, 23751, 16801, 31465, 30913, 40920, 40257, 52360, 46981, 66045, 64981, 82251, 80881, 101270, 84841, 123410, 121441
Offset: 1

Views

Author

N. J. A. Sloane, Bjorn Poonen (poonen(AT)math.princeton.edu)

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.
See also A101363, A292104, A292105.
See A290447 for an analogous problem on a line.

Programs

  • Maple
    delta:=(m,n) -> if (n mod m) = 0 then 1 else 0; fi;
f:=proc(n) global delta;
if n <= 2 then 0 else \
binomial(n,4)  \
+ (-5*n^3 + 45*n^2 - 70*n + 24)*delta(2,n)/24 \
- (3*n/2)*delta(4,n) \
+ (-45*n^2 + 262*n)*delta(6,n)/6  \
+ 42*n*delta(12,n) \
+ 60*n*delta(18,n) \
+ 35*n*delta(24,n) \
- 38*n*delta(30,n) \
- 82*n*delta(42,n) \
- 330*n*delta(60,n) \
- 144*n*delta(84,n) \
- 96*n*delta(90,n) \
- 144*n*delta(120,n) \
- 96*n*delta(210,n); fi; end;
[seq(f(n),n=1..100)]; # N. J. A. Sloane, Aug 09 2017
  • Mathematica
    del[m_,n_]:=If[Mod[n,m]==0,1,0]; Int[n_]:=If[n<4, 0, Binomial[n,4] + del[2,n](-5n^3+45n^2-70n+24)/24 - del[4,n](3n/2) + del[6,n](-45n^2+262n)/6 + del[12,n]*42n + del[18,n]*60n + del[24,n]*35n - del[30,n]*38n - del[42,n]*82n - del[60,n]*330n - del[84,n]*144n - del[90,n]*96n - del[120,n]*144n - del[210,n]*96n]; Table[Int[n], {n,1,1000}] (* T. D. Noe, Dec 21 2006 *)
  • PARI
    apply( {A006561(n)=binomial(n,4)+if(n%2==0, (n>2) + (-5*n^2+45*n-70)*n/24 + vecsum([t[2] | t<-[4,6,12,18,24,30,42,60,84,90,120,210;-3/2,(262-45*n)/6,42,60,35,-38,-82,-330,-144,-96,-144,-96], n%t[1]==0])*n)}, [1..44]) \\ M. F. Hasler, Aug 23 2017, edited Aug 06 2021
  • Python
    def d(n,m): return not n % m
def A006561(n): return 0 if n == 2 else n*(42*d(n,12) - 144*d(n,120) + 60*d(n,18) - 96*d(n,210) + 35*d(n,24)- 38*d(n,30) - 82*d(n,42) - 330*d(n,60) - 144*d(n,84) - 96*d(n,90)) + (n**4 - 6*n**3 + 11*n**2 - 6*n -d(n,2)*(5*n**3 - 45*n**2 + 70*n - 24) - 36*d(n,4)*n - 4*d(n,6)*n*(45*n - 262))//24 # Chai Wah Wu, Mar 08 2021

Formula

Let delta(m,n) = 1 if m divides n, otherwise 0.
For n >= 3, a(n) = binomial(n,4) + (-5*n^3 + 45*n^2 - 70*n + 24)*delta(2,n)/24
- (3*n/2)*delta(4,n) + (-45*n^2 + 262*n)*delta(6,n)/6 + 42*n*delta(12,n)
+ 60*n*delta(18,n) + 35*n*delta(24,n) - 38*n*delta(30,n)
- 82*n*delta(42,n) - 330*n*delta(60,n) - 144*n*delta(84,n)
- 96*n*delta(90,n) - 144*n*delta(120,n) - 96*n*delta(210,n). [Poonen and Rubinstein, Theorem 1] - N. J. A. Sloane, Aug 09 2017
For odd n, a(n) = binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24, see A053126. For even n, use this formula, but then subtract 2 for every 3-crossing, subtract 5 for every 4-crossing, subtract 9 for every 5-crossing, etc. The number to be subtracted for a d-crossing is (d-1)*(d-2)/2. - Graeme McRae, Dec 26 2004
a(n) = A007569(n) - n. - T. D. Noe, Dec 23 2006
a(2n+5) = A053126(n+4). - Philippe Deléham, Jun 07 2013

A053123 Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order).

Original entry on oeis.org

1, 1, -2, 1, -4, 3, 1, -6, 10, -4, 1, -8, 21, -20, 5, 1, -10, 36, -56, 35, -6, 1, -12, 55, -120, 126, -56, 7, 1, -14, 78, -220, 330, -252, 84, -8, 1, -16, 105, -364, 715, -792, 462, -120, 9, 1, -18, 136, -560, 1365, -2002, 1716, -792, 165, -10, 1, -20, 171, -816, 2380, -4368, 5005, -3432, 1287, -220, 11, 1
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

T(n,m) = A053122(n,n-m).
G.f. for row polynomials and row sums same as in A053122.
Unsigned column sequences are A000012, A005843, A014105, A002492 for m=0..3, resp. and A053126-A053131 for m=4..9.
This is also the coefficient triangle for Chebyshev's U(2*n+1,x) polynomials expanded in decreasing odd powers of (2*x): U(2*n+1,x) = Sum_{m=0..n} T(n,m)*(2*x)^(2*(n-m)+1). See the W. Lang link given in A053125.
Unsigned version is mirror image of A078812. - Philippe Deléham, Dec 02 2008

Examples

			Triangle begins:
  1;
  1,  -2;
  1,  -4,  3;
  1,  -6, 10,   -4;
  1,  -8, 21,  -20,   5;
  1, -10, 36,  -56,  35,  -6;
  1, -12, 55, -120, 126, -56, 7; ...
E.g. fourth row (n=3) {1,-6,10,-4} corresponds to polynomial S(3,x-2) = x^3-6*x^2+10*x-4.

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795
  • Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
  • Stephen Barnett, "Matrices: Methods and Applications", Oxford University Press, 1990, p. 132, 343.

Links

Crossrefs

Cf. A053124, A053122, A172431.

Programs

  • GAP
    Flat(List([0..10], n-> List([0..n], k-> (-1)^k*Binomial(2*n-k+1,k) ))); # G. C. Greubel, Jul 23 2019
  • Magma
    [(-1)^k*Binomial(2*n-k+1,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 23 2019
  • Maple
    A053123 := proc(n,m)
    (-1)^m*binomial(2*n+1-m,m) ;
end proc: # R. J. Mathar, Sep 08 2013
  • Mathematica
    T[n_, m_]:= (-1)^m*Binomial[2*n+1-m, m]; Table[T[n, m], {n, 0, 11}, {m, 0, n}]//Flatten (* Jean-François Alcover, Mar 05 2014, after R. J. Mathar *)
  • PARI
    for(n=0,10, for(k=0,n, print1((-1)^k*binomial(2*n-k+1,k), ", "))) \\ G. C. Greubel, Jul 23 2019
  • Sage
    [[(-1)^k*binomial(2*n-k+1,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 23 2019

Formula

T(n, m) = 0 if n
T(n, m) = -2*T(n-1, m-1) + T(n-1, m) - T(n-2, m-2), T(n, -2) = 0, T(-2, m) = 0, T(n, -1) = 0 = T(-1, m), T(0, 0) = 1, T(n, m) = 0 if n
G.f. for m-th column (signed triangle): ((-1)^m)*x^m*Po(m+1, x)/(1-x)^(m+1), with Po(k, x) := Sum_{j=0..floor(k/2)} binomial(k, 2*j+1)*x^j.
The n-th degree polynomial is the characteristic equation for an n X n tridiagonal matrix with (diagonal = all 2's, sub and superdiagonals all -1's and the rest 0's), exemplified by the 4X4 matrix M = [2 -1 0 0 / -1 2 -1 0 / 0 -1 2 -1 / 0 0 -1 2]. - Gary W. Adamson, Jan 05 2005
Sum_{m=0..n} T(n,m)*(c(n))^(2*n-2*m) = 1/c(n), where c(n) = 2*cos(Pi/(2*n+3)). - L. Edson Jeffery, Sep 13 2013

A128908 Riordan array (1, x/(1-x)^2).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 4, 10, 6, 1, 0, 5, 20, 21, 8, 1, 0, 6, 35, 56, 36, 10, 1, 0, 7, 56, 126, 120, 55, 12, 1, 0, 8, 84, 252, 330, 220, 78, 14, 1, 0, 9, 120, 462, 792, 715, 364, 105, 16, 1, 0, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1
Offset: 0

Views

Author

Philippe Deléham, Apr 22 2007

Keywords

Comments

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,2,-1/2,1/2,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
Row sums give A088305. - Philippe Deléham, Nov 21 2007
Column k is C(n,2k-1) for k > 0. - Philippe Deléham, Jan 20 2012
From R. Bagula's comment in A053122 (cf. Damianou link p. 10), this array gives the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - Tom Copeland, Oct 11 2014
T is the convolution triangle of the positive integers (see A357368). - Peter Luschny, Oct 19 2022

Examples

			The triangle T(n,k) begins:
   n\k  0    1    2    3    4    5    6    7    8    9   10
   0:   1
   1:   0    1
   2:   0    2    1
   3:   0    3    4    1
   4:   0    4   10    6    1
   5:   0    5   20   21    8    1
   6:   0    6   35   56   36   10    1
   7:   0    7   56  126  120   55   12    1
   8:   0    8   84  252  330  220   78   14    1
   9:   0    9  120  462  792  715  364  105   16    1
  10:   0   10  165  792 1716 2002 1365  560  136   18    1
  ... reformatted by _Wolfdieter Lang_, Jul 31 2017
From _Peter Luschny_, Mar 06 2022: (Start)
The sequence can also be seen as a square array read by upwards antidiagonals.
   1, 1,   1,    1,    1,     1,     1,      1,      1, ...  A000012
   0, 2,   4,    6,    8,    10,    12,     14,     16, ...  A005843
   0, 3,  10,   21,   36,    55,    78,    105,    136, ...  A014105
   0, 4,  20,   56,  120,   220,   364,    560,    816, ...  A002492
   0, 5,  35,  126,  330,   715,  1365,   2380,   3876, ... (A053126)
   0, 6,  56,  252,  792,  2002,  4368,   8568,  15504, ... (A053127)
   0, 7,  84,  462, 1716,  5005, 12376,  27132,  54264, ... (A053128)
   0, 8, 120,  792, 3432, 11440, 31824,  77520, 170544, ... (A053129)
   0, 9, 165, 1287, 6435, 24310, 75582, 203490, 490314, ... (A053130)
    A27,A292, A389, A580,  A582, A1288, A10966, A10968, A165817       (End)

Links

Crossrefs

Cf. A002450, A007318, A034008, A053122, A078812, A084938, A088305.
Cf. Columns : A000007, A000027, A000292, A000389, A000580, A000582, A001288, A010966 ..(+2).. A011000, A017713 ..(+2).. A017763.
Cf. A000007, A001352, A008574, A054888, A084099, A084103, A163810, A357368.
Cf. A165817 (the main diagonal of the array).

Programs

  • Maple
    # Computing the rows of the array representation:
S := proc(n,k) option remember;
if n = k then 1 elif k < 0 or k > n then 0 else
S(n-1, k-1) + 2*S(n-1, k) - S(n-2, k) fi end:
Arow := (n, len) -> seq(S(n+k-1, k-1), k = 0..len-1):
for n from 0 to 8 do Arow(n, 9) od; # Peter Luschny, Mar 06 2022
# Uses function PMatrix from A357368.
PMatrix(10, n -> n); # Peter Luschny, Oct 19 2022
  • Mathematica
    With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - x)^2/(1 - (2 + y)*x + x^2), {x, 0, nmax}, {y, 0, nmax}], x], y]] // Flatten (* G. C. Greubel, Nov 22 2017 *)
  • PARI
    for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, if(k==0, 0, binomial(n+k-1,2*k-1))), ", "))) \\ G. C. Greubel, Nov 22 2017
  • Python
    from functools import cache
@cache
def A128908(n, k):
    if n == k: return 1
    if (k <= 0 or k > n): return 0
    return A128908(n-1, k-1) + 2*A128908(n-1, k) - A128908(n-2, k)
for n in range(10):
    print([A128908(n, k) for k in range(n+1)]) # Peter Luschny, Mar 07 2022
  • Sage
    @cached_function
def T(k,n):
    if k==n: return 1
    if k==0: return 0
    return sum(i*T(k-1,n-i) for i in (1..n-k+1))
A128908 = lambda n,k: T(k,n)
for n in (0..10): print([A128908(n,k) for k in (0..n)]) # Peter Luschny, Mar 12 2016

Formula

T(n,0) = 0^n, T(n,k) = binomial(n+k-1, 2k-1) for k >= 1.
Sum_{k=0..n} T(n,k)*2^(n-k) = A002450(n) = (4^n-1)/3 for n>=1. - Philippe Deléham, Oct 19 2008
G.f.: (1-x)^2/(1-(2+y)*x+x^2). - Philippe Deléham, Jan 20 2012
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A001352(n), (-1)^(n+1)*A054888(n+1), (-1)^n*A008574(n), (-1)^n*A084103(n), (-1)^n*A084099(n), A163810(n), A000007(n), A088305(n) for x = -6, -5, -4, -3, -2, -1, 0, 1 respectively. - Philippe Deléham, Jan 20 2012
Riordan array (1, x/(1-x)^2). - Philippe Deléham, Jan 20 2012

A053134 Binomial coefficients C(2*n+4,4).

Original entry on oeis.org

1, 15, 70, 210, 495, 1001, 1820, 3060, 4845, 7315, 10626, 14950, 20475, 27405, 35960, 46376, 58905, 73815, 91390, 111930, 135751, 163185, 194580, 230300, 270725, 316251, 367290, 424270, 487635, 557845, 635376, 720720, 814385, 916895, 1028790, 1150626, 1282975
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Even-indexed members of fifth column of Pascal's triangle A007318.
Number of standard tableaux of shape (2n+1,1^4). - Emeric Deutsch, May 30 2004
Number of integer solutions to -n <= x <= y <= z <= w <= n. - Michael Somos, Dec 28 2011

Examples

			1 + 15*x + 70*x^2 + 210*x^3 + 495*x^4 + 1001*x^5 + 1820*x^6 + 3060*x^7 + ...

Links

Crossrefs

Cf. A000447, A002299, A053126, A000332.

Programs

  • Magma
    [Binomial(2*n+4,4): n in [0..30]]; // Vincenzo Librandi, Oct 07 2011
  • Mathematica
    Table[Binomial[2*n+4,4], {n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5 ,1}, {1, 15, 70, 210, 495}, 30] (* G. C. Greubel, Sep 03 2018 *)
  • PARI
    for(n=0,30, print1(binomial(2*n+4,4), ", ")) \\ G. C. Greubel, Sep 03 2018

Formula

a(n) = binomial(2*n+4, 4) = A000332(2*n+4).
G.f.: (1 + 10*x + 5*x^2)/(1-x)^5.
a(1 - n) = A053126(n). - Michael Somos, Dec 28 2011
E.g.f.: (6 + 84*x + 123*x^2 + 44*x^3 + 4*x^4)*exp(x)/6. - G. C. Greubel, Sep 03 2018
a(n) = (1/6)*(n + 1)*(n + 2)*(2*n + 1)*(2*n + 3). - Gerry Martens, Oct 13 2022
From Amiram Eldar, Oct 21 2022: (Start)
Sum_{n>=0} 1/a(n) = 16*log(2) - 10.
Sum_{n>=0} (-1)^n/a(n) = 10 - 2*Pi - 4*log(2). (End)

A112742 a(n) = n^2*(n^2 - 1)/3.

Original entry on oeis.org

0, 0, 4, 24, 80, 200, 420, 784, 1344, 2160, 3300, 4840, 6864, 9464, 12740, 16800, 21760, 27744, 34884, 43320, 53200, 64680, 77924, 93104, 110400, 130000, 152100, 176904, 204624, 235480, 269700, 307520, 349184, 394944, 445060, 499800, 559440
Offset: 0

Views

Author

Matthew T. Cornick (maruth(AT)gmail.com), Sep 16 2005

Keywords

Comments

Second derivative of the n-th Chebyshev polynomial (of the first kind) evaluated at x=1.
The second derivative at x=-1 is just (-1)^n * a(n).
The difference between two consecutive terms generates the sequence a(n+1) - a(n) = A002492(n).
Consider the partitions of 2n into two parts (p,q) where p <= q. Then a(n) is the total volume of the family of rectangular prisms with dimensions p, |q-p| and |q-p|. - Wesley Ivan Hurt, Apr 15 2018

Examples

			a(4)=80 because
C_4(x) = 1 - 8x^2 + 8x^4,
C'_4(x) = -16x + 32x^3,
C''_4(x) = -16 + 96x^2,
C''_4(1) = -16 + 96 = 80.

Links

Crossrefs

Cf. A000914, A002415, A002492.

Programs

  • Mathematica
    Table[D[ChebyshevT[n, x], {x, 2}], {n, 0, 100}] /. x -> 1
  • PARI
    a(n)=n^2*(n^2-1)/3 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (n-1)*n^2*(n+1)/3 = 4*A002415(n).
a(n) = 2*( A000914(n-1) + C(n+1,4) ). - David Scambler, Nov 27 2006
From Colin Barker, Jan 26 2012: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: 4*x^2*(1+x)/(1-x)^5. (End)
E.g.f.: exp(x)*x^2*(6 + 6*x + x^2)/3. - Stefano Spezia, Dec 11 2021
a(n) = A053126(n+2) - A006324(n-1). - Yasser Arath Chavez Reyes, Feb 22 2024

A053127 Binomial coefficients C(2*n-4,5).

Original entry on oeis.org

6, 56, 252, 792, 2002, 4368, 8568, 15504, 26334, 42504, 65780, 98280, 142506, 201376, 278256, 376992, 501942, 658008, 850668, 1086008, 1370754, 1712304, 2118760, 2598960, 3162510, 3819816, 4582116, 5461512, 6471002, 7624512
Offset: 5

Views

Author

Wolfdieter Lang

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Links

Crossrefs

Cf. A053123, A053132, A053126.

Programs

  • Haskell
    a053127 = (* 2) . a053132  -- Reinhard Zumkeller, Mar 03 2015
  • Magma
    [Binomial(2*n-4,5): n in [5..40]]; // Vincenzo Librandi, Oct 07 2011
  • Mathematica
    Binomial[2Range[5,40]-4,5] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{6,56,252,792,2002,4368},30] (* Harvey P. Dale, Jun 03 2013 *)
  • PARI
    for(n=5,50, print1(binomial(2*n-4,5), ", ")) \\ G. C. Greubel, Aug 26 2018

Formula

a(n) = binomial(2*n-4, 5) if n >= 5 else 0.
a(n) = -A053123(n,5), n >= 5; a(n) := 0, n=0..4 (sixth column of shifted Chebyshev's S-triangle, decreasing order).
G.f.: (6+20*x+6*x^2)/(1-x)^6.
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). - Harvey P. Dale, Jun 03 2013
E.g.f.: (-840 + 750*x - 330*x^2 + 95*x^3 - 20*x^4 + 4*x^5)*exp(x)/15. - G. C. Greubel, Aug 26 2018
a(n) = (2*n-8)*(2*n-7)*(2*n-6)*(2*n-5)*(2*n-4)/120. - Wesley Ivan Hurt, Mar 25 2020
From Amiram Eldar, Jan 03 2022: (Start)
Sum_{n>=5} 1/a(n) = 335/12 - 40*log(2).
Sum_{n>=5} (-1)^(n+1)/a(n) = 85/12 - 10*log(2). (End)

A335102 Irregular triangle read by rows: consider the regular n-gon defined in A007678. T(n,k) (n >= 1, k >= 4+2*t where t>=0) is the number of non-boundary vertices in the n-gon at which k polygons meet.

Original entry on oeis.org

0, 0, 0, 1, 5, 12, 1, 35, 40, 8, 1, 126, 140, 20, 0, 1, 330, 228, 60, 12, 0, 1, 715, 644, 112, 0, 0, 0, 1, 1365, 1168, 208, 0, 0, 0, 0, 1, 2380, 1512, 216, 54, 54, 0, 0, 0, 1, 3876, 3360, 480, 0, 0, 0, 0, 0, 0, 1, 5985, 5280, 660, 0, 0, 0, 0, 0, 0, 0, 1, 8855, 6144, 864, 264, 24, 0, 0, 0, 0, 0, 0, 12, 12650
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, May 23 2020

Keywords

Examples

			Table begins:
      0;
      0;
      0;
      1;
      5;
     12,    1;
     35;
     40,    8,   1;
    126;
    140,   20,   0,   1;
    330;
    228,   60,  12,   0,   1;
    715;
    644,  112,   0,   0,   0,  1;
   1365;
   1168,  208,   0,   0,   0,  0, 1;
   2380;
   1512,  216,  54,  54,   0,  0, 0, 1;
   3876;
   3360,  480,   0,   0,   0,  0, 0, 0, 1;
   5985;
   5280,  660,   0,   0,   0,  0, 0, 0, 0, 1;
   8855;
   6144,  864, 264,  24,   0,  0, 0, 0, 0, 0, 1;
  12650;
  11284, 1196,   0,   0,   0,  0, 0, 0, 0, 0, 0, 1;
  17550;
  15680, 1568,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 1;
  23751;
  13800, 2250, 420, 180, 120, 30, 0, 0, 0, 0, 0, 0, 0, 1;
  31465;
  28448, 2464,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  40920;
  37264, 2992,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  52360;

Links

Crossrefs

Columns give A292104, A101363 (2n-gon), A101364, A101365.
Row sums give A006561.
Cf. A007569, A007678, A053126, A292105, A333275.

Formula

If n = 2t+1 is odd then the n-th row has a single term, T(2t+1, 2t+4) = binomial(2t+1,4) (these values are given in A053126).

A292219 Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].

Original entry on oeis.org

1, 6, 1, 60, 20, 1, 840, 420, 42, 1, 15120, 10080, 1512, 72, 1, 332640, 277200, 55440, 3960, 110, 1, 8648640, 8648640, 2162160, 205920, 8580, 156, 1, 259459200, 302702400, 90810720, 10810800, 600600, 16380, 210, 1, 8821612800, 11762150400, 4116752640, 588107520, 40840800, 1485120, 28560, 272, 1
Offset: 0

Views

Author

Wolfdieter Lang, Sep 23 2017

Keywords

Comments

For the general L[d,a] triangles see A286724, also for references.
This is the generalized signless Lah number triangle L[4,3], the Sheffer triangle ((1 - 4*t)^(-3/2), t/(1 - 4*t)). It is defined as transition matrix
risefac[4,3](x, n) = Sum_{m=0..n} L[4,3](n, m)*fallfac[4,3](x, m), where risefac[4,3](x, n) := Product_{0..n-1} (x + (3 + 4*j)) for n >= 1 and risefac[4,3](x, 0) := 1, and fallfac[4,3](x, n):= Product_{0..n-1} (x - (3 + 4*j)) for n >= 1 and fallfac[4,3](x, 0) := 1.
In matrix notation: L[4,3] = S1phat[4,3]*S2hat[4,3] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations A225471 and A225469, respectively.
The a- and z-sequences for this Sheffer matrix have e.g.f.s Ea(t) = 1 + 4*t and Ez(t) = (1 + 4*t)*(1 - (1 + 4*t)^(-3/2))/t, respectively. That is, a = {1, 4, repeat(0)} and z(n) = 2*A292221(n). See the W. Lang link on a- and z-sequences there.
The inverse matrix T^(-1) = L^(-1)[4,3] is Sheffer ((1 + 4*t)^(-3/2), t/(1 + 4*t)). This means that T^(-1)(n, m) = (-1)^(n-m)*T(n, m).
fallfac[4,3](x, n) = Sum_{m=0..n} (-1)^(n-m)*T(n, m)*risefac[4,3](x, m), n >= 0.
Diagonal sequences have o.g.f. G(d, x) = A001813(d)*Sum_{m=0..d} A103327(d, m)*x^m/(1 - x)^(2*d + 1), for d >= 0 (d=0 main diagonal). G(d, x) generates {A001813(d)*binomial(2*(m + d) + 1, 2*d)}{m >= 0}. See the second W. Lang link on how to compute o.g.f.s of diagonal sequences of general Sheffer triangles. - _Wolfdieter Lang, Oct 12 2017

Examples

			The triangle T(n, m) begins:
  n\m          0           1          2         3        4       5     6   7  8
  0:           1
  1:           6           1
  2:          60          20          1
  3:         840         420         42         1
  4:       15120       10080       1512        72        1
  5:      332640      277200      55440      3960      110       1
  6:     8648640     8648640    2162160    205920     8580     156     1
  7:   259459200   302702400   90810720  10810800   600600   16380   210   1
  8:  8821612800 11762150400 4116752640 588107520 40840800 1485120 28560 272  1
  ...
Recurrence from a-sequence: T(4, 2) = (4/2)*T(3, 1) + 4*4*T(3, 2) = 2*420 + 16*42 = 1512.
Recurrence from z-sequence: T(4, 0) = 4*(z(0)*T(3, 0) + z(1)*T(3, 1) + z(2)*T(3, 2)+ z(3)*T(3, 3)) = 4*(6*840 - 6*420 + 40*42 -420*1) = 15120.
Meixner type identity for n = 2: (D_x - 4*(D_x)^2)*(60 + 20*x + 1*x^2 ) = (20 + 2*x) - 4*2 = 2*(6 + x).
Sheffer recurrence for R(3, x): [(6 + x) + 8*(3 + x)*D_x + 16*x*(D_x)^2] (60 + 20*x + 1*x^2) = (6 + x)*(60 + 20*x + x^2) + 8*(3 + x)*(20 + 2*x) + 16*2*x = 840 + 420*x + 42*x^2 + x^3 = R(3, x).
Boas-Buck recurrence for column m = 2 with n = 4: T(4, 2) = (2*4!/2)*(3 + 2*2)*(1*42/3! + 4*1/2!) = 1512.
Diagonal sequence d = 2: {60, 420, 1512, ...} has o.g.f. 12*(5 + 10*x + x^2)/(1 - x)^5 (see A001813(2) and row n=2 of A103327) generating {12*binomial(2*(m + 2) + 1, 4)}_{m >= 0}. - _Wolfdieter Lang_, Oct 12 2017

References

  • Steven Roman, The Umbral Calculus, Academic press, Orlando, London, 1984, p. 50.

Links

Crossrefs

Cf. A225469, A225471, A271703 L[1,0], A286724 L[2,1], A290596 L[3,1], A290597 L[3,2], A048854 L[4,1], A292221, A103327,
Diagonal sequences: A000012, 2*A014105(m+1), 12*A053126(m+4), 120*A053128(m+6), A053130(n+8), ... - Wolfdieter Lang, Oct 12 2017

Formula

T(n, m) = L[4,3](n,m) = Sum_{k=m..n} A225471(n, k)*A225469(k, m), 0 <= m <= n.
E.g.f. of row polynomials R(n, x) := Sum_{m=0..n} T(n, m)*x^m:
(1 - 4*t)^(-3/2)*exp(x*t/(1 - 4*t)) (this is the e.g.f. for the triangle).
E.g.f. of column m: (1 - 4*t)^(-3/2)*(t/(1 - 4*t))^m/m!, m >= 0.
Three term recurrence for column entries k >= 1: T(n, m) = (n/m)*T(n-1, m-1) + 4*n*T(n-1, m) with T(n, m) = 0 for n < m, and for the column m = 0: T(n, 0) = n*Sum_{j=0}^(n-1) z(j)*T(n-1, j), n >= 1, T(0, 0) = 0, from the a-sequence {1, 4 repeat(0)} and z(j) = 2*A292221(j) (see above).
Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(4*n - 1)*T(n-1, m) - 8*(n-1)*(2*n - 1)*T(n-2, m), n >= m >= 0, with T(0, 0) =1, T(-1, m) = 0, T(n, -1) = 0 and T(n, m) = 0 if n < m.
Meixner type identity for (monic) row polynomials: (D_x/(1 + 4*D_x)) * R(n, x) = n * R(n-1, x), n >= 1, with R(0, x) = 1 and D_x = d/dx. That is, Sum_{k=0..n-1} (-4)^k*{D_x)^(k+1)*R(n, x) = n*R(n-1, x), n >= 1.
General recurrence for Sheffer row polynomials (see the Roman reference, p. 50, Corollary 3.7.2, rewritten for the present Sheffer notation):
R(n, x) = [(6 + x)*1 + 8*(3 + x)*D_x + 16*x*(D_x)^2]*R(n-1, x), n >= 1, with R(0, x) = 1.
Boas-Buck recurrence for column m (see a comment in A286724 with references): T(n, m) = (2*n!/(n-m))*(3 + 2*m)*Sum_{p=0..n-1-m} 4^p*T(n-1-p, m)/(n-1-p)!, for n > m >= 0, with input T(m, m) = 1.
Explicit form (from the o.g.f.s of diagonal sequences): ((2*(n-m))!/(n-m)!)*binomial(2*n + 1, 2*(n-m)), n >= m >= 0, and vanishing for n < m. - Wolfdieter Lang, Oct 12 2017

A258582 a(n) = n*(2*n + 1)*(4*n + 1)/3.

Original entry on oeis.org

0, 5, 30, 91, 204, 385, 650, 1015, 1496, 2109, 2870, 3795, 4900, 6201, 7714, 9455, 11440, 13685, 16206, 19019, 22140, 25585, 29370, 33511, 38024, 42925, 48230, 53955, 60116, 66729, 73810, 81375, 89440, 98021, 107134, 116795, 127020, 137825, 149226, 161239, 173880
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 06 2015

Keywords

Comments

First bisection of the square pyramidal numbers (A000330).

Links

Crossrefs

Cf. A000330, A001477, A005408, A016813, A053126 (partial sums), A100157.

Programs

  • Magma
    [n*(2*n+1)*(4*n+1)/3: n in [0..50]]; // Wesley Ivan Hurt, Nov 17 2015
  • Maple
    A258582:=n->n*(2*n + 1)*(4*n + 1)/3: seq(A258582(n), n=0..50); # Wesley Ivan Hurt, Nov 17 2015
  • Mathematica
    Table[(1/3) n (2 n + 1) (4 n + 1), {n, 0, 45}]
  • PARI
    vector(100, n, n--; n*(2*n+1)*(4*n+1)/3) \\ Altug Alkan, Nov 06 2015
  • PARI
    concat(0, Vec((5*x + 10*x^2 + x^3)/(1 - x)^4 + O(x^50))) \\ Altug Alkan, Nov 06 2015

Formula

G.f.: x*(5 + 10*x + x^2)/(1 - x)^4.
a(n) = A000330(2*n).
Sum_{n>0} 1/a(n) = 3*(6 - Pi - 4*log(2)) = 0.25745587...
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Nov 18 2015
a(n) = A006918(4*n-1) = A053307(4*n-1) = A228706(4*n-1) for n>0. - Bruno Berselli, Nov 18 2015
a(n) = Sum_{k=1..2*n} k^2 (see the first comment). E.g.f. exp(x)*(5*x+ 20*x^2/2+16*x^3/3!). - Wolfdieter Lang, Mar 13 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2) + 6*sqrt(2)*log(1+sqrt(2)) + 3*(sqrt(2)-1/2)*Pi - 18. - Amiram Eldar, Sep 17 2022
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