A001788 -id:A001788 - cp's OEIS frontend

A002409 a(n) = 2^n*C(n+6,6). Number of 6D hypercubes in an (n+6)-dimensional hypercube.

1, 14, 112, 672, 3360, 14784, 59136, 219648, 768768, 2562560, 8200192, 25346048, 76038144, 222265344, 635043840, 1778122752, 4889837568, 13231325184, 35283533824, 92851404800, 241413652480, 620777963520, 1580162088960

Offset: 0

N. J. A. Sloane

If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>5, a(n-6) is equal to the number of (n+6)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

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 = 0..400
Herbert Izbicki, Über Unterbaeume eines Baumes, Monatshefte für Mathematik, Vol. 74 (1970), pp. 56-62.
Herbert Izbicki, Über Unterbaeume eines Baumes, Monatshefte für Mathematik, Vol. 74 (1970), pp. 56-62.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
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.
Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

First differences are in A006976.
Cf. A000079, A001787, A001788, A001789, A003472, A054849, A054851, A082140.
a(n) = A038207(n+6,6).

[2^n*Binomial(n+6, 6): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011

A002409:=-1/(2*z-1)**7; # Simon Plouffe in his 1992 dissertation
seq(binomial(n+6,6)*2^n,n=0..22); # Zerinvary Lajos, Jun 16 2008

CoefficientList[Series[1/(1-2x)^7,{x,0,40}],x] (* or *) LinearRecurrence[ {14,-84,280,-560,672,-448,128},{1,14,112,672,3360,14784,59136},40] (* Harvey P. Dale, Jan 24 2022 *)

G.f.: 1/(1-2*x)^7.
a(n) = 2*a(n-1) + A054849(n-1).
For n>0, a(n) = 2*A082140(n).
a(n) = Sum_{i=6..n+6} binomial(i,6)*binomial(n+6,i). Example: for n=5, a(5) = 1*462 + 7*330 + 28*165 + 84*55 + 210*11 + 462*1 = 14784. - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 47/5 - 12*log(2).
Sum_{n>=0} (-1)^n/a(n) = 2916*log(3/2) - 5907/5. (End)
n*a(n) +2*(-n-6)*a(n-1)=0. - R. J. Mathar, Jul 22 2025

More terms from Henry Bottomley and James Sellers, Apr 15 2000

Typo in definition corrected by Zerinvary Lajos, Jun 16 2008

A054849 a(n) = 2^(n-5)*binomial(n,5). Number of 5D hypercubes in an n-dimensional hypercube.

Original entry on oeis.org

1, 12, 84, 448, 2016, 8064, 29568, 101376, 329472, 1025024, 3075072, 8945664, 25346048, 70189056, 190513152, 508035072, 1333592064, 3451650048, 8820883456, 22284337152, 55710842880, 137950658560, 338606161920

Offset: 5

Henry Bottomley, Apr 14 2000

With 5 leading zeros, binomial transform of binomial(n,5). - Paul Barry, Apr 10 2003

If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>4, a(n) is equal to the number of (n+5)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

G. C. Greubel, Table of n, a(n) for n = 5..1000
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (12, -60, 160, -240, 192, -64).

Cf. A000079, A001787, A001788, A001789, A003472, A002409, A054851.
a(n) = A038207(n,5).
Equals 2 * A082139. First differences are in A006975.

List([5..30], n-> 2^(n-5)*Binomial(n,5)); # G. C. Greubel, Aug 27 2019

[2^(n-5)*Binomial(n,5): n in [5..30]]; // G. C. Greubel, Aug 27 2019

seq(binomial(n+5,5)*2^n,n=0..22); # Zerinvary Lajos, Jun 13 2008

Table[2^(n-5)*Binomial[n,5], {n,5,30}] (* G. C. Greubel, Aug 27 2019 *)

vector(25, n, 2^(n-1)*binomial(n+4,5)) \\ G. C. Greubel, Aug 27 2019

[lucas_number2(n, 2, 0)*binomial(n,5)/32 for n in range(5, 28)] # Zerinvary Lajos, Mar 10 2009

a(n) = 2*a(n-1) + A003472(n-1).
From Paul Barry, Apr 10 2003: (Start)
O.g.f.: x^5/(1-2*x)^6.
E.g.f.: exp(2*x)*(x^5/5!) (with 5 leading zeros). (End)
a(n) = Sum_{i=5..n} binomial(i,5)*binomial(n,i). Example: for n=8, a(8) = 1*56 + 6*28 + 21*8 + 56*1 = 448. - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=5} 1/a(n) = 10*log(2) - 35/6.
Sum_{n>=5} (-1)^(n+1)/a(n) = 810*log(3/2) - 655/2. (End)

More terms from James Sellers, Apr 15 2000

A059297 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 24, 12, 1, 0, 5, 80, 90, 20, 1, 0, 6, 240, 540, 240, 30, 1, 0, 7, 672, 2835, 2240, 525, 42, 1, 0, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 0, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 0, 10, 11520, 262440

Offset: 0

N. J. A. Sloane, Jan 25 2001

T(n,k) = C(n,k)*k^(n-k) is the number of functions f from domain [n] to codomain [n+1] such that f(x)=n+1 for exactly k elements x of [n] and f(f(x))=n+1 for the remaining n-k elements x of [n]. Subsequently, row sums of T(n,k) provide the number of functions f:[n]->[n+1] such that either f(x)=n+1 or f(f(x))=n+1 for every x in [n]. We note that there are C(n,k) ways to choose the k elements mapped to n+1 and there are k^(n-k) ways to map n-k elements to a set of k elements. - Dennis P. Walsh, Sep 05 2012
Conjecture: the matrix inverse is A137452. - R. J. Mathar, Mar 12 2013
The above conjecture is correct. This triangle is the exponential Riordan array [1, x*exp(x)]. Thus the  inverse array is the exponential Riordan array [ 1, W(x)], which equals A137452. - Peter Bala, Apr 08 2013

			Triangle begins:
1;
0,  1;
0,  2,   1;
0,  3,   6,    1;
0,  4,  24,   12,    1;
0,  5,  80,   90,   20,   1;
0,  6, 240,  540,  240,  30,  1;
0,  7, 672, 2835, 2240, 525, 42,  1;
Row 4. Expansion of x^4 in terms of Abel polynomials:
x^4 = -4*x+24*x*(x+2)-12*x*(x+3)^2+x*(x+4)^3.
O.g.f. for column 2: A(-2,1/x) = x^2/(1-2*x)^3 = x^2+6*x^3+24*x^4+80*x^5+....

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

Alois P. Heinz, Rows n = 0..140, flattened
Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics, arXiv:quant-ph/0401126, 2004.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
Bruce E. Sagan, A note on Abel polynomials and rooted labeled forests. Discrete Mathematics 44(3): 293-298 (1983).
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 96, [_Tom Copeland_, Dec 20 2018].
Eric Weisstein's World of Mathematics, Abel Polynomial
Eric Weisstein's World of Mathematics, Idempotent Number
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

There are 4 versions: A059297, A059298, A059299, A059300.
Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
Row sums are A000248.
Cf. A061356, A202017, A137452 (inverse array), A264428.

/* As triangle */ [[Binomial(n,k)*k^(n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015

T:= (n, k)-> binomial(n, k) *k^(n-k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Sep 05 2012

nn=10;f[list_]:=Select[list,#>0&];Prepend[Map[Prepend[#,0]&,Rest[Map[f,Range[0,nn]!CoefficientList[Series[Exp[y x Exp[x]],{x,0,nn}],{x,y}]]]],{1}]//Grid  (* Geoffrey Critzer, Feb 09 2013 *)
t[n_, k_] := Binomial[n, k]*k^(n - k); Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Mar 23 2013 *)

# uses[bell_transform from A264428]
def A059297_row(n):
    nat = [k for k in (1..n)]
    return bell_transform(n, nat)
[A059297_row(n)  for n in range(8)] # Peter Luschny, Dec 20 2015

E.g.f.: exp(x*y*exp(y)). - Vladeta Jovovic, Nov 18 2003
Up to signs, this is the triangle of connection constants expressing the monomials x^n as a linear combination of the Abel polynomials A(k,x) := x*(x+k)^(k-1), 0 <= k <= n. O.g.f. for the k-th column: A(-k,1/x) = x^k/(1-k*x)^(k+1). Cf. A061356. Examples are given below. - Peter Bala, Oct 09 2011
The o.g.f.'s for the diagonals of this triangle are the rational functions occurring in the expansion of the compositional inverse (with respect to x) (x-t*x*exp(x))^-1 = x/(1-t) + 2*t/(1-t)^3*x^2/2! + (3*t+9*t^2)/(1-t)^5*x^3/3! + (4*t+52*t^2+64*t^3)/(1-t)^7*x^4/4! + .... For example, the o.g.f. for second subdiagonal is (3*t+9*t^2)/(1-t)^5 = 3*t + 24*t^2 + 90*t^3 + 240*t^4 + .... See the Bala link. The coefficients of the numerator polynomials are listed in A202017. - Peter Bala, Dec 08 2011
Recurrence equation: T(n+1,k+1) = Sum_{j=0..n-k} (j+1)*binomial(n,j)*T(n-j,k). - Peter Bala, Jan 13 2015
The Bell transform of [1,2,3,...]. See A264428 for the Bell transform. - Peter Luschny, Dec 20 2015

A049611 a(n) = T(n,2), array T as in A049600.

Original entry on oeis.org

0, 1, 4, 13, 38, 104, 272, 688, 1696, 4096, 9728, 22784, 52736, 120832, 274432, 618496, 1384448, 3080192, 6815744, 15007744, 32899072, 71827456, 156237824, 338690048, 731906048, 1577058304, 3388997632, 7264534528, 15535702016

Offset: 0

Clark Kimberling

Refer to A089378 and A075729 for the definition of hierarchies, subhierarchies and one-step transitions. - Thomas Wieder, Feb 28 2004
We may ask for the number of one-step transitions (NOOST) between all unlabeled hierarchies of n elements with the restriction that no subhierarchies are allowed. As an example, consider n = 4 and the hierarchy H1 = [[2,2]] with two elements on level 1 and two on level 2. Starting from H1 the hierarchies [[1, 3]], [[2, 1, 1]], [[1, 2, 1]] can be reached by moving one element only, but [[1, 1, 2]] cannot be reached in a one-step transitition. The solution is n = 1, NOOST = 0; n = 2, NOOST = 1; n = 3, NOOST = 4; n = 4, NOOST = 13; n = 5, NOOST = 38; n = 6, NOOST = 104; n = 7, NOOST = 272; n = 8, NOOST = 688; n = 9, NOOST = 1696. This is sequence A049611. - Thomas Wieder, Feb 28 2004
If X_1,X_2,...,X_n are 2-blocks of a (2n+2)-set X then, for n>=1, a(n+1) is the number of (n+2)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
In each composition (ordered partition) of the integer n, circle the first summand once, circle the second summand twice, etc. a(n) is the total number of circles in all compositions of n (that is, add k*(k+1)/2 for each composition into k parts). Note the O.g.f. is B(A(x)) where A(x)= x/(1-x) and B(x)= x/(1-x)^3.
This is the Riordan transform with the Riordan matrix A097805 (of the associated type) of the triangular number sequence A000217. See a Feb 17 2017 comment on A097805. - Wolfdieter Lang, Feb 17 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Milan Janjic, Two Enumerative Functions
M. Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
S. Kitaev, J. Remmel, p-Ascent Sequences, arXiv:1503.00914 [math.CO], 2015.
Sergey Kitaev, J. B. Remmel, A note on p-Ascent Sequences, Preprint, 2016.
Igor Makhlin, Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties, arXiv:2003.02916 [math.CO], 2020.
Agustín Moreno Cañadas, Hernán Giraldo, Gabriel Bravo Rios, On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

a(n+1)= A055252(n, 0), n >= 0. Row sums of triangle A055249.

Cf. A001793, A058396, A075729, A089378, A000217.

CoefficientList[Series[x (1-x)^2/(1-2x)^3,{x,0,40}],x] (* Harvey P. Dale, Sep 24 2013 *)

concat(0, Vec(x*(1-x)^2/(1-2*x)^3+O(x^99))) \\ Charles R Greathouse IV, Jun 12 2015

G.f.: x*(1-x)^2/(1-2*x)^3.
Binomial transform of quarter squares A002620(n+1): a(n) = Sum_{k=0..n} binomial(n, k)*floor((k+1)^2/4). - Paul Barry, May 27 2003
a(n) = 2^(n-4)*(n^2+5*n+2) - 0^n/8. - Paul Barry, Jun 09 2003
a(n+2) = A001787(n+2) + A001788(n). - Creighton Dement, Aug 02 2005
a(n) = Hyper2F1([-n+1, 3], [1], -1) for n>0. - Peter Luschny, Aug 02 2014
a(n) = Sum_{k=0..n-1} Sum_{j=0..n-1} Sum_{i=0..n-1} binomial(n-1, i+j+k). - Yalcin Aktar, Aug 27 2023

A081135 5th binomial transform of (0,0,1,0,0,0, ...).

Original entry on oeis.org

0, 0, 1, 15, 150, 1250, 9375, 65625, 437500, 2812500, 17578125, 107421875, 644531250, 3808593750, 22216796875, 128173828125, 732421875000, 4150390625000, 23345947265625, 130462646484375, 724792480468750

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, three-fold convolution of A000351 (powers of 5).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (15,-75,125).

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), this sequence (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[5^(n-2)*Binomial(n, 2): n in [0..30]]; // Vincenzo Librandi, Aug 06 2013

seq(n*(n-1)*5^(n-2)/2, n=0..30); # Zerinvary Lajos, May 03 2007

CoefficientList[Series[x^2/(1-5x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{15,-75,125},{0,0,1},30] (* Harvey P. Dale, Sep 13 2017 *)

[5^(n-2)*binomial(n,2) for n in range(0, 30)] # Zerinvary Lajos, Mar 12 2009

a(n) = 15*a(n-1) - 75*a(n-2) + 125*a(n-3), a(0)=a(1)=0, a(2)=1.
a(n) = 5^(n-2)*binomial(n, 2).
G.f.: x^2/(1-5*x)^3.
E.g.f.: (x^2/2)*exp(5*x). - G. C. Greubel, May 14 2021
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=2} 1/a(n) = 10 - 40*log(5/4).
Sum_{n>=2} (-1)^n/a(n) = 60*log(6/5) - 10. (End)

A081136 6th binomial transform of (0,0,1,0,0,0, ...).

Original entry on oeis.org

0, 0, 1, 18, 216, 2160, 19440, 163296, 1306368, 10077696, 75582720, 554273280, 3990767616, 28298170368, 198087192576, 1371372871680, 9403699691520, 63945157902336, 431629815840768, 2894458765049856, 19296391766999040

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, three-fold convolution of A000400 (powers of 6).

Number of n-permutations of 7 objects: p, u, v, w, z, x, y with repetition allowed, containing exactly two u's. - Zerinvary Lajos, May 23 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (18,-108,216).

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), this sequence (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[6^n*Binomial(n+2,2): n in [-2..20]]; // Vincenzo Librandi, Oct 16 2011

seq(binomial(n, 2)*6^(n-2), n=0..19); # Zerinvary Lajos, May 23 2008

nn=20;Range[0,nn]!CoefficientList[Series[x^2/2! Exp[6x],{x,0,nn}],x] (* Geoffrey Critzer, Oct 03 2013 *)
LinearRecurrence[{18,-108,216},{0,0,1},30] (* Harvey P. Dale, Apr 20 2022 *)

[6^(n-2)*binomial(n,2) for n in range(0, 21)] # Zerinvary Lajos, Mar 13 2009

a(n) = 18*a(n-1) -108*a(n-2) +216*a(n-3), a(0)=a(1)=0, a(2)=1.
a(n) = 6^(n-2)*C(n, 2).
G.f.: x^2/(1-6*x)^3.
E.g.f.: exp(6*x) * x^2/2. - Geoffrey Critzer, Oct 03 2013
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=2} 1/a(n) = 12 - 60*log(6/5).
Sum_{n>=2} (-1)^n/a(n) = 84*log(7/6) - 12. (End)

A130809 If X_1, ..., X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 3-subsets of X containing none of X_i, (i=1,...,n).

Original entry on oeis.org

8, 32, 80, 160, 280, 448, 672, 960, 1320, 1760, 2288, 2912, 3640, 4480, 5440, 6528, 7752, 9120, 10640, 12320, 14168, 16192, 18400, 20800, 23400, 26208, 29232, 32480, 35960, 39680, 43648, 47872, 52360, 57120, 62160, 67488, 73112, 79040, 85280

Offset: 3

Milan Janjic, Jul 16 2007

Uncentered octahedral numbers: take a simple cubical grid of size n X n X n where n = 2k is an even number, n >= 6. Retain all points that are at Manhattan distance n or greater from all 8 corners of the cube, and discard all other points. The number of points that remain is a(k). If n were to be an odd number, the same operation would yield the centered octahedral numbers A001845. - Arun Giridhar, Mar 06 2014
For an (n+2)-dimensional Rubik's cube, the number of cubes that have exactly 3 exposed facets. - Phil Scovis, Aug 03 2009
a(n) is the number of 2-simplices in an n-cross polytope. - Arkadiusz Wesolowski, Oct 16 2012
a(n) is also the number of unit tetrahedra in an (n+1)-scaled octahedron composed of the tetrahedral-octahedral honeycomb. - Jason Pruski, Aug 31 2017

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Harlan J. Brothers, Pascal's Prism: Supplementary Material, 2012.
Milan Janjic, Two Enumerative Functions
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000079, A001787, A001788, A001789, A001845, A002409, A003472, A038207, A046092, A054849, A054851, A056220, A140325, A140354.

[(4/3)*n*(n-1)*(n-2): n in [3..60]]; // Vincenzo Librandi, Oct 03 2017

Maple
```
a:=n->4/3*n*(n-1)*(n-2);
```

Table[(4/3) n (n - 1) (n - 2), {n, 3, 41}] (* or *)
Table[Binomial[n, n - 3] 2^3, {n, 3, 41}] (* or *)
DeleteCases[#, 0] &@ CoefficientList[Series[8 x^3/(1 - x)^4, {x, 0, 41}], x] (* Michael De Vlieger, Aug 31 2017 *)

a(n) = 4*n*(n-1)*(n-2)/3; \\ Andrew Howroyd, Nov 06 2018

a(n) = (4/3)*n*(n-1)*(n-2).
a(n) = C(n,n-3)*8, n >= 3. - Zerinvary Lajos, Dec 07 2007
G.f.: 8*x^3/(1-x)^4. - Colin Barker, Apr 14 2012
For n>1, a(n) = a(n-1) + A056220(n-1) + A056220(n-2). - Bruce J. Nicholson, Feb 14 2018
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=3} 1/a(n) = 3/16.
Sum_{n>=3} (-1)^(n+1)/a(n) = 3*log(2)/2 - 15/16. (End)
E.g.f.: 4*x^3*exp(x)/3. - Stefano Spezia, Apr 02 2024

A000531 From area of cyclic polygon of 2n + 1 sides.

Original entry on oeis.org

1, 7, 38, 187, 874, 3958, 17548, 76627, 330818, 1415650, 6015316, 25413342, 106853668, 447472972, 1867450648, 7770342787, 32248174258, 133530264682, 551793690628, 2276098026922, 9373521044908, 38546133661492

Offset: 1

Simon Plouffe

Expected number of matches remaining in Banach's original matchbox problem (counted when empty box is chosen), multiplied by 2^(2*n-1). - Michael Steyer, Apr 13 2001

A conjectured definition: Let 0 < a_1 < a_2 <...

a(n) = total weight of upsteps in all Dyck n-paths (A000108) when each upstep is weighted with its position in the path. For example, the Dyck path UDUUDUDD has upsteps in positions 1,3,4,6 and contributes 1+3+4+6=14 to the weight for Dyck 4-paths. The summand (n-k)*binomial(2*n+1, k) in the Maple formula below is the total weight of upsteps terminating at height n-k, 0<=k<=n-1. - David Callan, Dec 29 2006

Catalan transform of binomial transform of squares. - Philippe Deléham, Oct 31 2008

a(n) is also the number of walks of length 2n in the quarter plane starting and ending at the origin using steps {(1,1),(1,0),(-1,0), (-1,-1)} (which appear in Gessel's conjecture) in which the steps (1,0) and (-1,0) appear exactly once each. - Arvind Ayyer, Mar 02 2009

Equals the Catalan sequence, A000108, convolved with A002457: (1, 6, 30, 140, ...). - Gary W. Adamson, May 14 2009

Total number of occurrences of the pattern 213 (or 132) in all skew-indecomposable (n+2)-permutations avoiding the pattern 123. For example, a(1) = 1, since there is one occurrence of the pattern 213 in the set {213, 132}. - Cheyne Homberger, Mar 13 2013

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I.

T. D. Noe, Table of n, a(n) for n = 1..100
Arvind Ayyer, Towards a human proof of Gessel's conjecture, arXiv:0902.2329 [math.CO], 2009, JIS 12 (2009) 09.4.2
Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, On the self-convolution of generalized Fibonacci numbers, arXiv:1703.00323 [math.CO], 2017.
F. Bowman, Cyclic pentagons, Math. Gaz. 36, (1952). 244-250. MR0051523.
A. Burstein and S. Elizalde, Total occurrence statistics on restricted permutations, arXiv preprint arXiv:1305.3177 [math.CO], 2013.
C. Homberger, Expected patterns in permutation classes, Electronic Journal of Combinatorics, 19(3) (2012), P43.
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
Yasuhiko Kamiyama, The Euler characteristic of the regular spherical polygon spaces, arXiv:1803.05559 [math.GT], 2018.
A. F. Moebius, Über die Gleichungen, mittelst welcher aus den Seiten eines in einen Kreis zu beschreibenden Vielecks der Halbmesser des Kreises und die Fläche des Vielecks gefunden werden, Gesammelte Werke, vol. 1., pp. 407-438.
D. P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly, 102 (1995), 523-530.
Eric Weisstein's World of Mathematics, Cyclic Polygon
Y. Q. Zhao, Introduction to Probability with Applications

Cf. A002457 (Banach's modified matchbox problem), A135404, A002457, A258431.

f := proc(n) sum((n-k)*binomial(2*n+1,k),k=0..n-1); end;

a[n_] := ((2n+1)!/n!^2-4^n)/2; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Dec 07 2011, after Pari *)

PARI
```
a(n)=if(n<1,0,((2*n+1)!/n!^2-4^n)/2)
```

a(n) = ((2n+1)!/((n!)^2)-4^n)/2. - Simon Norton (simon(AT)dpmms.cam.ac.uk), May 14 2001
na(n) = (8n-2)a(n-1) - (16n-8)a(n-2), n>1. - Michael Somos, Apr 18 2003
E.g.f.: 1/2*((1+4*x)*exp(2*x)*BesselI(0, 2*x) + 4*x*exp(2*x)*BesselI(1, 2*x) - exp(4*x)). - Vladeta Jovovic, Sep 22 2003
a(n-1) = 4^n*sum_{k=0..n} binomial(2*k+1, k)*4^(-k) = (2*n+1)*(2*n+3)*C(n) - 2^(2*n+1) (C(n) = Catalan); g.f.: x*c(x)/(1-4*x)^(3/2), c(x): g.f. of Catalan numbers A000108. - Wolfdieter Lang
a(n) = Sum_{k=0..n} A039599(n,k)*k^2, for n>=1. - Philippe Deléham, Jun 10 2007
a(n) = Sum_{k=0..n} A106566(n,k)*A001788(k). - Philippe Deléham, Oct 31 2008
(Conjecture) a(n)=2^(2*n)*sum_{k=1..n} cos(k*Pi/(2*n+1))^2*n. - L. Edson Jeffery, Jan 21 2012

Moebius reference from Michael Somos

A027474 a(n) = 7^(n-2) * C(n,2).

Original entry on oeis.org

1, 21, 294, 3430, 36015, 352947, 3294172, 29647548, 259416045, 2219448385, 18643366434, 154231485954, 1259557135291, 10173346092735, 81386768741880, 645668365352248, 5084638377148953, 39779817891812397, 309398583602985310

Offset: 2

Olivier Gérard

7th binomial transform of (0,0,1,0,0,0,........). Starting at 1, the three-fold convolution of A000420 (powers of 7). - Paul Barry, Mar 08 2003

Vincenzo Librandi, Table of n, a(n) for n = 2..400
Index entries for linear recurrences with constant coefficients, signature (21,-147,343).

Third column of A027466.

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), this sequence (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[7^(n-2)* Binomial(n, 2): n in [2..20]]; /* Vincenzo Librandi, Oct 12 2011 */

seq(binomial(n, 2)*7^(n-2), n=2..30); # Zerinvary Lajos, Jun 12 2008

Table[7^(n-2) Binomial[n,2], {n,2,20}] (* Harvey P. Dale, Sep 25 2011 *)

a(n)=7^(n-2)*n*(n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015

[7^(n-2)*binomial(n,2) for n in range(2, 21)] # Zerinvary Lajos, Mar 13 2009

From Paul Barry, Mar 08 2003: (Start)
G.f.: x^2 / (1-7*x)^3.
a(n) = 21*a(n-1) - 147*a(n-2) + 343*a(n-3), a(0) = a(1) = 0, a(2) = 1. (End)
Numerators of sequence a[3,n] in (a[i,j])^3 where a[i,j] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.
E.g.f.: (x^2/2)*exp(7*x). - G. C. Greubel, May 13 2021
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=2} 1/a(n) = 14 - 84*log(7/6).
Sum_{n>=2} (-1)^n/a(n) = 112*log(8/7) - 14. (End)

Edited by Ralf Stephan, Dec 30 2004

A081138 8th binomial transform of (0,0,1,0,0,0, ...).

Original entry on oeis.org

0, 0, 1, 24, 384, 5120, 61440, 688128, 7340032, 75497472, 754974720, 7381975040, 70866960384, 670014898176, 6253472382976, 57724360458240, 527765581332480, 4785074604081152, 43065671436730368, 385057768140177408

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, the three-fold convolution of A001018 (powers of 8).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (24,-192,512).

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), this sequence (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[8^n*Binomial(n+2, 2): n in [-2..20]]; // Vincenzo Librandi, Oct 16 2011

LinearRecurrence[{24,-192,512},{0,0,1},30] (* Harvey P. Dale, Jun 08 2014 *)

a(n) = 24*a(n-1) - 192*a(n-2) + 512*a(n-3) for n>2, a(0)=a(1)=0, a(2)=1.
a(n) = 8^(n-2)*binomial(n, 2).
G.f.: x^2/(1 - 8*x)^3.
E.g.f.: (x^2/2)*exp(8*x). - G. C. Greubel, May 13 2021
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=2} 1/a(n) = 16 - 112*log(8/7).
Sum_{n>=2} (-1)^n/a(n) = 144*log(9/8) - 16. (End)

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A002409 a(n) = 2^n*C(n+6,6). Number of 6D hypercubes in an (n+6)-dimensional hypercube.

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A054849 a(n) = 2^(n-5)*binomial(n,5). Number of 5D hypercubes in an n-dimensional hypercube.

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A059297 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1.

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A049611 a(n) = T(n,2), array T as in A049600.

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A081135 5th binomial transform of (0,0,1,0,0,0, ...).

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A081136 6th binomial transform of (0,0,1,0,0,0, ...).

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