A077042 -id:A077042 - cp's OEIS frontend

A005191 Central pentanomial coefficients: largest coefficient of (1 + x + ... + x^4)^n.

1, 1, 5, 19, 85, 381, 1751, 8135, 38165, 180325, 856945, 4091495, 19611175, 94309099, 454805755, 2198649549, 10651488789, 51698642405, 251345549849, 1223798004815, 5966636799745, 29125608152345, 142330448514875, 696235630761115, 3408895901222375

Offset: 0

N. J. A. Sloane

nonn

Coefficient of x^n in ((1-x^10)/((1-x^5)(1-x^2)(1-x)))^n. - Michael Somos, Sep 24 2003
Note that n divides a(n+1) - a(n). - T. D. Noe, Mar 16 2005
Terms that are not a multiple of 5 have zero density, namely, there are fewer than n^(log(4)/log(5)) such terms among A005191(1..n). In particular, A005191(5k+2) and A005191(5k+4) are multiples of 5 for every k. - Max Alekseyev, Apr 25 2005
Number of n-step 1-D walks ending at the origin with steps of size 0, 1 or 2. - David Scambler, Apr 09 2012
Number of compositions of 2n into exactly n nonnegative parts <= four.  a(2) = 5: [4,0], [3,1], [2,2], [1,3], [0,4]. - Alois P. Heinz, Sep 13 2018
Let f(m) = ceiling((q+log(q))/log(25)), where q = -log(log(5)/(2*m^2*Pi)) then f(a(n)) = n, for n > 0. - Miko Labalan, Oct 07 2024

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 603-604.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
Armen G. Bagdasaryan and Ovidiu Bagdasar, On some results concerning generalized arithmetic triangles, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77.
V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
Zagros Lalo, Formula for Central terms in triangle A035343 ((1 + x + x^2 + x^3 + x^4)^n).
Lyle E. Muller and Michelle Rudolph-Lilith, On a link between Dirichlet kernels and central multinomial coefficients, Discrete Mathematics, Volume 338, Issue 9, 6 September 2015, Pages 1567-1572.
Project Euler, Quintinomial coefficients, Problem 588
M. Rudolph-Lilith and L. E. Muller, On an explicit representation of central (2k+1)-nomial coefficients, arXiv preprint arXiv:1403.5942 [math.CO], 2014.
Index entries for sequences of k-nomial coefficients

Cf. A035343, A349936 (bisection).

Row 5 of A077042. Column 2 of A201551 (?)

List([0..25],n->Sum([0..Int(2*n/5)],k->Binomial(n,k)*Binomial(-n,2*n-5*k))); # Muniru A Asiru, Sep 26 2018

seq(coeff(series(((1-x^10)/((1-x^5)*(1-x^2)*(1-x)))^n,x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Sep 26 2018

Flatten[{1,Table[Coefficient[Expand[Sum[x^j,{j,0,4}]^n],x^(2*n)],{n,1,20}]}] (* Vaclav Kotesovec, Aug 09 2013 *)
a[n_] := a[n] = Sum[n!/((q - n)!*(j - 2*q + 2*n)!*(i - 2*j + q)!*(j - 2*i)!*i!), {i, 0, n/2}, {j, 0, n}, {q, n, 2*n}]; Table[a[n], {n, 0, 29}] (* Zagros Lalo, Sep 25 2018 *)
CoefficientList[Series[Sqrt[(-5x+2+2Sqrt[5x^2-6x+1])/(25x^3-10x^2-19x+4)],{x,0,30}],x] (* Harvey P. Dale, Aug 04 2021 *)

a(n)=if(n<0,0,polcoeff(((1-x^5)/(1-x)+x*O(x^(2*n)))^n,2*n))

a(n)=if(n<0,0,polcoeff(((1-x^10)/((1-x^5)*(1-x^2)*(1-x))+x*O(x^n))^n,n))

a(n) = sum(k=0,(2*n)\5,binomial(n,k)*binomial(-n,2*n-5*k)) /* Max Alekseyev */

a(n) = round((5^n+sum(j=1,2*n-1,(sin(5*Pi*j/2/n)/sin(Pi*j/2/n))^n))/2/n)-2 /* Max Alekseyev */

a(n) = vecmax(Vec(Pol(vector(5,k,1))^n)); \\ Michel Marcus, Jan 29 2017

a(n) = Sum_{k=0..floor(2n/5)} binomial(n,k)*binomial(-n, 2n-5k); a(n) = (5^n + Sum_{j=1..2n-1} (sin(5j*Pi/(2n))/sin(j*Pi/(2n)))^n)/(2n) - 2. - Max Alekseyev, Mar 04 2005
D-finite with recurrence: 2*n*(2*n-1)*(3*n-4)*a(n) - (3*n-1)*(19*n^2-38*n+18)*a(n-1) - 5*(n-1)*(3*n-4)*(2*n-1)*a(n-2) + 25*(n-1)*(n-2)*(3*n-1)*a(n-3) = 0. - R. J. Mathar, Feb 21 2010 [Proved using the Almkvist-Zeilberger algorithm in EKHAD. - Doron Zeilberger, Apr 02 2013]
G.f.: sqrt((-5*x+2+2*sqrt(5*x^2-6*x+1))/(25*x^3-10*x^2-19*x+4)). - Mark van Hoeij, May 06 2013
a(n) ~ 5^n/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 09 2013
a(n) = Sum_{i=0..n/2} Sum_{j=0..n} Sum_{q=n..2*n}(f); f=( n!/((q - n)!*(j - 2*q + 2*n)!*(i - 2*j + q)!*(j - 2*i)!*i!) ); f=0 for (j - 2*q + 2*n)<0 or (i - 2*j + q)<0 or (j - 2*i)<0. Also see formula in Links section. - Zagros Lalo, Sep 25 2018

A025012 Central heptanomial coefficients: largest coefficient of (1+x+...+x^6)^n.

Original entry on oeis.org

1, 1, 7, 37, 231, 1451, 9331, 60691, 398567, 2636263, 17538157, 117224317, 786588243, 5295520699, 35751527189, 241958082737, 1641010879207, 11150608945863, 75894449584849, 517331384963959, 3531097638576781, 24131083600660801, 165090433568378523

Offset: 0

David W. Wilson

Apparently: number of n-step 1-D walks ending at the origin with steps of size 0, 1, 2 or 3. - David Scambler, Apr 09 2012

Generally, largest coefficient of (1+x+...+x^k)^n is asymptotic to (k+1)^n * sqrt(6/(k*(k+2)*Pi*n)). - Vaclav Kotesovec, Aug 09 2013

Rudolph-Lilith, Michelle, and Lyle E. Muller. "On a link between Dirichlet kernels and central multinomial coefficients." Discrete Mathematics 338.9 (2015): 1567-1572.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
M. Rudolph-Lilith and L. E. Muller, On an explicit representation of central (2k+1)-nomial coefficients, arXiv preprint arXiv:1403.5942 [math.CO], 2014.
Doron Zeilberger, EKHAD; Local copy.
Index entries for sequences of k-nomial coefficients

Cf. column 3 of A201552. Row 7 of A077042.

g := RootOf((2*g+1)*(g^3+5*g^2+6*g+1)*x^2+g*(g^3+3+11*g+7*g^2)*x-g, g);
ogf := sqrt((-4*x*(7*x+6)*(1+2*x)*g^3-4*x*(77*x^2+117*x+43)*g^2-4*x*(119*x^2+188*x+72)*g-307*x^2-217*x^3+27-67*x)/((7*x-1)*(49*x^2-14*x-27)))/(1+x);
series(ogf,x=0,30); # Mark van Hoeij, May 06 2013

With[{poly=Plus@@(x^Range[0,6])},Table[Max[CoefficientList[Expand[poly^i],x]], {i,0,20}]] (* Harvey P. Dale, Mar 09 2011 *)

D-finite with recurrence: 3*n*(3*n-1)*(3*n-2)*a(n) +(41*n^3-723*n^2+1132*n-456)*a(n-1) +7*(-383*n^3+2607*n^2-5992*n+4608)*a(n-2) +49*(-83*n^3+1317*n^2-6706*n +10512)*a(n-3) +343*(199*n^3-2487*n^2+10394*n-14496)*a(n-4) +2401*(n-4)*(43*n^2-437*n+1116)*a(n-5) -16807*(n-4)*(n-5)*(5*n-24)*a(n-6) -117649*(n-5)*(n-6)*(n-4)*a(n-7) = 0. - R. J. Mathar, Feb 21 2010
I checked that Mathar's conjectured formula fits the 200 terms of the b-file. Note that the coefficients are powers of 7: 49=7^2, 343=7^3, 2401=7^4, 16807=7^5, 117649=7^6. - T. D. Noe, Nov 15 2010
Comment from Doron Zeilberger, Apr 12 2011: The recurrence can be rigorously proved (and also discovered at the same time, no need for a guessing program) automatically via the Almkvist-Zeilberger algorithm (procedure AZd in the EKHAD link). In fact the program finds a 4th-order recurrence, not a 7th-order one: 343/3*(4*n+15)*(n+3)*(n+2)*(n+1)/(3*n+7)/(4*n+7)/(11+3*n)/(n+4)*a(n)+98/3*(4*n+15)*(n+3)*(n+2)*(2*n+7) /(11+3*n)/(3*n+10)/(4*n+11)/(n+4)*a(n+1) -14/3*(n+3)*(92*n^3+713*n^2+1747*n+1372)/(3*n+7)/(4*n+7)/(11+3*n)/(n+4)*a(n+2) -2/3*(4*n+15)*(2*n+7)*(37*n^2 +222*n+332)/(11+3*n)/(3*n+10)/(4*n+1)/(n+4)*a(n+3) +a(n+4) = 0.
G.f.: sqrt((-4*x*(7*x+6)*(1+2*x)*g^3-4*x*(77*x^2+117*x+43)*g^2-4*x*(119*x^2+188*x+72)*g-307*x^2-217*x^3+27-67*x)/((7*x-1)*(49*x^2-14*x-27)))/(1+x)  where (2*g+1)*(g^3+5*g^2+6*g+1)*x^2+g*(g^3+3+11*g+7*g^2)*x-g = 0. - Mark van Hoeij, May 06 2013
a(n) ~ 7^n / sqrt(8*Pi*n). - Vaclav Kotesovec, Aug 09 2013

A077043 "Three-quarter squares": a(n) = n^2 - A002620(n).

Original entry on oeis.org

0, 1, 3, 7, 12, 19, 27, 37, 48, 61, 75, 91, 108, 127, 147, 169, 192, 217, 243, 271, 300, 331, 363, 397, 432, 469, 507, 547, 588, 631, 675, 721, 768, 817, 867, 919, 972, 1027, 1083, 1141, 1200, 1261, 1323, 1387, 1452, 1519, 1587, 1657, 1728, 1801, 1875, 1951

Offset: 0

Henry Bottomley, Oct 22 2002

Triangular numbers plus quarter squares: (n+1)*(n+2)/2 + floor(n^2/4) (i.e., A000217(n+1) + A002620(n)).
Largest coefficient in the expansion of (1+x+x^2+...+x^(n-1))^3=((1-x^n)/(1-x))^3, i.e., the coefficient of x^floor[3(n-1)/2] and of x^ceiling[3(n-1)/2]; also number of compositions of [3(n+1)/2] into exactly 3 positive integers each no more than n.
A set of n independent statements a,b,c,d..., produces n^2 conditional statements of the form "If a, then b" (including self-implications such as "If a, then a"). If such statements are taken as equivalent to "It is not the case that the first statement is true and the second is false" (material implication), A077043(n) is the minimum number of the conditional statements that can be true. (The maximum number of false conditional statements is A002620(n), the maximum product of two integers whose sum is n.) - Matthew Vandermast, Mar 04 2003
This is also the maximum number of triple intersections between three sets of n lines, where the lines in each set are parallel to each other. E.g., for n=3:
\.\.\.../././
.\.\.\./././.
..\.\.x././..
---+-*-*-+---
----*-*-*----
---+-*-*-+---
.././.x.\.\..
./././.\.\.\.
/././...\.\.\
where '*' = triple intersection, '+' and 'x' = double intersection.
I am pretty sure that the hexagonal configuration of intersections shown above is the optimum and I get the formulas a(n) = (3n^2)/4 for n even and (3n^2+1)/4 for n odd. - Gabriel Nivasch (gnivasch(AT)yahoo.com), Jan 13 2004
For n > 1 the sequence represents the maximum number of points that can be placed in a plane such that the largest distances between any two points does not exceed the shortest of the distances between any two points by more than a factor n-1. - Johannes Koelman (Joc_kay(AT)hotmail.com), Apr 27 2006
This is also the number of distinct noncongruent isosceles triangles with side length up to n. - Patrick Hurst (patrick(AT)imsa.edu), May 14 2008
Also concentric triangular numbers. A033428 and A003215 interleaved. - Omar E. Pol, Sep 28 2011
Number of (w,x,y) with all terms in {0,...,n} and w=x>range{w,x,y}. - Clark Kimberling, Jun 02 2012
Number of pairs (x,y) with x in {0,...,n}, y even in {0,...,2n}, and x<=y. - Clark Kimberling, Jul 02 2012
From Bob Selcoe, Aug 05 2013: (Start)
a(n) is the number of 3-member sets with non-repeating positive integer values (x,y,z) whose sums equal 3(n+1). Example: a(4)=12; thus there are 12 sets where x+y+z = 15: (1,2,12), (1,3,11), (1,4,10), (1,5,9), (1,6,8), (2,3,10), (2,4,9), (2,5,8), (2,6,7), (3,4,8), (3,5,7) and (4,5,6).
From above, the number of sets sharing minimum values (minvals) equals a(1)-a(0), a(2)-a(1), a(3)-a(2),... a(n)-a(n-1) which are the numbers not divisible by 3, in sequence (A001651), range n to 1. So in the above example, there is one set with minval 4, two sets with minval 3, four sets with minval 2 and five sets with minval 1. (End)
Number of partitions of 3n into exactly 3 parts. - Wesley Ivan Hurt, Jan 21 2014
Number of partitions of 3(n-1) into at most 3 parts. - Colin Barker, Mar 31 2015
Number of possible positions after n-1 steps on the lines of a hexagonal grid. - Reg Robson, Mar 08 2014
12*a(n) is a perfect square when n is even and 12*a(n) - 3 is a perfect square when n is odd. - Miquel Cerda, Jun 30 2016
Square of largest Euclidean distance from start point reachable by an n-step walk on a honeycomb lattice. - Hugo Pfoertner, Jun 21 2018

			G.f. = x + 3*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 27*x^6 + 37*x^7 + 48*x^8 + ...
a(4)=12 since the compositions of floor(3*(4+1)/2) = 7 into exactly 3 positive integers each no more than 4 are 1+2+4, 1+3+3, 1+4+2, 2+1+4, 2+2+3, 2+3+3, 2+4+1, 3+1+3, 3+2+2, 3+3+1, 4+1+2, 4+2+1.
From _Philippe Deléham_, Dec 17 2011: (Start)
a(1) = 1 = 1^3;
a(1) + a(3) = 1 + 7 = 2^3;
a(1) + a(3) + a(5) = 1 + 7 + 19 = 3^3;
a(1) + a(3) + a(5) + a(7) = 1 + 7 + 19 + 37 = 4^3;
a(1) + a(3) + a(5) + a(7) + a(9) = 1 + 7 + 19 + 37 + 61 = 5^3; ... (End)

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Ralf Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi, Section 4.4.4.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7 (2004), Article 04.1.6.
Gabriel Nivasch and Eyal Lev, Nonattacking Queens on a Triangle, Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 399-403. See page 402.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index entries for sequences related to compositions.

Column 3 of A195040. - Omar E. Pol, Sep 28 2011
Cf. A019298 (partial sums).
Cf. A002620, A033428, A003215.
Equals one more than A331952 and one less than A084684. - Greg Dresden, Feb 22 2020

a077043 n = a077043_list !! n
a077043_list = scanl (+) 0 a001651_list
-- Reinhard Zumkeller, Jan 06 2014

[Ceiling(n^2*3/4): n in [0..60]]; // Vincenzo Librandi, Jun 29 2011

A077043:=n->ceil(3*n^2/4); seq(A077043(n), n=0..60); # Wesley Ivan Hurt, Jan 21 2014

Table[Ceiling[(3n^2)/4], {n,0,60}] (* or *) LinearRecurrence[{2,0,-2,1}, {0,1,3,7}, 60] (* Harvey P. Dale, Dec 16 2012 *)

{a(n) = n^2 - (n^2 \ 4)}; /* Michael Somos, Jun 29 2011 */

a(n) = ceiling(n^2*3/4) = A077042(n, 3); a(-n) = a(n).
Also can be computed from 1 * C(n,0) + 2 * C(n,1) + 2 * C(n,2) - Sum((-2)^(k-3) C(n, k)). - Joshua Zucker, Nov 10 2002
a(n) = A002620(n-1) + A002620(n) + A002620(n+1). - Jon Perry, May 29 2003
From Jon Perry, May 29 2003: (Start)
a(2k)   = a(2k-2) + 6k - 3,
a(2k+1) = a(2k-1) + 6k,
a(4n)   = 12n^2,
a(4n+1) = a(4n)   + 6n + 1,
a(4n+2) = a(4n+1) + 6n + 2,
a(4n+3) = a(4n+2) + 6n + 4,
a(4n+4) = a(4n+3) + 6n + 5.
Differences between alternate terms give 3, 6, 9, 12, ... (End)
a(n+1) - a(n) = A001651(n), partial sums of A001651. - Reinhard Zumkeller, Dec 28 2007
From R. J. Mathar, Nov 10 2008: (Start)
G.f.: x*(1+x+x^2)/((1+x)*(1-x)^3).
a(n) + a(n+1) = A005448(n+1).
The inverse binomial transform yields 0 followed by A141531. (End)
Euler transform of length 3 sequence [3, 1, -1]. - Michael Somos, Jun 29 2011
a(n) = 3*n^2/4 - ((-1)^n-1)/8. - Omar E. Pol, Sep 28 2011
Sum_{k=0..n} a(2k+1) = partial sums of A003215 = (n+1)^3 (see example). - Philippe Deléham, Dec 17 2011
a(0)=0, a(1)=1, a(2)=3, a(3)=7, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Dec 16 2012
a(0)=0, a(1)=1, a(n) = 3*(n-1) + a(n-2). - Reg Robson, Mar 08 2014
a(2k) = 3k^2 = A033428(k), a(2k+1) = 3k^2 + 3k + 1 = A003215(k). - Jon Perry, Oct 25 2014
a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n)/2). - Wesley Ivan Hurt, Mar 12 2015
a(n) = (3*n)^2/12 for n even and a(n) = ((3*n)^2 + 3)/12 for n odd. - Miquel Cerda, Jun 30 2016
a(n) = Sum_{k=1..n} floor((n+k)/2). - Wesley Ivan Hurt, Mar 31 2017
0 = 1 +a(n)*(+a(n+1) -a(n+2)) +a(n+1)*(-3 -a(n+1) +a(n+2)) for all n in Z. - Michael Somos, Apr 02 2017
E.g.f.: (1/8)*exp(-x)*(-1 + exp(2*x)*(1 + 6*x + 6*x^2)). - Stefano Spezia, Nov 29 2019
Sum_{n>=1} 1/a(n) = Pi^2/18 + tanh(Pi/(2*sqrt(3)))*Pi/sqrt(3). - Amiram Eldar, Jan 16 2023

A025014 Central "nonomial" coefficient: largest coefficient of (1+x+...+x^8)^n.

Original entry on oeis.org

1, 1, 9, 61, 489, 3951, 32661, 273127, 2306025, 19610233, 167729959, 1441383219, 12434998005, 107632809909, 934263293679, 8129320828911, 70886845397481, 619288973447049, 5419332253680705, 47494787636620701, 416800775902696839

Offset: 0

David W. Wilson

Generally, largest coefficient of (1+x+...+x^k)^n is asymptotic to (k+1)^n * sqrt(6/(k*(k+2)*Pi*n)). - Vaclav Kotesovec, Aug 09 2013

T. D. Noe, Table of n, a(n) for n=0..200
Michelle Rudolph-Lilith and Lyle E. Muller, On a link between Dirichlet kernels and central multinomial coefficients, Discrete Mathematics 338.9 (2015): 1567-1572.
Index entries for sequences of k-nomial coefficients

Column 4 of A201552. Row 9 of A077042.

seq(add((-1)^k * binomial(n, k)*binomial(5*n-9*k-1, n-1), k = 0..floor(4*n/9)), n = 0..20) ; # Peter Bala, Oct 16 2024

Flatten[{1,Table[Coefficient[Expand[Sum[x^j,{j,0,8}]^n],x^(4*n)],{n,1,20}]}] (* Vaclav Kotesovec, Aug 09 2013 *)

The Almkvist-Zeilberger algorithm in EKHAD establishes the following recurrence:
-6561*(4*n+17)*(4*n+13)*(5*n+24)*(5*n+19)*(5*n+14)*(5*n+23)*(n+4)*(n+3)*(n+2)*(n+1)*a(n)+1458*(5*n+24)*(5*n+19)*(4*n+17)*(5*n+9)*(4*n+9)*(5*n+18)*(2*n+9)*(n+4)*(
n+3)*(n+2)*a(n+1)+162*(5*n+24)*(5*n+14)*(4*n+13)*(5*n+23)*(n+4)*(n+3)*(1020*n^4+12291*n^3+53378*n^2+98617*n+65610)*a(n+2)-18*(4*n+17)*(4*n+9)*(5*n+19)*(2*n+9)*(5
*n+9)*(5*n+18)*(n+4)*(385*n^3+4158*n^2+14551*n+16610)*a(n+3)-(5*n+23)*(4*n+13)*(4*n+9)*(5*n+24)*(5*n+14)*(5*n+9)*(2101*n^4+33616*n^3+201391*n^2+535416*n+532980)*
a(n+4)+8*(4*n+19)*(5*n+19)*(5*n+14)*(5*n+9)*(2*n+9)*(4*n+17)*(4*n+13)*(4*n+9)*(5*n+18)*(n+5)*a(n+5) = 0. - Doron Zeilberger, Apr 02 2013.
a(n) ~ 9^n * sqrt(3/(40*Pi*n)). - Vaclav Kotesovec, Aug 09 2013
a(n) = Sum_{k = 0..floor(4*n/9)} (-1)^k * binomial(n, k)*binomial(5*n-9*k-1, n-1). - Peter Bala, Oct 16 2024

A005190 Central quadrinomial coefficients: largest coefficient of (1 + x + x^2 + x^3)^n.

Original entry on oeis.org

1, 1, 4, 12, 44, 155, 580, 2128, 8092, 30276, 116304, 440484, 1703636, 6506786, 25288120, 97181760, 379061020, 1463609356, 5724954544, 22187304112, 86981744944, 338118529539, 1327977811076, 5175023913008, 20356299454276

Offset: 0

N. J. A. Sloane

The maximal coefficient is that of x^[3n/2]. - M. F. Hasler, Jul 23 2007

Let f(m) = ceiling((q+log(q))/log(16)), where q = -log(log(256)/(5*m^2*Pi)) then f(a(n)) = n, for n > 0. - Miko Labalan, Oct 07 2024

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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..200
V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
T. Neuschel, A Note on Extended Binomial Coefficients, J. Int. Seq. 17 (2014) # 14.10.4.
Claudia Smith and Verner E. Hoggatt, Jr., A Study of the Maximal Values in Pascal's Quadrinomial Triangle, Fibonacci Quart. 17 (1979), no. 3, 264-269.
Index entries for sequences of k-nomial coefficients

Cf. A005721 (bisection), A005723 (bisection).

Row 4 of A077042.

P:=PolynomialRing(Integers()); [Max(Coefficients((1+x+x^2+x^3)^n)): n in [0..26]]; // Vincenzo Librandi, Aug 09 2014

With[{exp=Total[x^Range[0,3]]},Table[Max[CoefficientList[Expand[exp^n], x]],{n,0,30}]] (* Harvey P. Dale, Nov 24 2011 *)

a(n)=vecmax(vector(3*n,i,polcoeff((1+x+x^2+x^3)^n,i,x)))

A005190(n)=polcoeff((1+x+x^2+x^3)^n,(3*n)>>1) \\ M. F. Hasler, Jul 23 2007

Limit_{n -> infinity} a(n+1)/a(n) = 4; for n>2, a(n+1) < 4*a(n). - Benoit Cloitre, Sep 28 2002
a(n) ~ 4^n * sqrt(2/(5*Pi*n)). - Vaclav Kotesovec, Aug 09 2013
Recurrence: 3*n*(3*n-1)*(3*n+1)*(75*n^3 - 390*n^2 + 635*n - 348)*a(n) = 12*(675*n^5 - 4095*n^4 + 8405*n^3 - 7925*n^2 + 3548*n - 664)*a(n-1) + 16*(n-1)*(2175*n^5 - 13335*n^4 + 29275*n^3 - 27707*n^2 + 11334*n - 2814)*a(n-2) - 640*(n-2)*(n-1)*(15*n^3 - 66*n^2 + 52*n - 15)*a(n-3) - 512*(n-3)*(n-2)*(n-1)*(75*n^3 - 165*n^2 + 80*n - 28)*a(n-4). - Vaclav Kotesovec, Aug 09 2013

A018901 Central hexanomial coefficients: largest coefficient of (1 + x + ... + x^5)^n.

Original entry on oeis.org

1, 1, 6, 27, 146, 780, 4332, 24017, 135954, 767394, 4395456, 25090131, 144840476, 833196442, 4836766584, 27981391815, 163112472594, 947712321234, 5542414273884, 32312202610863, 189456975899496, 1107575676600876

Offset: 0

Jonn Dalton jdalton(AT)vnet.ibm.com

nonn

Greatest multiplicity of one- or two-dimensional standard representation of Lie algebras sl(2) in decomposition of tensor power F6^k, where F6 is the standard 6-dimensional irreducible representation of sl(2). - Leonid Bedratyuk, Jul 22 2004

Sum_{k=0..floor(5*n/12)} (-1)^k*binomial(n,k)*binomial(n + floor(5*n/2) - 6*k - 1, n-1). - Warut Roonguthai, May 21 2006

			Number of ways of getting most likely sum using n 6-sided dice (e.g., for n=2, 7 is the most prevalent sum and there are 6 different ways to get it: 1+6, 2+5, 3+4, 4+3, 5+2, 6+1).

T. D. Noe, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Recurrence
Index entries for sequences of k-nomial coefficients

Cf. A063419 (bisection). Row 6 of A077042.

sum((-1)^(k)*binomial(n,k)*binomial(n+floor(5*n/2)-6*k-1, n-1), k=0..floor(5*n/12)); # Warut Roonguthai, May 21 2006

Flatten[{1,Table[Coefficient[Expand[Sum[x^j,{j,0,5}]^n],x^Floor[5*n/2]],{n,1,20}]}] (* Vaclav Kotesovec, Aug 09 2013 *)

a(n) ~ 6^n * sqrt(6/(35*Pi*n)). - Vaclav Kotesovec, Aug 09 2013

A025015 Central decanomial coefficients: largest coefficient of (1 + x + ... + x^9)^n.

Original entry on oeis.org

1, 1, 10, 75, 670, 6000, 55252, 512365, 4816030, 45433800, 432457640, 4123838279, 39581170420, 380242296850, 3671331273480, 35460394945125, 343900019857310, 3335361909606710, 32458256583753952, 315825118347405835

Offset: 0

David W. Wilson

Number of integers in [0, 10^n-1] whose sums of digits are equal to the most common value, which is 9*n/2 for even n and (9*n +/- 1)/2 for odd n > 1. E.g., the most common value of sums of digits of numbers from 0 to 9999 is 9*4/2 = 18, so there are a(4)=670 numbers in this range whose sums of digits are 18. - Warut Roonguthai, Jun 08 2006
Generally, largest coefficient of (1 + x + ... + x^k)^n is asymptotic to (k+1)^n * sqrt(6/(k*(k+2)*Pi*n)). - Vaclav Kotesovec, Aug 09 2013
a(n) is the largest coefficients of the n-th row of A213651. - Miquel Cerda, Jul 19 2017

T. D. Noe, Table of n, a(n) for n=0..200
Vaclav Kotesovec, Recurrence
Index entries for sequences of k-nomial coefficients

Row 10 of A077042.

Flatten[{1,Table[Coefficient[Expand[Sum[x^j,{j,0,9}]^n],x^Floor[9*n/2]],{n,1,20}]}] (* Vaclav Kotesovec, Aug 09 2013 *)

a(n) = Sum_{k=0..floor(9*n/20)}(-1)^(k)*binomial(n, k)*binomial(n+floor(9*n/2)-10*k-1, n-1). - Warut Roonguthai, Jun 08 2006

a(n) ~ 10^n * sqrt(2/(33*Pi*n)). - Vaclav Kotesovec, Aug 09 2013

A071816 Number of ordered solutions to x+y+z = u+v+w, 0 <= x, y, z, u, v, w < n.

Original entry on oeis.org

1, 20, 141, 580, 1751, 4332, 9331, 18152, 32661, 55252, 88913, 137292, 204763, 296492, 418503, 577744, 782153, 1040724, 1363573, 1762004, 2248575, 2837164, 3543035, 4382904, 5375005, 6539156, 7896825, 9471196, 11287235, 13371756

Offset: 1

Graeme McRae, Jun 07 2002

Number of 6-digit numbers in base n (with leading zeros allowed) such that the sum of the first three digits equals the sum of the last three digits.

a(n) = largest coefficient of (1+...+x^(n-1))^6. - R. H. Hardin, Jul 23 2009

			For n = 2 there are 20 ordered solutions (x,y,z,u,v,w) to x+y+z = u+v+w: (0,0,0,0,0,0), (0,0,1,0,0,1), (0,0,1,0,1,0), (0,0,1,1,0,0), (0,1,0,0,0,1), (0,1,0,0,1,0), (0,1,0,1,0,0), (0,1,1,0,1,1), (0,1,1,1,0,1), (0,1,1,1,1,0), (1,0,0,0,0,1), (1,0,0,0,1,0), (1,0,0,1,0,0), (1,0,1,0,1,1), (1,0,1,1,0,1), (1,0,1,1,1,0), (1,1,0,0,1,1), (1,1,0,1,0,1), (1,1,0,1,1,0), (1,1,1,1,1,1).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. B. Nathanson, Growth polynomials for additive quadruples and (h, k)-tuples, arXiv preprint arXiv:1305.7172 [math.NT], 2013.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A008528, A071817, A077042.

First differences are in A070302.

[n*(11*n^4+5*n^2+4)/20: n in [1..30]]; // Vincenzo Librandi, Sep 05 2011

A071816 := proc(n) n*(11*n^4+5*n^2+4)/20 ; end proc: # R. J. Mathar, Sep 04 2011

The sum of the squares of the number of different 3-digit numbers that add up to k (summed over all possible k's) - cf. A071817.
a(n) = A077042(n,6).
a(n) = n*(11*n^4+5*n^2+4)/20. Recurrence: 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). G.f.: x*(1+14*x+36*x^2+14*x^3+x^4)/(1-x)^6. - Vladeta Jovovic, Jun 09 2002

New definition from Vladeta Jovovic, Jun 09 2002
Comment revised by Franklin T. Adams-Watters, Jul 27 2009
Edited by N. J. A. Sloane, Jul 28 2009

A077045 Doubly restricted composition numbers: number of compositions of 1+2+3+...+n = n(n+1)/2 into exactly n positive integers each no more than n.

Original entry on oeis.org

1, 1, 2, 7, 44, 381, 4332, 60691, 1012664, 19610233, 432457640, 10701243741, 293661065788, 8851373201919, 290711372717976, 10334165623697259, 395320344293410544, 16192709833199300337, 707125993042984343136, 32795665902734099555845, 1609908874238209683872480

Offset: 0

Henry Bottomley, Oct 22 2002

			a(3) = 7 since the compositions of 1+2+3=6 into exactly 3 positive integers each no more than 3 are: 1+2+3, 1+3+2, 2+1+3, 2+2+2, 2+3+1, 3+1+2, 3+2+1.

Alois P. Heinz, Table of n, a(n) for n = 0..100
Index entries for sequences related to compositions

Cf. A077042, A077046, A077047, A077048, A362967.

b:= proc(n, h, t) option remember;
      `if`(t*h b(n*(n-1)/2, n-1, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 10 2012

f[n_] := Union[ CoefficientList[ Expand[ Sum[x^j, {j, n}]^n], x]][[-1]]; Join[{1}, Array[f, 18]] (* Robert G. Wilson v, Apr 06 2012 *)
f[n_] := Block[{ip = IntegerPartitions[n (n + 1)/2, {n}, Range@ n], k = 1, s = 0}, len = Length[ip] + 1; While[k < len, s = s + Length@ Permutations[ ip[[k]]]; k++]; s]; Array[f, 11, 0] (* CAUTION, very slow and requires a lot of resources beyond 10, Robert G. Wilson v, Apr 09 2012 *)
b[n_, h_, t_] := b[n, h, t] = If[t*h < n, 0, If[n == 0, 1, Sum[b[n-j, Min[h, n-j], t-1], {j, 0, Min[n, h]}]]]; a[n_] := b[n*(n-1)/2, n-1, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Sep 16 2013, after Alois P. Heinz *)
Table[Sum[Binomial[n, k] Binomial[n + Binomial[n, 2] - n k - 1, n - 1] (-1)^k, {k, 0, Floor[(n-1)/2] + If[n == 0, 1, 0]}], {n, 0, 100}] (* Emanuele Munarini, Jul 15 2016 *)

makelist(sum(binomial(n,k)*binomial(n+binomial(n,2)-n*k-1,n-1)*(-1)^k,k,0,floor((n-1)/2)), n, 1, 12); /* for n >= 1; Emanuele Munarini, Jul 15 2016 */

a(n) = if(n<1,n==0,sum(k=0, (n-1)\2, binomial(n,k)*binomial(n + binomial(n,2) - n*k - 1, n-1)*(-1)^k)); \\ Andrew Howroyd, Feb 27 2018

a(n) = A077042(n, n). Roughly n^(n-3/2)*sqrt(6/Pi) by the central limit theorem and something like n^n*sqrt(6/(Pi*(n^3+0.3*n^2-0.91*n+0.3))) seems to be even closer.
From Emanuele Munarini, Jul 15 2016: (Start)
a(n) = [x^binomial(n,2)](1+x+x^2+...+x^(n-1))^n.
a(n) = Sum_{k = 0..floor((n-1)/2)} binomial(n,k)*binomial(n + binomial(n,2) - n*k - 1, n-1)*(-1)^k for n >=1. (End)

cp's OEIS Frontend

A219876 Duplicate of A077042 with alternating up/down antidiagonals.

Views

Author

Keywords

A005191 Central pentanomial coefficients: largest coefficient of (1 + x + ... + x^4)^n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

Formula

A025012 Central heptanomial coefficients: largest coefficient of (1+x+...+x^6)^n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A077043 "Three-quarter squares": a(n) = n^2 - A002620(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

A025014 Central "nonomial" coefficient: largest coefficient of (1+x+...+x^8)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A005190 Central quadrinomial coefficients: largest coefficient of (1 + x + x^2 + x^3)^n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A018901 Central hexanomial coefficients: largest coefficient of (1 + x + ... + x^5)^n.

Views

Author

Keywords