A005190 -id:A005190 - cp's OEIS frontend

A295870 a(n) = binomial(3n,n)*CQC(n), where CQC(n) = A005721(n) = A005190(2n) is a central quadrinomial coefficient.

1, 12, 660, 48720, 4005540, 349260912, 31626298704, 2940502593600, 278788387440420, 26831860080682800, 2613367831568654160, 257012469788428710720, 25479526081439438845200, 2543092744417831625342400, 255292245777771431285140800, 25755871314484468746363582720

Offset: 0

Bradley Klee, Feb 23 2018

nonn

Compare with EllipticK A002894 and the Ramanujan period-energy functions A113424, A006480, A000897. The series expansion "T(x) = 2*Pi*Sum_{n>=0} a_n*x^n" determines the real period T of elliptic curves in the family "x=p^2+q^2-4*(q^2-p^2)*q, 0 < x < 1/108". This sequence serves as a counterexample to the naive idea that elliptic integrals will always evaluate to a hypergeometric function such as 2F1(a,b;c;x).
A300058 is the complex period-energy function, after scaling energy and time dimensions such that all a(n) are integers and a(0)=1. The Picard-Fuchs equation is "(12-288*x+9216*x^2)*T(x) + (-1+232*x-8160*x^2+82944*x^3)*T'(x) + (-x+164*x^2-6432*x^3+41472*x^4)*T''(x)".
Although the sequence is not generated by a hypergeometric function, it can be formulated in terms of Hypergeometric numbers, specifically the binomial coefficients. Then Zeilberger's algorithm outputs a second order recurrence with polynomial coefficients.
The contour plot is nice to look at, with reflection symmetry, three critical points, and two separatrices dividing the phase plane into eight distinct regions.
Hyperbolic Critical points are located at (q,p) locations (1/6,0) and (-1/4,sqrt(5)/4) and (-1/4,-sqrt(5)/4). Is it possible to use chord-and-tangent addition rules to produce an exponentially-convergent Diophantine approximation to sqrt(5) that moves along the upper separatrix x=1/8?
Does there exist a period-preserving transformation that takes any one of the curves with 0 < x < 1/108 into a particular Weierstrass curve from the L-function and Modular Forms Database?

D. Husemöller, Elliptic Curves, 2nd ed., New York: Springer, 2004.
J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., New York: Springer, 2009.

J. Cremona, Elliptic Curves over Q, LMFDB 2017.
B. Klee, The Virtues of X_{n+1} = (4+3*X_{n})/(3+2*X_{n}), seqfans mailing list, 2017.
B. Klee, Geometric G.F. for Ramanujan Periods, seqfans mailing list, 2017.
Brad Klee, Deriving Hypergeometric Picard-Fuchs Equations, Wolfram Demonstrations Project (2018).
Bradley Klee, Phase Plane Geometry.
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22.
P. Paule and M. Schorn, FastZeil: the Paule/Schorn implementation of Gosper's and Zeilberger's algorithm, RISC 2017; Local copy, pdf file only, no active links
D. Zeilberger, The Method of Creative Telescoping, Journal of Symbolic Computation, 11.3 (1991), 195-204.

Cf. A000897, A002894, A006480, A113424.
Factors: A005190, A005809, A005721.
Complex Period: A300058.

b[NN_]:=Total/@Table[((-1)^k)*Binomial[3*n,n]*Binomial[2*n,k]*Binomial[5*n-4*k-1,3*n-4*k],{n,0,NN},{k,0,Floor[3*n/4]}];
c1=8*(-30+201*n-319*n^2+145*n^3);c2=-8640*(n-5/3)*(n-4/3)*(n-1/5);c3=10*(n-6/5)*n^2;a[0]=1;a[1]=12;a[n0_]:=ReplaceAll[(c1/c3)*a[n0-1]+(c2/c3)*a[n0-2],{n->n0}];
({#,SameQ[a/@Range[0, 15],#]}&@b[15])[[1]]

a(n) = A005809(n)*A005721(n).
a(n) = Sum_{k=0..floor(3n/4)} ((-1)^k)*binomial(3*n,n)*binomial(2 *n, k)*binomial(5*n - 4*k - 1, 3*n - 4*k).
c1 = 8 *(-30 + 201*n - 319*n^2 + 145*n^3); c2 = -8640*(n - 5/3)*(n - 4/3)*(n - 1/5); c3 = 10*(n - 6/5)*n^2; a(0)=1; a(1)=12; a(n) = (c1/c3)*a(n-1) + (c2/c3)*a(n-2).

A087186 Q(p)/p where p runs through the primes and Q(k) is the k-th central quadrinomial coefficient (A005190).

Original entry on oeis.org

2, 4, 31, 304, 40044, 500522, 86094668, 1167752848, 225001039696, 652498288154820, 9451735761626880, 29731605314969017772, 6529673496702876605072, 97300805803445759460364, 21809340225815485957990464, 74647602337610140872493303186, 260456466719887195671579173952976

Offset: 1

Benoit Cloitre, Oct 19 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..263 (terms 1..100 from M. F. Hasler)

Cf. A005190.

f[n_] := Max[CoefficientList[Expand[Sum[x^k, {k, 0, 3}]^n], x]]; Table[f[p]/p, {p, Prime[Range[15]]}] (* Amiram Eldar, Apr 25 2025 *)

A077042 Square array read by falling antidiagonals of central polynomial coefficients: largest coefficient in expansion of (1 + x + x^2 + ... + x^(n-1))^k = ((1-x^n)/(1-x))^k, i.e., the coefficient of x^floor(k(n-1)/2) and of x^ceiling(k(n-1)/2); also number of compositions of floor(k*(n+1)/2) into exactly k positive integers each no more than n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 1, 1, 0, 1, 6, 7, 4, 1, 1, 0, 1, 10, 19, 12, 5, 1, 1, 0, 1, 20, 51, 44, 19, 6, 1, 1, 0, 1, 35, 141, 155, 85, 27, 7, 1, 1, 0, 1, 70, 393, 580, 381, 146, 37, 8, 1, 1, 0, 1, 126, 1107, 2128, 1751, 780, 231, 48, 9, 1, 1, 0, 1, 252, 3139

Offset: 0

Henry Bottomley, Oct 22 2002

From Michel Marcus, Dec 01 2012: (Start)
A pair of numbers written in base n are said to be comparable if all digits of the first number are at least as big as the corresponding digit of the second number, or vice versa. Otherwise, this pair will be defined as uncomparable. A set of pairwise uncomparable integers will be called anti-hierarchic.
T(n,k) is the size of the maximal anti-hierarchic set of integers written with k digits in base n.
For example, for base n=2 and k=4 digits:
- 0 (0000) and 15 (1111) are comparable, while 6 (0110) and 9 (1001) are uncomparable,
- the maximal antihierarchic set is {3 (0011), 5 (0101), 6 (0110), 9 (1001), 10 (1010), 12 (1100)} with 6 elements that are all pairwise uncomparable. (End)

			Rows of square array start:
  1,    0,    0,    0,    0,    0,    0, ...
  1,    1,    1,    1,    1,    1,    1, ...
  1,    1,    2,    3,    6,   10,   20, ...
  1,    1,    3,    7,   19,   51,  141, ...
  1,    1,    4,   12,   44,  155,  580, ...
  1,    1,    5,   19,   85,  381, 1751, ...
  ...
Read by antidiagonals:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 2, 1, 1;
  0, 1, 3, 3, 1, 1;
  0, 1, 6, 7, 4, 1, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Denis Bouyssou, Thierry Marchant, Marc Pirlot, The size of the largest antichains in products of linear orders, arXiv:1903.07569 [math.CO], 2019.
J. W. Sander, On maximal antihierarchic sets of integers, Discrete Mathematics, Volume 113, Issues 1-3, 5 April 1993, Pages 179-189.
Index entries for sequences related to compositions

Rows include A000007, A000012, A001405, A002426, A005190, A005191, A018901, A025012, A025013, A025014, A025015, A201549, A225779, A201550. Columns include A000012, A000012, A001477, A077043, A005900, A077044, A071816, A133458.
Central diagonal is A077045, with A077046 and A077047 either side. Cf. A067059, A270918, A201552.
Cf. A273975.

t[n_, k_] := Max[ CoefficientList[ Series[ ((1-x^n) / (1-x))^k, {x, 0, k*(n-1)}], x]]; t[0, 0] = 1; t[0, ] = 0; Flatten[ Table[ t[n-k, k], {n, 0, 12}, {k, n, 0, -1}]] (* _Jean-François Alcover, Nov 04 2011 *)

T(n,k)=if(n<1 || k<1,k==0,vecmax(Vec(((1-x^n)/(1-x))^k)))

By the central limit theorem, T(n,k) is roughly n^(k-1)*sqrt(6/(Pi*k)).

T(n,k) = Sum{j=0,h/n} (-1)^j*binomial(k,j)*binomial(k-1+h-n*j,k-1) with h=floor(k*(n-1)/2), k>0. - Michel Marcus, Dec 01 2012

A005721 Central quadrinomial coefficients.

Original entry on oeis.org

1, 4, 44, 580, 8092, 116304, 1703636, 25288120, 379061020, 5724954544, 86981744944, 1327977811076, 20356299454276, 313095240079600, 4829571309488760, 74683398325804080, 1157402982351003420, 17971185794898859248

Offset: 0

N. J. A. Sloane

Sum of squares of entries in the n-th row of triangle of quadrinomial coefficients A008287 (Pascal triangle of order 4). - Adi Dani, Jul 03 2011

Central coefficients in triangle A008287 ((1 + x + x^2 + x^3)^n), see link. - Zagros Lalo, Sep 25 2018

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 601, 602.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 0..597 (first 101 terms from T. D. Noe)
Hacène Belbachir and Yassine Otmani, Quadrinomial-Like Versions for Wolstenholme, Morley and Glaisher Congruences, Integers (2023) Vol. 23.
Adi Dani, Restricted compositions of natural numbers-section: Generalized Pascal triangle
R. K. Guy, editor, Western Number Theory Problems, 1985-12-21 & 23, Typescript, Jul 13 1986, Dept. of Math. and Stat., Univ. Calgary, 11 pages. Annotated scan of pages 1, 3, 7, 9, with permission. See Problem 85:02.
R. K. Guy, Letter to N. J. A. Sloane, 1987
Zagros Lalo, Formula for the Central coefficients in triangle A008287 ((1 + x + x^2 + x^3)^n).

Cf. A002426, A005190, A008287.

List([0..20],n->Sum([0..Int(3*n/4)],k->(-1)^k*Binomial(2*n,k)*Binomial(5*n-4*k-1,3*n-4*k))); # Muniru A Asiru, Sep 26 2018

[(&+[(-1)^k*Binomial(2*n,k)*Binomial(5*n-4*k-1,3*n-4*k): k in [0..Floor(3*n/4)]]): n in [0..30]]; // G. C. Greubel, Oct 06 2018

F := (t^2-1)*(6*t+t^2+1)^(1/2)/(3*t^3+13*t^2+t-1); G := t/((t+1)^2*(6*t+t^2+1));
Ginv := RootOf(numer(G-x),t); series(eval(F,t=Ginv),x=0,20);
seq(coeff((1+x+x^2+x^3)^(2*n),x,3*n),n=0..50); # Robert Israel, Nov 01 2015

Table[Sum[(-1)^k*Binomial[2*n,k]*Binomial[5*n-4*k-1,3*n-4*k],{k,0,3*n/4}],{n,0,25}] (* Adi Dani, Jul 03 2011 *)
RecurrenceTable[{128*(n-1)*(2*n-3)*(2*n-1)*(5*n-1)*a[n-2] -8*(2*n-1)*(145*n^3-319*n^2+201*n-30)*a[n-1]+3*n*(3*n-2)*(3*n-1)*(5*n-6)*a[n]==0,
a[0]==1,a[1]==4},a,{n,0,5000}] (* Bradley Klee, Jun 25 2018 *)
a[n_] := a[n] = Sum[(2*n)!/((j - n)!*(3*n + i - 2*j)!*(j - 2*i)!*i!), {i, 0, n}, {j, n, 2*n}]; Table[a[n], {n, 0, 20}] (* Zagros Lalo, Sep 25 2018 *)

a(n)={local(v=Vec((1+x+x^2+x^3)^n));sum(k=1,#v, v[k]^2);}

a(n)=sum(k=0,3*n/4, (-1)^k*binomial(2*n,k)*binomial(5*n-4*k-1,3*n-4*k));

vector(30, n, n--; polcoeff((1+x+x^2+x^3)^(2*n), (6*n)>>1)) \\ Altug Alkan, Nov 01 2015

a(n) = A005190(2*n) = A008287(2*n, 3*n).
G.f.:  Let Z(x) be a solution of (-1+16*x)*(32*x-27)^2*Z^6+9*(-9+64*x)*(32*x-27)*Z^4+81*(80*x-27)*Z^2+729 = 0, with Z(0)=1. Compute a Puiseux series for Z(x) at x=0, then Z(x) in C[[x^(1/3)]].  Remove all non-integer powers of x.  The result is the generating function for A005721.  - Mark van Hoeij, Oct 29 2011
G.f.: F(G^(-1)(x)) where F(t) = (t^2-1)*(6*t+t^2+1)^(1/2)/(3*t^3+13*t^2+t-1) and G(t) = t/((t+1)^2*(6*t+t^2+1)). - Mark van Hoeij, Oct 30 2011
From Bradley Klee, Jun 25 2018: (Start)
128*(n-1)*(2*n-3)*(2*n-1)*(5*n-1)*a(n-2) - 8*(2*n-1)*(145*n^3-319*n^2+201*n-30)*a(n-1) + 3*n*(3*n-2)*(3*n-1)*(5*n-6)*a(n) = 0.
G.f. G(x) satisfies a Picard-Fuchs type differential equation, 0 = Sum_{m=0..5, n=0..3} M_{m,n} x^m*(d^n/dx^n G(x)), with integer matrix:
M={{  24,     -6,      0,     0},
   {-768,   1488,    -54,     0},
   {6144, -16128,   2520,   -27},
   {   0,  55296, -29568,   896},
   {   0,      0,  49152, -7936},
   {   0,      0,      0,  8192}}(End)
a(n) = sum_{k=0..floor(3n/4)} (-1)^k binomial(2n,k) * binomial(5n-4k-1,3n-4k). - Muniru A Asiru, Sep 26 2018
a(n) = Sum_{i=0..n} Sum_{j=n..2n}(f); f= ( (2*n)!/((j - n)!*(3*n + i - 2*j)!*(j - 2*i)!*i!) ); f=0 for (3*n + i - 2*j)<0 or (j - 2*i)<0. See also formula in Links section. - Zagros Lalo, Sep 27 2018

A273975 Three-dimensional array written by antidiagonals in k,n: T(k,n,h) with k >= 1, n >= 0, 0 <= h <= n*(k-1) is the coefficient of x^h in the polynomial (1 + x + ... + x^(k-1))^n = ((x^k-1)/(x-1))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 6, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10

Offset: 1

Andrey Zabolotskiy, Nov 10 2016

Equivalently, T(k,n,h) is the number of ordered sets of n nonnegative integers < k with the sum equal to h.
From Juan Pablo Herrera P., Nov 21 2016: (Start)
T(k,n,h) is the number of possible ways of randomly selecting h cards from k-1 sets, each with n different playing cards. It is also the number of lattice paths from (0,0) to (n,h) using steps (1,0), (1,1), (1,2), ..., (1,k-1).
Shallow diagonal sums of each triangle with fixed k give the k-bonacci numbers. (End)
T(k,n,h) is the number of n-dimensional grid points of a k X k X ... X k grid, which are lying in the (n-1)-dimensional hyperplane which is at an L1 distance of h from one of the grid's corners, and normal to the corresponding main diagonal of the grid. - Eitan Y. Levine, Apr 23 2023

			For first few k and for first few n, the rows with h = 0..n*(k-1) are given:
k=1:  1;  1;  1;  1;  1; ...
k=2:  1;  1, 1;  1, 2, 1;  1, 3, 3, 1;  1, 4, 6, 4, 1; ...
k=3:  1;  1, 1, 1;  1, 2, 3, 2, 1;  1, 3, 6, 7, 6, 3, 1; ...
k=4:  1;  1, 1, 1, 1;  1, 2, 3, 4, 3, 2, 1; ...
For example, (1 + x + x^2)^3 = 1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6, hence T(3,3,2) = T(3,3,4) = 6.
From _Eitan Y. Levine_, Apr 23 2023: (Start)
Example for the repeated cumulative sum formula, for (k,n)=(3,3) (each line is the cumulative sum of the previous line, and the first line is the padded, alternating 3rd row from Pascal's triangle):
  1  0  0 -3  0  0  3  0  0 -1
  1  1  1 -2 -2 -2  1  1  1
  1  2  3  1 -1 -3 -2 -1
  1  3  6  7  6  3  1
which is T(3,3,h). (End)

Andrey Zabolotskiy, slices with k+n = 1..20, flattened
Florentin Smarandache, K-Nomial Coefficients, arXiv:math/0612062 [math.GM], 2006 (originally published in French in: F. Smarandache, Généralisations et Généralités, Ed. Nouvelle, 1984, pp. 24-26).

k-nomial arrays for fixed k=1..10: A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.
Arrays for fixed n=0..6: A000012, A000012, A004737, A109439, A277949, A277950, A277951.
Central n-nomial coefficients for n=1..9, i.e., sequences with h=floor(n*(k-1)/2) and fixed n: A000012, A000984 (A001405), A002426, A005721 (A005190), A005191, A063419 (A018901), A025012, (A025013), A025014, A174061 (A025015), A201549, (A225779), A201550. Arrays: A201552, A077042, see also cfs. therein.
Triangle n=k-1: A181567. Triangle n=k: A163181.

a = Table[CoefficientList[Sum[x^(h-1),{h,k}]^n,x],{k,10},{n,0,9}];
Flatten@Table[a[[s-n,n+1]],{s,10},{n,0,s-1}]
(* alternate program *)
row[k_, n_] := Nest[Accumulate,Upsample[Table[((-1)^j)*Binomial[n,j],{j,0,n}],k],n][[;;n*(k-1)+1]] (* Eitan Y. Levine, Apr 23 2023 *)

T(k,n,h) = Sum_{i = 0..floor(h/k)} (-1)^i*binomial(n,i)*binomial(n+h-1-k*i,n-1). [Corrected by Eitan Y. Levine, Apr 23 2023]
From Eitan Y. Levine, Apr 23 2023: (Start)
(T(k,n,h))_{h=0..n*(k-1)} = f(f(...f(g(P))...)), where:
(x_i)_{i=0..m} denotes a tuple (in particular, the LHS contains the values for 0 <= h <= n*(k-1)),
f repeats n times,
f((x_i){i=0..m}) = (Sum{j=0..i} x_j)_{i=0..m} is the cumulative sum function,
g((x_i){i=0..m}) = (x(i/k) if k|i, otherwise 0)_{i=0..m*k} is adding k-1 zeros between adjacent elements,
and P=((-1)^i*binomial(n,i))_{i=0..n} is the n-th row of Pascal's triangle, with alternating signs. (End)
From Eitan Y. Levine, Jul 27 2023: (Start)
Recurrence relations, the first follows from the sequence's defining polynomial as mentioned in the Smarandache link:
T(k,n+1,h) = Sum_{i = 0..s-1} T(k,n,h-i)
T(k+1,n,h) = Sum_{i = 0..n} binomial(n,i)*T(k,n-i,h-i*k) (End)

A177882 Trisection of A001317.

Original entry on oeis.org

1, 15, 85, 771, 4369, 65535, 327685, 3342387, 16843009, 252645135, 1431655765, 12884901891, 73014444049, 1095216660735, 5519032976645, 56294136361779, 281479271743489, 4222189076152335, 23925738098196565

Offset: 0

Vladimir Shevelev, Dec 14 2010

nonn

For n>=1, all terms are in A001969.

Or rows of triangle A008287 mod 2 converted to decimal.

Gheorghe Coserea, Table of n, a(n) for n = 0..200

Cf. A001317, A001969, A005190, A007318, A008287.

f[n_] := BitXor[n, BitShiftLeft[n, 1]]; Table[Nest[f, 1, x], {x, 0, 54, 3}]

a(n) = subst(lift(Pol(Mod([1, 1, 1, 1], 2), 'x)^n), 'x, 2);
vector(19, n, a(n-1))  \\ Gheorghe Coserea, Jun 12 2016

def A177882(n): return sum((bool(~(3*n)&3*n-k)^1)<Chai Wah Wu, May 02 2023

a(n) = A001317(3*n).

Definition rewritten by N. J. A. Sloane, Jan 01 2011

A270918 Largest coefficient of (1+x+...+x^n)^(2*n).

Original entry on oeis.org

1, 2, 19, 580, 38165, 4395456, 786588243, 202384723528, 70886845397481, 32458256583753952, 18832730699014127291, 13507852690353224821652, 11738630472138500287398379, 12155701820213424461220851360, 14790850878997102285050287114419

Offset: 0

Vaclav Kotesovec, Mar 26 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..210

Cf. A077047.

Cf. A002426, A005190, A005191, A018901, A025012, A025013, A025014, A025015.

Table[Max[CoefficientList[Expand[Sum[x^k, {k, 0, n}]^(2n)], x]], {n, 0, 20}]

a(n) = vecmax(Vec((sum(k=0,n,x^k))^(2*n))); \\ Michel Marcus, Apr 01 2016

a(n) ~ exp(2) * sqrt(3/Pi) * n^(2*n - 3/2).

Typo in formula corrected by Vaclav Kotesovec, Dec 10 2021

A005723 Quadrinomial coefficients.

Original entry on oeis.org

1, 12, 155, 2128, 30276, 440484, 6506786, 97181760, 1463609356, 22187304112, 338118529539, 5175023913008, 79492847013100, 1224838471521240, 18922450356489780, 293003808610433280, 4546150487318508156, 70662280030419277200, 1100069396653853657564

Offset: 1

N. J. A. Sloane

nonn

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).

R. K. Guy, Letter to N. J. A. Sloane, 1987

Bisection of A005190.

cp's OEIS Frontend

A295870 a(n) = binomial(3n,n)*CQC(n), where CQC(n) = A005721(n) = A005190(2n) is a central quadrinomial coefficient.

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A087186 Q(p)/p where p runs through the primes and Q(k) is the k-th central quadrinomial coefficient (A005190).

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A005721 Central quadrinomial coefficients.

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A273975 Three-dimensional array written by antidiagonals in k,n: T(k,n,h) with k >= 1, n >= 0, 0 <= h <= n*(k-1) is the coefficient of x^h in the polynomial (1 + x + ... + x^(k-1))^n = ((x^k-1)/(x-1))^n.

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A177882 Trisection of A001317.

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A270918 Largest coefficient of (1+x+...+x^n)^(2*n).

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