A005712 -id:A005712 - cp's OEIS frontend

A002426 Central trinomial coefficients: largest coefficient of (1 + x + x^2)^n.

1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, 212941, 616227, 1787607, 5196627, 15134931, 44152809, 128996853, 377379369, 1105350729, 3241135527, 9513228123, 27948336381, 82176836301, 241813226151, 712070156203, 2098240353907, 6186675630819

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of ordered trees with n + 1 edges, having root of odd degree and nonroot nodes of outdegree at most 2. - Emeric Deutsch, Aug 02 2002
Number of paths of length n with steps U = (1,1), D = (1,-1) and H = (1,0), running from (0,0) to (n,0) (i.e., grand Motzkin paths of length n). For example, a(3) = 7 because we have HHH, HUD, HDU, UDH, DUH, UHD and DHU. - Emeric Deutsch, May 31 2003
Number of lattice paths from (0,0) to (n,n) using steps (2,0), (0,2), (1,1). It appears that 1/sqrt((1 - x)^2 - 4*x^s) is the g.f. for lattice paths from (0,0) to (n,n) using steps (s,0), (0,s), (1,1). - Joerg Arndt, Jul 01 2011
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2). - Joerg Arndt, Jul 05 2011
Binomial transform of A000984, with interpolated zeros. - Paul Barry, Jul 01 2003
Number of leaves in all 0-1-2 trees with n edges, n > 0. (A 0-1-2 tree is an ordered tree in which every vertex has at most two children.) - Emeric Deutsch, Nov 30 2003
a(n) is the number of UDU-free paths of n + 1 upsteps (U) and n downsteps (D) that start U. For example, a(2) = 3 counts UUUDD, UUDDU, UDDUU. - David Callan, Aug 18 2004
Diagonal sums of triangle A063007. - Paul Barry, Aug 31 2004
Number of ordered ballots from n voters that result in an equal number of votes for candidates A and B in a three candidate election. Ties are counted even when candidates A and B lose the election. For example, a(3) = 7 because ballots of the form (voter-1 choice, voter-2 choice, voter-3 choice) that result in equal votes for candidates A and B are the following: (A,B,C), (A,C,B), (B,A,C), (B,C,A), (C,A,B), (C,B,A) and (C,C,C). - Dennis P. Walsh, Oct 08 2004
a(n) is the number of weakly increasing sequences (a_1,a_2,...,a_n) with each a_i in [n]={1,2,...,n} and no element of [n] occurring more than twice. For n = 3, the sequences are 112, 113, 122, 123, 133, 223, 233. - David Callan, Oct 24 2004
Note that n divides a(n+1) - a(n). In fact, (a(n+1) - a(n))/n = A007971(n+1). - T. D. Noe, Mar 16 2005
Row sums of triangle A105868. - Paul Barry, Apr 23 2005
Number of paths of length n with steps U = (1,1), D = (1,-1) and H = (1,0), starting at (0,0), staying weakly above the x-axis (i.e., left factors of Motzkin paths) and having no H steps on the x-axis. Example: a(3) = 7 because we have UDU, UHD, UHH, UHU, UUD, UUH and UUU. - Emeric Deutsch, Oct 07 2007
Equals right border of triangle A152227; starting with offset 1, the row sums of triangle A152227. - Gary W. Adamson, Nov 29 2008
Starting with offset 1 = iterates of M * [1,1,1,...] where M = a tridiagonal matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. - Gary W. Adamson, Jan 07 2009
Hankel transform is 2^n. - Paul Barry, Aug 05 2009
a(n) is prime for n = 2, 3 and 4, with no others for n <= 10^5 (E. W. Weisstein, Mar 14 2005). It has apparently not been proved that no [other] prime central trinomials exist. - Jonathan Vos Post, Mar 19 2010
a(n) is not divisible by 3 for n whose base-3 representation contains no 2 (A005836).
a(n) = number of (n-1)-lettered words in the alphabet {1,2,3} with as many occurrences of the substring (consecutive subword) [1,2] as those of [2,1]. See the papers by Ekhad-Zeilberger and Zeilberger. - N. J. A. Sloane, Jul 05 2012
a(n) = coefficient of x^n in (1 + x + x^2)^n. - L. Edson Jeffery, Mar 23 2013
a(n) is the number of ordered pairs (A,B) of subsets of {1,2,...,n} such that (i.) A and B are disjoint and (ii.) A and B contain the same number of elements. For example, a(2) = 3 because we have: ({},{}) ; ({1},{2}) ; ({2},{1}). - Geoffrey Critzer, Sep 04 2013
Also central terms of A082601. - Reinhard Zumkeller, Apr 13 2014
a(n) is the number of n-tuples with entries 0, 1, or 2 and with the sum of entries equal to n. For n=3, the seven 3-tuples are (1,1,1), (0,1,2), (0,2,1), (1,0,2), (1,2,0), (2,0,1), and (2,1,0). - Dennis P. Walsh, May 08 2015
The series 2*a(n) + 3*a(n+1) + a(n+2) = 2*A245455(n+3) has Hankel transform of L(2n+1)*2^n, offset n = 1, L being a Lucas number, see A002878 (empirical observation). - Tony Foster III, Sep 05 2016
The series (2*a(n) + 3*a(n+1) + a(n+2))/2 = A245455(n+3) has Hankel transform of L(2n+1), offset n=1, L being a Lucas number, see A002878 (empirical observation). - Tony Foster III, Sep 05 2016
Conjecture: An integer n > 3 is prime if and only if a(n) == 1 (mod n^2). We have verified this for n up to 8*10^5, and proved that a(p) == 1 (mod p^2) for any prime p > 3 (cf. A277640). - Zhi-Wei Sun, Nov 30 2016
This is the analog for Coxeter type B of Motzkin numbers (A001006) for Coxeter type A. - F. Chapoton, Jul 19 2017
a(n) is also the number of solutions to the equation x(1) + x(2) + ... + x(n) = 0, where x(1), ..., x(n) are in the set {-1,0,1}. Indeed, the terms in (1 + x + x^2)^n that produce x^n are of the form x^i(1)*x^i(2)*...*x^i(n) where i(1), i(2), ..., i(n) are in {0,1,2} and i(1) + i(2) + ... + i(n) = n. By setting j(t) = i(t) - 1 we obtain that j(1), ..., j(n) satisfy j(1) + ... + j(n) =0 and j(t) in {-1,0,1} for all t = 1..n. - Lucien Haddad, Mar 10 2018
If n is a prime greater than 3 then a(n)-1 is divisible by n^2. - Ira M. Gessel, Aug 08 2021
Let f(m) = ceiling((q+log(q))/log(9)), where q = -log(log(27)/(2*m^2*Pi)) then f(a(n)) = n, for n > 0. - Miko Labalan, Oct 07 2024
Diagonal of the rational function 1 / (1 - x^2 - y^2 - x*y). - Ilya Gutkovskiy, Apr 23 2025

			For n = 2, (x^2 + x + 1)^2 = x^4 + 2*x^3 + 3*x^2 + 2*x + 1, so a(2) = 3. - _Michael B. Porter_, Sep 06 2016

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Index entries for sequences of k-nomial coefficients
Index entries for "core" sequences

INVERT transform is A007971. Partial sums are A097893. Squares are A168597.
Main column of A027907. Column k=2 of A305161. Column k=0 of A328347. Column 1 of A201552(?).
Cf. A001006, A002878, A005043, A005717, A082758 (bisection), A273055 (bisection), A102445, A113302, A113303, A113304, A113305 (divisibility of central trinomial coefficients), A152227, A277640.

a002426 n = a027907 n n  -- Reinhard Zumkeller, Jan 22 2013

P:=PolynomialRing(Integers()); [Max(Coefficients((1+x+x^2)^n)): n in [0..26]]; // Bruno Berselli, Jul 05 2011

A002426 := proc(n) local k;
    sum(binomial(n, k)*binomial(n-k, k), k=0..floor(n/2));
end proc: # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
# Alternatively:
a := n -> simplify(GegenbauerC(n,-n,-1/2)):
seq(a(n), n=0..29); # Peter Luschny, May 07 2016

Table[ CoefficientList[ Series[(1 + x + x^2)^n, {x, 0, n}], x][[ -1]], {n, 0, 27}] (* Robert G. Wilson v *)
a=b=1; Join[{a,b}, Table[c=((2n-1)b + 3(n-1)a)/n; a=b; b=c; c, {n,2,100}]]; Table[Sqrt[-3]^n LegendreP[n,1/Sqrt[-3]],{n,0,26}] (* Wouter Meeussen, Feb 16 2013 *)
a[ n_] := If[ n < 0, 0, 3^n Hypergeometric2F1[ 1/2, -n, 1, 4/3]]; (* Michael Somos, Jul 08 2014 *)
Table[4^n *JacobiP[n,-n-1/2,-n-1/2,-1/2], {n,0,29}] (* Peter Luschny, May 13 2016 *)
a[n_] := a[n] = Sum[n!/((n - 2*i)!*(i!)^2), {i, 0, n/2}]; Table[a[n], {n, 0, 29}] (* Shara Lalo and Zagros Lalo, Oct 03 2018 *)

trinomial(n,k):=coeff(expand((1+x+x^2)^n),x,k);
makelist(trinomial(n,n),n,0,12); /* Emanuele Munarini, Mar 15 2011 */

makelist(ultraspherical(n,-n,-1/2),n,0,12); /* Emanuele Munarini, Dec 20 2016 */

{a(n) = if( n<0, 0, polcoeff( (1 + x + x^2)^n, n))};

/* as lattice paths: same as in A092566 but use */
steps=[[2, 0], [0, 2], [1, 1]];
/* Joerg Arndt, Jul 01 2011 */

a(n)=polcoeff(sum(m=0, n, (2*m)!/m!^2 * x^(2*m) / (1-x+x*O(x^n))^(2*m+1)), n) \\ Paul D. Hanna, Sep 21 2013

from math import comb
def A002426(n): return sum(comb(n,k)*comb(k,n-k) for k in range(n+1)) # Chai Wah Wu, Nov 15 2022

A002426 = lambda n: hypergeometric([-n/2, (1-n)/2], [1], 4)
[simplify(A002426(n)) for n in (0..29)]
# Peter Luschny, Sep 17 2014

def A():
    a, b, n = 1, 1, 1
    yield a
    while True:
        yield b
        n += 1
        a, b = b, ((3 * (n - 1)) * a + (2 * n - 1) * b) // n
A002426 = A()
print([next(A002426) for  in range(30)])  # _Peter Luschny, May 16 2016

G.f.: 1/sqrt(1 - 2*x - 3*x^2).
E.g.f.: exp(x)*I_0(2x), where I_0 is a Bessel function. - Michael Somos, Sep 09 2002
a(n) = 2*A027914(n) - 3^n. - Benoit Cloitre, Sep 28 2002
a(n) is asymptotic to d*3^n/sqrt(n) with d around 0.5.. - Benoit Cloitre, Nov 02 2002, d = sqrt(3/Pi)/2 = 0.4886025119... - Alec Mihailovs (alec(AT)mihailovs.com), Feb 24 2005
D-finite with recurrence: a(n) = ((2*n - 1)*a(n-1) + 3*(n - 1)*a(n-2))/n; a(0) = a(1) = 1; see paper by Barcucci, Pinzani and Sprugnoli.
Inverse binomial transform of A000984. - Vladeta Jovovic, Apr 28 2003
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k, k/2)*(1 + (-1)^k)/2; a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*binomial(2*k, k). - Paul Barry, Jul 01 2003
a(n) = Sum_{k>=0} binomial(n, 2*k)*binomial(2*k, k). - Philippe Deléham, Dec 31 2003
a(n) = Sum_{i+j=n, 0<=j<=i<=n} binomial(n, i)*binomial(i, j). - Benoit Cloitre, Jun 06 2004
a(n) = 3*a(n-1) - 2*A005043(n). - Joost Vermeij (joost_vermeij(AT)hotmail.com), Feb 10 2005
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k, n-k). - Paul Barry, Apr 23 2005
a(n) = (-1/4)^n*Sum_{k=0..n} binomial(2*k, k)*binomial(2*n-2*k, n-k)*(-3)^k. - Philippe Deléham, Aug 17 2005
a(n) = A111808(n,n). - Reinhard Zumkeller, Aug 17 2005
a(n) = Sum_{k=0..n} (((1 + (-1)^k)/2)*Sum_{i=0..floor((n-k)/2)} binomial(n, i)*binomial(n-i, i+k)*((k + 1)/(i + k + 1))). - Paul Barry, Sep 23 2005
a(n) = 3^n*Sum_{j=0..n} (-1/3)^j*C(n, j)*C(2*j, j); follows from (a) in A027907. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006
a(n) = (1/2)^n*Sum_{j=0..n} 3^j*binomial(n, j)*binomial(2*n-2*j, n) = (3/2)^n*Sum_{j=0..n} (1/3)^j*binomial(n, j)*binomial(2*j, n); follows from (c) in A027907. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006
a(n) = (1/Pi)*Integral_{x=-1..3} x^n/sqrt((3 - x)*(1 + x)) is moment representation. - Paul Barry, Sep 10 2007
G.f.: 1/(1 - x - 2x^2/(1 - x - x^2/(1 - x - x^2/(1 - ... (continued fraction). - Paul Barry, Aug 05 2009
a(n) = sqrt(-1/3)*(-1)^n*hypergeometric([1/2, n+1], [1], 4/3). - Mark van Hoeij, Nov 12 2009
a(n) = (1/Pi)*Integral_{x=-1..1} (1 + 2*x)^n/sqrt(1 - x^2) = (1/Pi)*Integral_{t=0..Pi} (1 + 2*cos(t))^n. - Eli Wolfhagen, Feb 01 2011
In general, g.f.: 1/sqrt(1 - 2*a*x + x^2*(a^2 - 4*b)) = 1/(1 - a*x)*(1 - 2*x^2*b/(G(0)*(a*x - 1) + 2*x^2*b)); G(k) = 1 - a*x - x^2*b/G(k+1); for g.f.: 1/sqrt(1 - 2*x - 3*x^2) = 1/(1 - x)*(1 - 2*x^2/(G(0)*(x - 1) + 2*x^2)); G(k) = 1 - x - x^2/G(k+1), a = 1, b = 1; (continued fraction). - Sergei N. Gladkovskii, Dec 08 2011
a(n) = Sum_{k=0..floor(n/3)} (-1)^k*binomial(2*n-3*k-1, n-3*k)*binomial(n, k). - Gopinath A. R., Feb 10 2012
G.f.: A(x) = x*B'(x)/B(x) where B(x) satisfies B(x) = x*(1 + B(x) + B(x)^2). - Vladimir Kruchinin, Feb 03 2013 (B(x) = x*A001006(x) - Michael Somos, Jul 08 2014)
G.f.: G(0), where G(k) = 1 + x*(2 + 3*x)*(4*k + 1)/(4*k + 2 - x*(2 + 3*x)*(4*k + 2)*(4*k + 3)/(x*(2 + 3*x)*(4*k + 3) + 4*(k + 1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013
E.g.f.: exp(x) * Sum_{k>=0} (x^k/k!)^2. - Geoffrey Critzer, Sep 04 2013
G.f.: Sum_{n>=0} (2*n)!/n!^2*(x^(2*n)/(1 - x)^(2*n+1)). - Paul D. Hanna, Sep 21 2013
0 = a(n)*(9*a(n+1) + 9*a(n+2) - 6*a(n+3)) + a(n+1)*(3*a(n+1) + 4*a(n+2) - 3*a(n+3)) + a(n+2)*(-a(n+2) + a(n+3)) for all n in Z. - Michael Somos, Jul 08 2014
a(n) = hypergeometric([-n/2, (1-n)/2], [1], 4). - Peter Luschny, Sep 17 2014
a(n) = A132885(n,0), that is, a(n) = A132885(A002620(n+1)). - Altug Alkan, Nov 29 2015
a(n) = GegenbauerC(n,-n,-1/2). - Peter Luschny, May 07 2016
a(n) = 4^n*JacobiP[n,-n-1/2,-n-1/2,-1/2]. - Peter Luschny, May 13 2016
From Alexander Burstein, Oct 03 2017: (Start)
G.f.: A(4*x) = B(-x)*B(3*x), where B(x) is the g.f. of A000984.
G.f.: A(2*x)*A(-2*x) = B(x^2)*B(9*x^2).
G.f.: A(x) = 1 + x*M'(x)/M(x), where M(x) is the g.f. of A001006. (End)
a(n) = Sum_{i=0..n/2} n!/((n - 2*i)!*(i!)^2). [Cf. Lalo and Lalo link. It is Luschny's terminating hypergeometric sum.] - Shara Lalo and Zagros Lalo, Oct 03 2018
From Peter Bala, Feb 07 2022: (Start)
a(n)^2 = Sum_{k = 0..n} (-3)^(n-k)*binomial(2*k,k)^2*binomial(n+k,n-k) and has g.f. Sum_{n >= 0} binomial(2*n,n)^2*x^n/(1 + 3*x)^(2*n+1). Compare with the g.f. for a(n) given above by Hanna.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all prime p and positive integers n and k.
Conjecture: The stronger congruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all prime p >= 5 and positive integers n and k. (End)
a(n) = A005043(n) + A005717(n) for n >= 1. - Amiram Eldar, May 17 2024
For even n, a(n) = (n-1)!!* 2^{n/2}/ (n/2)!* 2F1(-n/2,-n/2;1/2;1/4). For odd n, a(n) = n!! *2^(n/2-1/2) / (n/2-1/2)! * 2F1(1/2-n/2,1/2-n/2;3/2;1/4). - R. J. Mathar, Mar 19 2025

A027907 Triangle of trinomial coefficients T(n,k) (n >= 0, 0 <= k <= 2*n), read by rows: n-th row is obtained by expanding (1 + x + x^2)^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 10, 16, 19, 16, 10, 4, 1, 1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1, 1, 6, 21, 50, 90, 126, 141, 126, 90, 50, 21, 6, 1, 1, 7, 28, 77, 161, 266, 357, 393, 357, 266, 161, 77, 28, 7, 1, 1, 8, 36, 112, 266

Offset: 0

N. J. A. Sloane

When the rows are centered about their midpoints, each term is the sum of the three terms directly above it (assuming the undefined terms in the previous row are zeros). - N. J. A. Sloane, Dec 23 2021
T(n,k) = number of integer strings s(0),...,s(n) such that s(0)=0, s(n)=k, s(i) = s(i-1) + c, where c is 0, 1 or 2. Columns of T include A002426, A005717 and A014531.
Also number of ordered trees having n+1 leaves, all at level three and n+k+3 edges. Example: T(3,5)=3 because we have three ordered trees with 4 leaves, all at level three and 11 edges: the root r has three children; from one of these children two paths of length two are hanging (i.e., 3 possibilities) while from each of the other two children one path of length two is hanging. Diagonal sums are the tribonacci numbers; more precisely: Sum_{i=0..floor(2*n/3)} T(n-i,i) = A000073(n+2). - Emeric Deutsch, Jan 03 2004
T(n,k) = A111808(n,k) for 0 <= k <= n and T(n, 2*n-k) = A111808(n,k) for 0 <= k < n. - Reinhard Zumkeller, Aug 17 2005
The trinomial coefficients, T(n,i), are the absolute value of the coefficients of the chromatic polynomial of P_2 X P_n factored with x*(x-1)^i terms. Example: The chromatic polynomial of P_2 X P_2 is: x*(x-1) - 2*x*(x-1)^2 + x*(x-1)^3 and so T(1,0)=1, T(1,1)=2 and T(1,1) = 1. - Thomas J. Pfaff (tpfaff(AT)ithaca.edu), Oct 02 2006
T(n,k) is the number of distinct ways in which k unlabeled objects can be distributed in n labeled urns allowing at most 2 objects to fall into each urn. - N-E. Fahssi, Mar 16 2008
T(n,k) is the number of compositions of k into n parts p, each part 0 <= p <= 2. Adding 1 to each part, as a corollary, T(n,k) is the number of compositions of n+k into n parts p where 1 <= p <= 3. E.g., T(2,3)=2 since 5 = 3+2 = 2+3. - Steffen Eger, Jun 10 2011
Number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (1,2). - Joerg Arndt, Jul 05 2011
Number of lattice paths from (0,0) to (2*n-k,k) using steps (2,0), (1,1), (0,2). - Werner Schulte, Jan 25 2017
T(n,k) is number of distinct ways to sum the integers -1, 0 , and 1 n times to obtain n-k, where T(n,0) = T(n,2*n+1) = 1. - William Boyles, Apr 23 2017
T(n-1,k-1) is the number of 2-compositions of n with 0's having k parts; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 15 2020
T(n,k) is the number of ways to obtain a sum of n+k when throwing a 3-sided die n times. Follows from the "T(n,k) is the number of compositions of n+k into n parts p where 1 <= p <= 3" comment above. - Feryal Alayont, Dec 30 2024

			The triangle T(n, k) begins:
  n\k 0   1   2   3   4   5   6   7   8   9 10 11 12
  0:  1
  1:  1   1   1
  2:  1   2   3   2   1
  3:  1   3   6   7   6   3   1
  4:  1   4  10  16  19  16  10   4   1
  5:  1   5  15  30  45  51  45  30  15   5  1
  6:  1   6  21  50  90 126 141 126  90  50 21  6  1
Concatenated rows:
G.f. = 1 + (x^2+x+1)*x + (x^2+x+1)^2*x^4 + (x^2+x+1)^3*x^9 + ...
     = 1 + (x + x^2 + x^3) + (x^4 + 2*x^5 + 3*x^6 + 2*x^7 + x^8) +
  (x^9 + 3*x^10 + 6*x^11 + 7*x^12 + 6*x^13 + 3*x^14 + x^15) + ... .
As a centered triangle, this begins:
           1
        1  1  1
     1  2  3  2  1
  1  3  6  7  6  3  1

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Columns of T include A002426, A005717, A014531, A005581, A005712, etc. See also A035000, A008287.
First differences are in A025177. Pairwise sums are in A025564.
Cf. A007318 (Pascal's triangle); A000073, A001006, A123531, A055217, A027908, A027913, A027914, A027023.

a027907 n k = a027907_tabf !! n !! k
a027907_row n = a027907_tabf !! n
a027907_tabf = [1] : iterate f [1, 1, 1] where
   f row = zipWith3 (((+) .) . (+))
                    (row ++ [0, 0]) ([0] ++ row ++ [0]) ([0, 0] ++ row)
a027907_list = concat a027907_tabf
-- Reinhard Zumkeller, Jul 06 2014, Jan 22 2013, Apr 02 2011

A027907 := proc(n,k) expand((1+x+x^2)^n) ; coeftayl(%,x=0,k) ; end proc:
seq(seq(A027907(n,k),k=0..2*n),n=0..5) ; # R. J. Mathar, Jun 13 2011
T := (n,k) -> simplify(GegenbauerC(`if`(kPeter Luschny, May 08 2016

Table[CoefficientList[Series[(Sum[x^i, {i, 0, 2}])^n, {x, 0, 2 n}], x], {n, 0, 10}] // Grid (* Geoffrey Critzer, Mar 31 2010 *)
Table[Sum[Binomial[n, i]Binomial[n - i, k - 2i], {i, 0, n}], {n, 0, 10}, {k, 0, 2n}] (* Adi Dani, May 07 2011 *)
T[ n_, k_] := If[ n < 0, 0, Coefficient[ (1 + x + x^2)^n, x, k]]; (* Michael Somos, Nov 08 2016 *)
Flatten[DeleteCases[#,0]&/@CellularAutomaton[{Total[#] &, {}, 1}, {{1}, 0}, 8] ] (* Giorgos Kalogeropoulos, Nov 09 2021 *)

trinomial(n,k):=coeff(expand((1+x+x^2)^n),x,k);
create_list(trinomial(n,k),n,0,8,k,0,2*n); /* Emanuele Munarini, Mar 15 2011 */

create_list(ultraspherical(k,-n,-1/2),n,0,6,k,0,2*n); /* Emanuele Munarini, Oct 18 2016 */

{T(n, k) = if( n<0, 0, polcoeff( (1 + x + x^2)^n, k))}; /* Michael Somos, Jun 27 2003 */

G.f.: 1/(1-z*(1+w+w^2)).
T(n,k) = Sum_{r=0..floor(k/3)} (-1)^r*binomial(n, r)*binomial(k-3*r+n-1, n-1).
Recurrence: T(0,0) = 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k-0), with T(n,k) = 0 if k < 0 or k > 2*n:
T(i,0) = T(i, 2*i) = 1 for i >= 0, T(i, 1) = T(i, 2*i-1) = i for i >= 1 and for i >= 2 and 2 <= j <= i-2, T(i, j) = T(i-1, j-2) + T(i-1, j-1) + T(i-1, j).
The row sums are powers of 3 (A000244). - Gerald McGarvey, Aug 14 2004
T(n,k) = Sum_{i=0..floor(k/2)} binomial(n, 2*i+n-k) * binomial(2*i+n-k, i). - Ralf Stephan, Jan 26 2005
T(n,k) = Sum_{j=0..n} binomial(n, j) * binomial(j, k-j). - Paul Barry, May 21 2005
T(n,k) = Sum_{j=0..n} binomial(k-j, j) * binomial(n, k-j). - Paul Barry, Nov 04 2005
From Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006: (Start)
T(n,k) = Sum_{j=0..n} (-1)^j * binomial(n,j) * binomial(2*n-2*j, k-j); (G. E. Andrews (1990)) obtained by expanding ((1+x)^2 - x)^n.
T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(n-j, k-2*j); obtained by expanding ((1+x) + x^2)^n.
T(n,k) = (-1)^k*Sum_{j=0..n} (-3)^j * binomial(n,j) * binomial(2*n-2*j, k-j); obtained by expanding ((1-x)^2 + 3*x)^n.
T(n,k) = (1/2)^k * Sum_{j=0..n} 3^j * binomial(n,j) * binomial(2*n-2*j, k-2*j); obtained by expanding ((1+x/2)^2 + (3/4)*x^2)^n.
T(n,k) = (2^k/4^n) * Sum_{j=0..n} 3^j * binomial(n,j) * binomial(2*n-2*j, k); obtained by expanding ((1/2+x)^2 + 3/4)^n using T(n,k) = T(2*n-k). (End)
From Paul D. Hanna, Apr 18 2012: (Start)
Let A(x) be the g.f. of the flattened sequence, then:
G.f.: A(x) = Sum_{n>=0} x^(n^2) * (1+x+x^2)^n.
G.f.: A(x) = Sum_{n>=0} x^n*(1+x+x^2)^n * Product_{k=1..n} (1 - (1+x+x^2) * x^(4*k-3)) / (1 - (1+x+x^2)*x^(4*k-1)).
G.f.: A(x) = 1/(1 - x*(1+x+x^2)/(1 + x*(1-x^2)*(1+x+x^2)/(1 - x^5*(1+x+x^2)/(1 + x^3*(1-x^4)*(1+x+x^2)/(1 - x^9*(1+x+x^2)/(1 + x^5*(1-x^6)*(1+x+x^2)/(1 - x^13* (1+x+x^2)/(1 + x^7*(1-x^8)*(1+x+x^2)/(1 - ...))))))))), a continued fraction.
(End)
Triangle: G.f. = Sum_{n>=0} (1+x+x^2)^n * x^(n^2) * y^n. - Daniel Forgues, Mar 16 2015
From Peter Luschny, May 08 2016: (Start)
T(n+1,n)/(n+1) = A001006(n) (Motzkin) for n>=0.
T(n,k) = H(n, k) if k < n else H(n, 2*n-k) where H(n,k) = binomial(n,k)*hypergeom([(1-k)/2, -k/2], [n-k+1], 4).
T(n,k) = GegenbauerC(m, -n, -1/2) where m=k if k < n else 2*n-k. (End)
T(n,k) = (-1)^k * C(2*n,k) * hypergeom([-k, -(2*n-k)], [-n+1/2], 3/4), for all k with 0 <= k <= 2n. - Robert S. Maier, Jun 13 2023
T(n,n) = Sum_{k=0..2*n} (-1)^k*(T(n,k))^2 and T(2*n,2*n) = Sum_{k=0..2*n} (T(n,k))^2 for n >= 0. - Werner Schulte, Nov 08 2016
T(n,n) = A002426(n), central trinomial coefficients. - M. F. Hasler, Nov 02 2019
Sum_{k=0..n-1} T(n, 2*k) = (3^n-1)/2. - Tony Foster III, Oct 06 2020

A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

Original entry on oeis.org

1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90

Offset: 1

Clark Kimberling, Jun 14 2012

Principal diagonal: A002412.
Antidiagonal sums: A002415.
Row 1:  (1,2,3,...)**(1,2,3,...) = A000292.
Row 2:  (1,2,3,...)**(2,3,4,...) = A005581.
Row 3:  (1,2,3,...)**(3,4,5,...) = A006503.
Row 4:  (1,2,3,...)**(4,5,6,...) = A060488.
Row 5:  (1,2,3,...)**(5,6,7,...) = A096941.
Row 6:  (1,2,3,...)**(6,7,8,...) = A096957.
...
In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples:  let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.
...
In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.
b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
h .......... h .......... A213500 . A002412 . A002415
h .......... h^2 ........ A212891 . A213436 . A024166
h^2 ........ h .......... A213503 . A117066 . A033455
h^2 ........ h^2 ........ A213505 . A213546 . A213547
h .......... h*(h+1)/2 .. A213548 . A213549 . A051836
h*(h+1)/2 .. h .......... A213550 . A002418 . A005585
h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923
h .......... h^3 ........ A213553 . A213554 . A101089
h^3 ........ h .......... A213555 . A213556 . A213547
h^3 ........ h^3 ........ A213558 . A213559 . A213560
h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563
h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094
2^(h-1) .... h .......... A213568 . A213569 . A047520
2^(h-1) .... h^2 ........ A213573 . A213574 . A213575
h .......... Fibo(h) .... A213576 . A213577 . A213578
Fibo(h) .... h .......... A213579 . A213580 . A053808
Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988
Fibo(h+1) .. h .......... A213584 . A213585 . A213586
Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589
h^2 ........ Fibo(h) .... A213590 . A213504 . A213557
Fibo(h) .... h^2 ........ A213566 . A213567 . A213570
h .......... -1+2^h ..... A213571 . A213572 . A213581
-1+2^h ..... h .......... A213582 . A213583 . A156928
-1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749
h .......... 2*h-1 ...... A213750 . A007585 . A002417
2*h-1 ...... h .......... A213751 . A051662 . A006325
2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238
2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755
-1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758
2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764
2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767
Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770
Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776
Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881
h .......... 1+[h/2] .... A213778 . A213779 . A213780
1+[h/2] .... h .......... A213781 . A213782 . A005712
1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760
h .......... 3*h-2 ...... A213761 . A172073 . A002419
3*h-2 ...... h .......... A213771 . A213772 . A132117
3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818
h .......... 3*h-1 ...... A213819 . A213820 . A153978
3*h-1 ...... h .......... A213821 . A033431 . A176060
3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824
3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827
3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830
2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560
3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834
h .......... 4*h-3 ...... A213835 . A172078 . A051797
4*h-3 ...... h .......... A213836 . A213837 . A071238
4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840
2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843
2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846
4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324
[(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850
h .......... C(2*h-2,h-1) A213853
...
Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.

			Northwest corner (the array is read by southwest falling antidiagonals):
  1,  4, 10, 20,  35,  56,  84, ...
  2,  7, 16, 30,  50,  77, 112, ...
  3, 10, 22, 40,  65,  98, 140, ...
  4, 13, 28, 50,  80, 119, 168, ...
  5, 16, 34, 60,  95, 140, 196, ...
  6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A000027.

b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213500 *)

t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017

def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
    print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017

T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
T(n,k) = 2*T(n-1,k) - T(n-2,k).
G.f. for row n:  x*(n - (n - 1)*x)/(1 - x)^4.

A000581 a(n) = binomial coefficient C(n,8).

Original entry on oeis.org

1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758, 75582, 125970, 203490, 319770, 490314, 735471, 1081575, 1562275, 2220075, 3108105, 4292145, 5852925, 7888725, 10518300, 13884156, 18156204, 23535820, 30260340, 38608020, 48903492, 61523748, 76904685

Offset: 8

N. J. A. Sloane

Figurate numbers based on 8-dimensional regular simplex. - Jonathan Vos Post, Nov 28 2004
Just as A005712 and A000574 are described as the coefficients of x^4 and x^5 in the expansion of (1+x+x^2)^n, so should this sequence be described as the coefficients of x^3 therein. - R. K. Guy, Oct 19 2007
Product of 8 consecutive numbers divided by 8!. - Artur Jasinski, Dec 02 2007
In this sequence there are no primes. - Artur Jasinski, Dec 02 2007
a(n) = number of (n-8)-digit numbers with nondescending digits. E.g., a(9) = 9 = {1,2,3,..,9}, a(10) = 45 = {11-19, 22-29, 33-39, ..., 99} [0 is counted as a zero-digit number rather than a 1-digit number]. - Toby Gottfried, Feb 14 2012
a(n) =fallfac(n, 8)/8! = binomial(n, 8) is also the number of independent components of an antisymmetric tensor of rank 8 and dimension n >= 8 (for n = 1..7 this becomes 0). Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
Number of compositions (ordered partitions) of n+1 into exactly 9 parts. - Juergen Will, Jan 02 2016
Number of weak compositions (ordered weak partitions) of n-8 into exactly 9 parts. - Juergen Will, Jan 02 2016
Partial sums of A000580. - Art Baker, Mar 26 2019

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.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
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.
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).

T. D. Noe, Table of n, a(n) for n = 8..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 9.
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 258.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - _Juergen Will_, Jan 02 2016
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
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
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Composition.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000217, A000292, A000332, A000389, A000579, A000580, A053130, A053137, A254142, A001787, A242091.

[Binomial(n,8): n in [8..100]]; // Vincenzo Librandi, Apr 08 2011

ZL := [S, {S=Prod(B, B, B, B, B, B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n+1), n=8..40); # Zerinvary Lajos, Mar 13 2007
A000581:=-1/(z-1)**9; # Simon Plouffe in his 1992 dissertation, with offset 0
seq(binomial(n,8),n=8..40); # Zerinvary Lajos, Jun 23 2008

Table[Binomial[n,8],{n,8,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

a(n)=binomial(n,8) \\ Charles R Greathouse IV, Feb 14 2012

G.f.: x^8/(1-x)^9.
a(n) = A110555(n+1,8). - Reinhard Zumkeller, Jul 27 2005
a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)/8!. - Artur Jasinski, Dec 02 2007
Sum_{k>=8} 1/a(k) = 8/7. - Tom Edgar, Sep 10 2015
Sum_{n>=8} (-1)^n/a(n) = A001787(8)*log(2) - A242091(8)/7! = 1024*log(2) - 74432/105 = 0.9065224171... - Amiram Eldar, Dec 10 2020

More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000
Some formulas referring to other offsets rewritten by R. J. Mathar, Jul 07 2009
3 more terms from William Boyles, Aug 06 2015

A005581 a(n) = (n-1)n(n+4)/6.

Original entry on oeis.org

0, 0, 2, 7, 16, 30, 50, 77, 112, 156, 210, 275, 352, 442, 546, 665, 800, 952, 1122, 1311, 1520, 1750, 2002, 2277, 2576, 2900, 3250, 3627, 4032, 4466, 4930, 5425, 5952, 6512, 7106, 7735, 8400, 9102, 9842, 10621, 11440, 12300, 13202, 14147, 15136, 16170

Offset: 0

N. J. A. Sloane

A class of Boolean functions of n variables and rank 2.
Also, number of inscribable triangles within a (n+4)-gon sharing with them its vertices but not its sides. - Lekraj Beedassy, Nov 14 2003
a(n) = A111808(n,3) for n > 2. - Reinhard Zumkeller, Aug 17 2005
If X is an n-set and Y a fixed 2-subset of X then a(n-2) is equal to the number of (n-3)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
The sequence starting with offset 2 = binomial transform of [2, 5, 4, 1, 0, 0, 0, ...]. - Gary W. Adamson, Mar 20 2009
Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 4, a(n-4) is the number of (0,1) n X n matrices A <= P^(-1) + I + P having exactly two 1's in every row and column with perA=8. - Vladimir Shevelev, Apr 12 2010
Also arises as the number of triples of edges which can be chosen as the cut-points in the "three-opt" heuristic for a traveling salesman problem on (n+4) nodes. - James McDermott, Jul 10 2015
a(n) = risefac(n, 3)/3! - n is for n >= 1 also the number of independent components of a symmetric traceless tensor of rank 3 and dimension n. Here risefac is the rising factorial. - Wolfdieter Lang, Dec 10 2015
For n >= 2, a(n) is the number of characters in a word Q formed by concatenating all 'directed' ( left to right or vice versa), unrearranged subwords, from length 1 to (n-1), of a length (n-1) word q- allowing for the appearance of repeated subwords- and simply inserting an extra character for all subwords thus concatenated. - Christopher Hohl, May 30 2019

			In hexagon ABCDEF, the "interior" triangles are ACE and BDF, and a(6-4)=a(2)=2. - _Toby Gottfried_, Nov 12 2011
G.f. = 2*x^2 + 7*x^3 + 16*x^4 + 30*x^5 + 50*x^6 + 77*x^7 + 112*x^8 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
Joseph D. Konhauser, Dan Velleman and Stan Wagon,, Which Way Did the Bicycle Go?, MAA, 1996, p. 177.
V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, Vol. 3 (1992), pp. 15-19. - Vladimir Shevelev, Apr 12 2010
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #51 (the case k=3) (First published: San Francisco: Holden-Day, Inc., 1964).

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972. [Alternative scanned copy]
Armen G. Bagdasaryan and Ovidiu Bagdasar, On some results concerning generalized arithmetic triangles, Electronic Notes in Discrete Mathematics, Vol. 67 (2018), pp. 71-77.
Beáta Bényi, Miguel Méndez, José L. Ramírez and Tanay Wakhare, Restricted r-Stirling Numbers and their Combinatorial Applications, arXiv:1811.12897 [math.CO], 2018.
John Elias, Illustration: Triangular Staircasing
Richard K. Guy, Letter to N. J. A. Sloane, 1987.
Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart., Vol. 49, No. 3 (2011), pp. 231-243.
Milan Janjic, Two Enumerative Functions.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, Vol. 11, No. 4 (1999), pp. 127-138.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, Vol. 9, No. 6 (1999), pp. 593-605.
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Alice McLeod and William Moser, Counting cyclic binary strings, Math. Mag., Vol. 80, No. 1 (2007), pp. 29-37.
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.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Dead link]
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Cached copy, May 15 2013]
Eric Weisstein's World of Mathematics, Trinomial Coefficient.
Index entries for sequences related to Boolean functions.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000217, A000292, A005582, A005586, A111808.
Cf. A176222, A000211, A052928, A128209.
Cf. A127672 (Chebyshev C).

[(n-1)*n*(n+4)/6 : n in [0..50]]; // Wesley Ivan Hurt, Jul 10 2015

A005581 := n->(n-1)*n*(n+4)/6: seq(A005581(n), n=0..50);
a:=n->sum ((j+3)*j/2,j=0..n): seq(a(n),n=-1..49); # Zerinvary Lajos, Dec 17 2006
seq((n+3)*binomial(n,3)/n, n=1..46); # Zerinvary Lajos, Feb 28 2007
A005581:=-(-2+z)/(z-1)**4; # Simon Plouffe in his 1992 dissertation
seq(sum(binomial(n,m), m=1..3)+n^2,n=-1..44); # Zerinvary Lajos, Jun 19 2008
A005581 := n -> GegenbauerC(`if`(3A005581(n)), n=0..50); # Peter Luschny, May 10 2016

Table[(n-1)*n*(n+4)/6, {n,0,50}] (* Stefan Steinerberger, Apr 10 2006 *)
LinearRecurrence[{4,-6,4,-1},{0,0,2,7},50] (* Harvey P. Dale, Sep 22 2012 *)

A005581(n):=(n-1)*n*(n+4)/6$ makelist(A005581(n),n,0,50); /* Martin Ettl, Dec 18 2012 */

{a(n) = n * (n+4) * (n-1) / 6}; /* Michael Somos, Apr 13 2007 */

concat([0, 0], Vec((x^2)*(2-x)/(1-x)^4 + O(x^50))) \\ Altug Alkan, Dec 10 2015

[(n-1)*n*(n+4)/6 for n in range(50)] # Danny Rorabaugh, Apr 20 2015

G.f.: (x^2)*(2-x)/(1-x)^4.
a(n) = binomial(n+1, n-2) + binomial(n, n-2).
a(n) = A027907(n, 3), n >= 0 (fourth column of trinomial coefficients). - N. J. A. Sloane, May 16 2003
Convolution of {1, 2, 3, ...} with {2, 3, 4, ...}. - Jon Perry, Jun 25 2003
a(n+2) = 2*te(n) - te(n-1), e.g., a(5) = 2*te(3) - te(2) = 2*20 - 10 = 30, where te(n) are the tetrahedral numbers A000292. - Jon Perry, Jul 23 2003
a(n) is the coefficient of x^3 in the expansion of (1+x+x^2)^n. For example, a(1)=0 since (1+x+x^2)^1=1+x+x^2. - Peter C. Heinig (algorithms(AT)gmx.de), Apr 09 2007
E.g.f.: (x^2 + x^3/6) * exp(x). - Michael Somos, Apr 13 2007
a(n) = - A005586(-4-n) for all n in Z. - Michael Somos, Apr 13 2007
a(n) = C(4+n,3)-(n+4)*(n+1), since C(4+n,3) = number of all triangles in (n+4)-gon, and (n+4)*(n+1)=number of triangles with at least one of the edges included. Example: n=0,in a square, all 4 possible triangles include some of the square's edges and C(4+n,3)-(n+4)*(n+1)=4-4*1=0 = number of other triangles = a(0). - Toby Gottfried, Nov 12 2011
a(n) = 2*binomial(n,2) + binomial(n,3). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(0)=0, a(1)=0, a(2)=2, a(3)=7, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Sep 22 2012
a(n) = A000292(n-1) + A000217(n-1) for all n in Z. - Michael Somos, Jul 29 2015
a(n+2) = -A127672(6+n, n), n >= 0, with A127672 giving the coefficients of Chebyshev's C polynomials. See the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 10 2015
a(n) = GegenbauerC(N, -n, -1/2) where N = 3 if 3Peter Luschny, May 10 2016
From Amiram Eldar, Jan 09 2022: (Start)
Sum_{n>=2} 1/a(n) = 163/200.
Sum_{n>=2} (-1)^n/a(n) = 12*log(2)/5 - 253/200. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000

A005717 Construct triangle in which n-th row is obtained by expanding (1 + x + x^2)^n and take the next-to-central column.

Original entry on oeis.org

1, 2, 6, 16, 45, 126, 357, 1016, 2907, 8350, 24068, 69576, 201643, 585690, 1704510, 4969152, 14508939, 42422022, 124191258, 363985680, 1067892399, 3136046298, 9217554129, 27114249960, 79818194925, 235128465026, 693085098852, 2044217638456, 6032675068061

Offset: 1

N. J. A. Sloane

Number of ordered trees with n+1 edges, having root of even degree and nonroot nodes of outdegree at most 2. - Emeric Deutsch, Aug 02 2002
The connection to Motzkin numbers comes from the Lagrange inversion formula. - Michael Somos, Oct 10 2003
Number of horizontal steps in all Motzkin paths of length n. - Emeric Deutsch, Nov 09 2003
Number of UHD's in all Motzkin paths of length n+2 (here U=(1,1), H=(1,0) and D=(1,-1)). Example: a(2)=2 because in the nine Motzkin paths of length 4, HHHH, HHUD, HUDH, H(UHD), UDHH, UDUD, (UHD)H, UHHD and UUDD, we have altogether two UHD's (shown between parentheses). - Emeric Deutsch, Dec 26 2003
Number of ordered trees with n+1 edges, having exactly one leaf at even height. Number of Dyck path of semilength n+1, having exactly one peak at even height. Example: a(3)=6 because we have uuu(ud)ddd, u(ud)dudud, udu(ud)dud, ududu(ud)d, u(ud)uuddd and uuudd(ud)d (here u=(1,1),d=(1,-1) and the unique peak at even height is shown between parentheses). - Emeric Deutsch, Mar 10 2004
a(n) is the number of Dyck (n+1)-paths containing exactly one UDU. - David Callan, Jul 15 2004
Number of peaks in all Motzkin paths of length n+1. - Emeric Deutsch, Sep 01 2004
This is a kind of Motzkin transform of A059841 because the substitution x -> x*A001006(x) in the independent variable of the g.f. of A059841 generates 1,0,1,2,6,16,... that is 1,0 followed by this sequence here. - R. J. Mathar, Nov 08 2008
a(n) is the number of lattice paths avoiding N^(>=3) from (0,0) to (n,n). - Shanzhen Gao, Apr 20 2010
a(n+1) is the number of binary strings having n 0's and n 1's and no appearance of 000.  For example, for n = 1, there 2 strings: 01 and 10.  For n = 2, there are 6: 0011, 0101, 0110, 1001, 1010, 1100. - Toby Gottfried, Sep 12 2011
a(n) is the number of paths in the half-plane x>=0, from (0,0) to (n,1), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=3, we have the 6 paths HHU, HUH, UDU, UUD, UHH, DUU. - José Luis Ramírez Ramírez, Apr 19 2015
a(n) is the number of ways to tile a strip of length 2*n+1 with squares, dominos, and trominos, where the number of trominos is always one more than the number of squares. - Greg Dresden and Anna Kalynchuk, Jul 30 2025

			G.f. = x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 126*x^6 + 357*x^7 + ...

Louis 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).

G. C. Greubel, Table of n, a(n) for n = 1..1000 (Terms 1 to 200 computed by T. D. Noe; terms 201 to 1000 by G. C. Greubel, Jan 15 2017)
Kassie Archer and Christina Graves, A new statistic on Dyck paths for counting 3-dimensional Catalan words, arXiv:2205.09686 [math.CO], 2022.
Jean-Luc Baril, Sergey Kirgizov, José L. Ramírez, and Diego Villamizar, The Combinatorics of Motzkin Polyominoes, arXiv:2401.06228 [math.CO], 2024. See page 13.
Emeric Deutsch, Ordered trees with prescribed root degrees, node degrees, and branch lengths, Discrete Mathematics 282 (2004), 89-94.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Richard K. Guy, Letter to N. J. A. Sloane, 1987.
Stanislav Krymski and Alexander Okhotin, Longer Shortest Strings in Two-Way Finite Automata, in: Jirásková G., Pighizzini G. (eds) Descriptional Complexity of Formal Systems. DCFS 2020. Lecture Notes in Computer Science, vol 12442. Springer, Cham.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
Mark Shattuck, Subword Patterns in Smooth Words, Enum. Comb. Appl. (2024) Vol. 4, No. 4, Art. No. S2R32. See p. 6.
Chenying Wang, Piotr Miska, and István Mező, The r-derangement numbers, Discrete Mathematics 340(7) (2017), 1681-1692.
Eric Weisstein's World of Mathematics, Trinomial Coefficient.

A diagonal of A027907.

Cf. A001006, A002426, A005043, A005773, A076540 (binomial transform).

seq(add(binomial(i, k) *binomial(i-k, k+1), k=0..floor(i/2)), i=1..30); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
M:= proc(n) option remember; `if` (n<2, 1, (3*(n-1)*M(n-2) +(2*n+1) *M(n-1))/ (n+2)) end: A005717 := n -> n*M(n-1):
seq(A005717(i), i=1..27); # Peter Luschny, Sep 12 2011
a := n -> simplify(GegenbauerC(n,-n-1,-1/2)):
seq(a(n), n=0..28); # Peter Luschny, May 07 2016

Table[Coefficient[Expand[(1+x+x^2)^n], x, n-1], {n, 1, 40}]
Table[n*Hypergeometric2F1[(1 - n)/2, 1 - n/2, 2, 4], {n, 29}] (* Arkadiusz Wesolowski, Aug 13 2012 *)
Table[GegenbauerC[n,-n-1,-1/2],{n,0,100}] (* Emanuele Munarini, Oct 20 2016 *)

makelist(ultraspherical(n,-n-1,-1/2),n,0,12); /* Emanuele Munarini, Oct 20 2016 */

{a(n) = if( n<0, 0, polcoeff( (1 + x + x^2)^n, n-1))}; /* Michael Somos, Sep 09 2002 */

{a(n) = if( n<0, 0, n * polcoeff( serreverse( x / (1 + x + x^2) + x * O(x^n)), n))}; /* Michael Somos, Oct 10 2003 */

N=10^3;  x='x+'x*O('x^N);
gf = 2*x/(1-2*x-3*x^2+(1-x)*sqrt(1-2*x-3*x^2));
v005717 = Vec(gf);
/* Joerg Arndt, Aug 16 2012 */

def A():
    a, b, n = 0, 1, 1
    while True:
        yield b
        n += 1
        a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1))
A005717 = A()
print([next(A005717) for  in range(29)]) # _Peter Luschny, May 16 2016

a(n) = Sum_{k=1..n} T(k, k-1), where T is the array defined in A025177.
G.f.: 2*x/(1-2*x-3*x^2+(1-x)*sqrt(1-2*x-3*x^2)). - Emeric Deutsch, Aug 14 2002
E.g.f.: exp(x) * I_1(2x), where I_1 is the Bessel function. - Michael Somos, Sep 09 2002
a(n) = A111808(n,n-1). - Reinhard Zumkeller, Aug 17 2005
a(n) = Sum_{k=0..floor((n-1)/3)} (-1)^k * binomial(n,k) * binomial(2n-2-3k, n-1). - David Callan, Jul 03 2006
From Paul Barry, Feb 05 2007: (Start)
a(n) = n*Sum_{k=0..floor((n-1)/2), C(n-1,2k)*C(k)}, C(n) = A000108(n).
a(n) = Sum_{k=0..floor((n-1)/2)} (2k+1)*C(n,2k+1)*C(k).
a(n) = Sum_{k=0..n-1} ( Sum_{j=0..floor(k/2)} C(k,2j)*C(2j+1,j) ). (End)
a(n) = (A002426(n+1) - A002426(n))/2. - Paul Barry, May 22 2008
a(n) = n*A001006(n-1). - Paul Barry, Oct 05 2009
a(n) = Sum_{i=0..floor(n/2)} C(n+1,n-i) * C(n-i,i). - Shanzhen Gao, Apr 20 2010
D-finite with recurrence: (n+1)*a(n) - 3*n*a(n-1) - (n+3)*a(n-2) + 3*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 28 2011
a(n) ~ 3^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 09 2013
0 = a(n) * 3*(n+1)*(n+2) + a(n+1) * (n+2)*(2*n+3) - a(n+2) * (n+1)*(n+3) for all n in Z. - Michael Somos, Apr 03 2014
G.f.: z*M(z)/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths. - José Luis Ramírez Ramírez, Apr 19 2015
Working with an offset of 0, a(n) = [x^n](1 + x + x^2)^(n+1); binomial transform is A076540. - Peter Bala, Jun 15 2015
a(n) = GegenbauerC(n,-n-1,-1/2). - Peter Luschny, May 07 2016
a(n) = (-1)^(n+1) * n * hypergeom([3/2, 1-n], [3], 4). - Vladimir Reshetnikov, Sep 28 2016
a(n) = Sum_{k=0..n-1} binomial(n,k)*binomial(n-k, k+1) [Krymski and Okhotin]. - Michel Marcus, Dec 04 2020
a(n) = (1/2)*(A005773(n+1) - A005043(n)). - Peter Bala, Feb 11 2022
a(n) = A002426(n) - A005043(n). - Amiram Eldar, May 17 2024

More terms from Erich Friedman, Jun 01 2001

A008287 Triangle of quadrinomial coefficients, row n is the sequence of coefficients of (1 + x + x^2 + x^3)^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1, 1, 5, 15, 35, 65, 101, 135, 155, 155, 135, 101, 65, 35, 15, 5, 1, 1, 6, 21, 56, 120, 216, 336, 456, 546, 580, 546, 456, 336, 216, 120, 56, 21, 6, 1

Offset: 0

N. J. A. Sloane

Coefficient of x^k in (1 + x + x^2 + x^3)^n is the number of distinct ways in which k unlabeled objects can be distributed in n labeled urns allowing at most 3 objects to fall in each urn. - N-E. Fahssi, Mar 16 2008
Rows of A008287 mod 2 converted to decimal equals A177882. - Vladimir Shevelev, Jan 02 2011
T(n,k) is the number of compositions of k into n parts p, each part 0<=p<=3. Adding 1 to each part, as a corollary, T(n,k) is the number of compositions of n+k into n parts p where 1<=p<=4. E.g., T(2,3)=4 since 3=0+3=3+0=1+2=2+1. In general, the entry (n,k) of the (l+1)-nomial triangle gives the number of compositions of k into n parts p, each part 0<=p<=l. - Steffen Eger, Jun 18 2011
Number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (1,2), (1,3). - Joerg Arndt, Jul 05 2011
T(n-1,k-1) is the number of 3-compositions of n with zeros having k parts; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020
T(n,k) is the number of ways to obtain a sum of n+k when throwing a 4-sided die n times. Follows from the "T(n,k) is the number of compositions of n+k into n parts p where 1 <= p <= 4" comment above. - Feryal Alayont, Dec 30 2024

			Triangle begins
  1;
  1,1,1,1;
  1,2,3,4,3,2,1;
  1,3,6,10,12,12,10,6,3,1; ...

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
D. C. Fielder and C. O. Alford, Pascal's triangle: top gun or just one of the gang?, in G E Bergum et al., eds., Applications of Fibonacci Numbers Vol. 4 1991 pp. 77-90 (Kluwer).

Alois P. Heinz, Rows n = 0..100, flattened (first 26 rows from T. D. Noe)
Moussa Ahmia and Hacene Belbachir, Preserving log-convexity for generalized Pascal triangles, Electronic Journal of Combinatorics, 19(2) (2012), #P16. - _N. J. A. Sloane_, Oct 13 2012
Said Amrouche and Hacène Belbachir, Asymmetric extension of Pascal-Dellanoy triangles, arXiv:2001.11665 [math.CO], 2020.
Armen G. Bagdasaryan and Ovidiu Bagdasar, On some results concerning generalized arithmetic triangles, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77.
Hacène Belbachir and Oussama Igueroufa, Combinatorial interpretation of bisnomial coefficients and Generalized Catalan numbers, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 47-54.
Hacène Belbachir and Yassine Otmani, Quadrinomial-Like Versions for Wolstenholme, Morley and Glaisher Congruences, Integers (2023) Vol. 23.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 17.
Ji Young Choi, Digit Sums Generalizing Binomial Coefficients, J. Int. Seq., Vol. 22 (2019), Article 19.8.3.
Spiros D. Dafnis, Frosso S. Makri, and Andreas N. Philippou, Restricted occupancy of s kinds of cells and generalized Pascal triangles, Fibonacci Quart. 45 (2007), no. 4, 347-356.
L. Euler, On the expansion of the power of any polynomial (1+x+x^2+x^3+x^4+etc.)^n, arXiv:math/0505425 [math.HO], 2005.
L. Euler, De evolutione potestatis polynomialis cuiuscunque (1+x+x^2+x^3+x^4+etc.)^n, E709.
Nour-Eddine Fahssi, Polynomial Triangles Revisited, arXiv:1202.0228 [math.CO], (25-July-2012).
D. C. Fielder and C. O. Alford, Pascal's triangle: top gun or just one of the gang?, Applications of Fibonacci Numbers 4 (1991), 77-90. (Annotated scanned copy)
J. E. Freund, Restricted Occupancy Theory - A Generalization of Pascal's Triangle, American Mathematical Monthly, Vol. 63, No. 1 (1956), pp. 20-27.
S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.
W. Florek and T. Lulek, Combinatorial analysis of magnetic configurations, Séminaire Lotharingien de Combinatoire, B26d (1991), 12 pp.
R. K. Guy, Letter to N. J. A. Sloane, 1987
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
Kantaphon Kuhapatanakul and Anantakitpaisal, The k-nacci triangle and applications, Cogent Math. 4, Article ID 1333293, 13 p. (2017).
Thorsten Neuschel, A Note on Extended Binomial Coefficients, J. Int. Seq. 17 (2014) # 14.10.4.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 4.
Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.]
Boualam Rezig and Moussa Ahmia, Combinatorics of three-Catalan numbers and some positivities, arXiv:2502.03615 [math.CO], 2025. See Table 1 p. 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.
Bao-Xuan Zhu, Linear transformations and strong q-log-concavity for certain combinatorial triangle, arXiv preprint arXiv:1605.00257 [math.CO], 2016.

Cf. A007318, A027907, A177882.

a008287 n = a008287_list !! n
a008287_list = concat $ iterate ([1,1,1,1] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

#Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 4-nomials as a table
r := 4:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do;
# Peter Bala, Sep 07 2013
# second Maple program:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))((1+x+x^2+x^3)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 17 2018

Flatten[Table[CoefficientList[(1 + x + x^2 + x^3)^n, x], {n, 0, 10}]] (* T. D. Noe, Apr 04 2011 *)
T[n_, k_] := Sum[Binomial[n, i] Binomial[n, k-2i], {i, 0, k/2}]; Table[T[n, k], {n, 0, 6}, {k, 0, 3n}] // Flatten (* Jean-François Alcover, Feb 02 2018 *)

quadrinomial(n,k):=coeff(expand((1+x+x^2+x^3)^n),x,k);
create_list(quadrinomial(n,k),n,0,8,k,0,3*n); /* Emanuele Munarini, Mar 15 2011 */

n-th row is formed by expanding (1+x+x^2+x^3)^n.
From Vladimir Shevelev, Dec 15 2010: (Start)
T(n,0) = 1; T(n,3*n) = 1; T(n,k) = T(n,3*n-k);
T(n,k) = 0, iff k<0 or k>3*n; Sum{k=0..3*n} T(n,k) = 4^n; Sum{k=0..3*n}((-1)^k)*T(n,k)=0 for n > 0; [corrected by Werner Schulte, Sep 09 2015]
T(n,k) = Sum{i=0..floor(k/2)} C(n,i)*C(n,k-2*i);
T(n+1,k) = T(n,k-3)+T(n,k-2)+T(n,k-1)+T(n,k). (End)
T(n,k) = Sum_{i = 0..floor(k/4)} (-1)^i*C(n,i)*C(n+k-1-4*i,n-1) for n >= 0 and 0 <= k <= 3*n. - Peter Bala, Sep 07 2013
G.f.: 1/(1 - ( x + y*x + y^2*x +y^3*x )). - Geoffrey Critzer, Feb 05 2014
T(n,k) = Sum_{j=0..k} (-2)^j*binomial(n,j)*binomial(3*n-2*j,k-j) for n >= 0 and 0 <= k <= 3*n (conjectured). - Werner Schulte, Sep 09 2015

A026729 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 3, 1, 0, 0, 0, 3, 4, 1, 0, 0, 0, 1, 6, 5, 1, 0, 0, 0, 0, 4, 10, 6, 1, 0, 0, 0, 0, 1, 10, 15, 7, 1, 0, 0, 0, 0, 0, 5, 20, 21, 8, 1, 0, 0, 0, 0, 0, 1, 15, 35, 28, 9, 1, 0, 0, 0, 0, 0, 0, 6, 35, 56, 36, 10, 1, 0, 0, 0, 0, 0, 0, 1, 21, 70, 84, 45, 11, 1, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Jan 19 2003

The signed triangular matrix T(n,k)*(-1)^(n-k) is the inverse matrix of the triangular Catalan convolution matrix A106566(n,k), n=k>=0, with A106566(n,k) = 0 if nPhilippe Deléham, Aug 01 2005
As a number triangle: unsigned version of A109466. - Philippe Deléham, Oct 26 2008
A063967*A130595 as infinite lower triangular matrices. - Philippe Deléham, Dec 11 2008
Modulo 2, this sequence becomes A106344. - Philippe Deléham, Dec 18 2008
Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th antidiagonal of the array. Then s_k(n) = Sum_{i=0..k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A111808. For example, s_1(n) = binomial(n,1) = n is the first column of A111808 for n>1, s_2(n) = binomial(n,1) + binomial(n,2) is the second column of A111808 for n>1, etc. Therefore, in cases k=3,4,5,6,7,8, s_k(n) is A005581(n), A005712(n), A000574(n), A005714(n), A005715(n), A005716(n), respectively. Besides, s_k(n+5) = A064054(n). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
As a triangle, T(n,k) = binomial(k,n-k). - Peter Bala, Nov 27 2015
For all n >= 0, k >= 0, the k-th homology group of the n-torus H_k(T^n) is the free abelian group of rank T(n,k) = binomial(n,k). See the Math Stack Exchange link below. - Jianing Song, Mar 13 2023

			Array begins
  1 0 0 0 0 0 ...
  1 1 0 0 0 0 ...
  1 2 1 0 0 0 ...
  1 3 3 1 0 0 ...
  1 4 6 4 1 0 ...
As a triangle, this begins
  1
  0 1
  0 1 1
  0 0 2 1
  0 0 1 3 1
  0 0 0 3 4 1
  0 0 0 1 6 5 1
  ...
Production array is
  0    1
  0    1   1
  0   -1   1   1
  0    2  -1   1  1
  0   -5   2  -1  1  1
  0   14  -5   2 -1  1  1
  0  -42  14  -5  2 -1  1  1
  0  132 -42  14 -5  2 -1  1  1
  0 -429 132 -42 14 -5  2 -1  1  1
  ... (Cf. A000108)

Muniru A Asiru, Rows n=0..50 of triangle, flattened
Tom Copeland, Addendum to Elliptic Lie Triad
Math Stack Exchange, Homology of the n-torus using the Künneth Formula
Lili Mu and Sai-nan Zheng, On the Total Positivity of Delannoy-Like Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.6.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

The official entry for Pascal's triangle is A007318. See also A052553 (the same array read by upward antidiagonals).
Cf. A030528 (subtriangle for 1<=k<=n).
Cf. A109466, A102426.

nmax:=15;; T:=List([0..nmax],n->List([0..nmax],k->Binomial(n,k)));;
b:=List([2..nmax],n->OrderedPartitions(n,2));;
a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->T[b[i][j][1]][b[i][j][2]]))); # Muniru A Asiru, Jul 17 2018

/* As triangle: */ [[Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 29 2015

seq(seq(binomial(k,n-k),k=0..n),n=0..12); # Peter Luschny, May 31 2014

Table[Binomial[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 28 2015 *)

As a number triangle, this is defined by T(n,0) = 0^n, T(0,k) = 0^k, T(n,k) = T(n-1,k-1) + Sum_{j, j>=0} (-1)^j*T(n-1,k+j)*A000108(j) for n>0 and k>0. - Philippe Deléham, Nov 07 2005
As a triangle read by rows, it is [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 22 2006
As a number triangle, this is defined by T(n, k) = Sum_{i=0..n} (-1)^(n+i)*binomial(n, i)*binomial(i+k, i-k) and is the Riordan array ( 1, x*(1+x) ). The row sums of this triangle are F(n+1). - Paul Barry, Jun 21 2004
Sum_{k=0..n} x^k*T(n,k) = A000007(n), A000045(n+1), A002605(n), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for n=0,1,2,3,4,5,6,7,8,9,10. - Philippe Deléham, Oct 16 2006
T(n,k) = A109466(n,k)*(-1)^(n-k). - Philippe Deléham, Dec 11 2008
G.f. for the triangular interpretation: -1/(-1+x*y+x^2*y). - R. J. Mathar, Aug 11 2015
For T(0,0) = 0, the triangle below has the o.g.f. G(x,t) = [t*x(1+x)]/[1-t*x(1+x)]. See A109466 for a signed version and inverse, A030528 for reverse and A102426 for a shifted version. - Tom Copeland, Jan 19 2016

cp's OEIS Frontend

A002426 Central trinomial coefficients: largest coefficient of (1 + x + x^2)^n.

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A027907 Triangle of trinomial coefficients T(n,k) (n >= 0, 0 <= k <= 2*n), read by rows: n-th row is obtained by expanding (1 + x + x^2)^n.

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A213500 Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.

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A000581 a(n) = binomial coefficient C(n,8).

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A005581 a(n) = (n-1)*n*(n+4)/6.

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A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

A005581 a(n) = (n-1)n(n+4)/6.

A008778 a(n) = (n+1)(n^2 +8n +6)/6. Number of n-dimensional partitions of 4. Number of terms in 4th derivative of a function composed with itself n times.