A027907 -id:A027907 - cp's OEIS frontend

A111808 Left half of trinomial triangle (A027907), triangle read by rows.

1, 1, 1, 1, 2, 3, 1, 3, 6, 7, 1, 4, 10, 16, 19, 1, 5, 15, 30, 45, 51, 1, 6, 21, 50, 90, 126, 141, 1, 7, 28, 77, 161, 266, 357, 393, 1, 8, 36, 112, 266, 504, 784, 1016, 1107, 1, 9, 45, 156, 414, 882, 1554, 2304, 2907, 3139, 1, 10, 55, 210, 615, 1452, 2850, 4740, 6765, 8350

Offset: 1

Reinhard Zumkeller, Aug 17 2005

Consider a doubly infinite chessboard with squares labeled (n,k), ranks or rows n in Z, files or columns k in Z (Z denotes ...,-2,-1,0,1,2,... ); number of king-paths of length n from (0,0) to (n,k), 0 <= k <= n, is T(n,n-k). - Harrie Grondijs, May 27 2005. Cf. A026300, A114929, A114972.

Triangle of numbers C^(2)(n-1,k), n>=1, of combinations with repetitions from elements {1,2,...,n} over k, such that every element i, i=1,...,n, appears in a k-combination either 0 or 1 or 2 times (cf. also A213742-A213745). - Vladimir Shevelev and Peter J. C. Moses, Jun 19 2012

Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Trinomial Triangle
Eric Weisstein's World of Mathematics, Trinomial Coefficient

Row sums give A027914; central terms give A027908;
T(n, 0) = 0;
T(n, 1) = n for n>1;
T(n, 2) = A000217(n) for n>1;
T(n, 3) = A005581(n) for n>2;
T(n, 4) = A005712(n) for n>3;
T(n, 5) = A000574(n) for n>4;
T(n, 6) = A005714(n) for n>5;
T(n, 7) = A005715(n) for n>6;
T(n, 8) = A005716(n) for n>7;
T(n, 9) = A064054(n-5) for n>8;
T(n, n-5) = A098470(n) for n>4;
T(n, n-4) = A014533(n-3) for n>3;
T(n, n-3) = A014532(n-2) for n>2;
T(n, n-2) = A014531(n-1) for n>1;
T(n, n-1) = A005717(n) for n>0;
T(n, n) = central terms of A027907 = A002426(n).

T := (n,k) -> simplify(GegenbauerC(k, -n, -1/2)):
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, May 09 2016

Table[GegenbauerC[k, -n, -1/2], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

(1 + x + x^2)^n = Sum(T(n,k)*x^k: 0<=k<=n) + Sum(T(n,k)*x^(2*n-k): 0<=k

T(n, k) = A027907(n, k) = Sum_{i=0,..,(k/2)} binomial(n, n-k+2*i) * binomial(n-k+2*i, i), 0<=k<=n.

T(n, k) = GegenbauerC(k, -n, -1/2). - Peter Luschny, May 09 2016

Corrected and edited by Johannes W. Meijer, Oct 05 2010

A025177 Triangular array, read by rows: first differences in n,n direction of trinomial array A027907.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 2, 4, 4, 4, 2, 1, 1, 3, 7, 10, 12, 10, 7, 3, 1, 1, 4, 11, 20, 29, 32, 29, 20, 11, 4, 1, 1, 5, 16, 35, 60, 81, 90, 81, 60, 35, 16, 5, 1, 1, 6, 22, 56, 111, 176, 231, 252, 231, 176, 111, 56, 22, 6, 1, 1, 7, 29, 84, 189, 343, 518, 659, 714, 659, 518, 343

Offset: 0

Clark Kimberling

The Motzkin transforms of the rows starting (1, 2), (1, 3) and (1, 4), extended by zeros after their last element, are apparently in A026134, A026109 and A026110. - R. J. Mathar, Dec 11 2008

			               1
            1  0  1
         1  1  2  1  1
      1  2  4  4  4  2  1
   1  3  7 10 12 10  7  3  1
1  4 11 20 29 32 29 20 11  4  1

Georg Fischer, Table of n, a(n) for n = 0..2499 [rows 0..49; first 676 terms from _G. C. Greubel_]

Columns include A025178, A025179, A025180, A025181, A025182.

Cf. A024996, A025192 (row sums).

A025177 := proc(n,k)
    option remember;
    if k < 0 or k > 2*n then
        0;
    elif n = 0 then
        1 ;
    elif n = 1 then
        op(k+1,[1,0,1]) ;
    else
        procname(n-1,k-2)+procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc:
seq(seq(A025177(n,k),k=0..2*n),n=0..20)  ; # R. J. Mathar, Feb 25 2015

nmax = 10; CoefficientList[CoefficientList[Series[(1 - y*x)/(1 - x*(1 + y + y^2)), {x, 0, nmax}, {y, 0, 2*nmax}], x], y] // Flatten (* G. C. Greubel, May 22 2017; amended by Georg Fischer, Jun 24 2020 *)

{T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, if( n==1, [1,0,1][k+1], if( n==2, [1,1,2,1,1][k+1], T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)))))};

T(n,k)=polcoeff(Ser(polcoeff(Ser((1-y*z)/(1-z*(1+y+y^2)),y),k,y),z),n,z)

{T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, polcoeff( (1 + x + x^2)^n, k) - polcoeff( (1 + x + x^2)^(n-1), k-1)))};

g=matrix(33,65);
for(n=0,32,for(k=0,2*n,g[n+1,k+1]=0));
g[1,1]=1;
g[2,1]=1;g[2,2]=0;g[2,3]=1;
g[3,1]=1;g[3,2]=1;g[3,3]=2;g[3,4]=1;g[3,5]=1;
for(n=0,2,for(k=0,2*n,print(n," ",k," ",g[n+1,k+1])))
for(n=3,32,g[n+1,1]=1;print(n," 1 1");g[n+1,2]=n-1;print(n," 2 ",n-1);for(k=2,2*n,g[n+1,k+1]=g[n,k-1]+g[n,k]+g[n,k+1];print(n," ",k," ",g[n+1,k+1])))
\\ Michael B. Porter, Feb 02 2010

T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 0, 1].

G.f.: (1-y*z)/[1-z*(1+y+y^2)].

Edited by Ralf Stephan, Jan 09 2005

Offset corrected by R. J. Mathar, Feb 25 2015

A024996 Triangular array, read by rows: second differences in n,n direction of trinomial array A027907.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 1, 1, 3, 2, 3, 1, 1, 1, 2, 5, 6, 8, 6, 5, 2, 1, 1, 3, 8, 13, 19, 20, 19, 13, 8, 3, 1, 1, 4, 12, 24, 40, 52, 58, 52, 40, 24, 12, 4, 1, 1, 5, 17, 40, 76, 116, 150, 162, 150, 116, 76, 40, 17, 5, 1, 1, 6, 23, 62, 133, 232, 342, 428, 462, 428, 342, 232, 133, 62, 23, 6

Offset: 0

Clark Kimberling

For n > 2, T(n,k) is the number of integer strings s(0), ..., s(n) such that s(n) = n - k, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2 and <= 1 for i >= 3.

			                  1
               1  0  1
            1  0  2  0  1
         1  1  3  2  3  1  1
      1  2  5  6  8  6  5  2  1
   1  3  8 13 19 20 19 13  8  3  1

G. C. Greubel, Table of n, a(n) for n = 0..1874 [a(676) ff. corrected by _Georg Fischer_, Jun 24 2020]

First differences in n, n direction of array A025177.
Central column is essentially A024997, other columns are A024998, A026069, A026070, A026071. Row sums are in A025579.
Cf. A027907, A026552, A024072.

using Nemo
function A024996Expansion(prec)
    R, t = PolynomialRing(ZZ, "t")
    S, x = PowerSeriesRing(R, prec+1, "x")
    ser = divexact(x^2*t^3 + x^2*t + x*t - 1, x*t^2 + x*t + x - 1)
    L = zeros(ZZ, prec^2)
    for k ∈ 0:prec-1, n ∈ 0:2*k
        L[k^2+n+1] = coeff(coeff(ser, k), n)
    end
    L
end
A024996Expansion(8) |> println # Peter Luschny, Jun 25 2020

A024996 := proc(n,k)
    option remember;
    if n < 0 or k < 0 or k > 2*n then
        0 ;
    elif n <= 2 then
        if k = 2*n or k = 0 then
            1;
        elif k = 2*n-1 or k = 1 then
            0;
        elif k =2 then
            2;
        end if;
    else
        procname(n-1,k-1)+procname(n-1,k-2)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Jun 23 2013
seq(seq(A024996(n,k), k=0..2*n), n=0..11); # added by Georg Fischer, Jun 24 2020

nmax = 10; CoefficientList[CoefficientList[Series[y*x + (1 - y*x)^2/(1 - x*(1 + y + y^2)), {x, 0, nmax}, {y, 0, 2*nmax}], x], y] // Flatten (* G. C. Greubel, May 22 2017; amended by Georg Fischer, Jun 24 2020 *)

T(n,k)=if(n<0||k<0||k>2*n,0,if(n==0,1,if(n==1,[1,0,1][k+1],if(n==2,[1,0,2,0,1][k+1],T(n-1,k-2)+T(n-1,k-1)+T(n-1,k))))) \\ Ralf Stephan, Jan 09 2004
nmax=8; for(n=0, nmax, for(k=0, 2*n, print1(T(n,k),","))) \\ added by _Georg Fischer, Jun 24 2020

T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 0, 1], [1, 0, 2, 0, 1].

G.f.: y*z + (1-y*z)^2 / (1-z*(1+y+y^2)). - Ralf Stephan, Jan 09 2005 [corrected by Peter Luschny, Jun 25 2020]

Edited by Ralf Stephan, Jan 09 2004

Offset corrected by R. J. Mathar, Jun 23 2013

A027914 T(n,0) + T(n,1) + ... + T(n,n), T given by A027907.

Original entry on oeis.org

1, 2, 6, 17, 50, 147, 435, 1290, 3834, 11411, 34001, 101400, 302615, 903632, 2699598, 8068257, 24121674, 72137547, 215786649, 645629160, 1932081885, 5782851966, 17311097568, 51828203475, 155188936431, 464732722872

Offset: 0

Clark Kimberling

Let b(n)=a(n) mod 2; then b(n)=1/2+(-1)^n*(1/2-A010060(floor(n/2))). - Benoit Cloitre, Mar 23 2004
Binomial transform of A027306. Inverse binomial transform of = A032443. Hankel transform is {1, 2, 3, 4, ..., n, ...}. - Philippe Deléham, Jul 20 2005
Sums of rows of the triangle in A111808. - Reinhard Zumkeller, Aug 17 2005
Number of 3-ary words of length n in which the number of 1's does not exceed the number of 0's. - David Scambler, Aug 14 2012
The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 07 2022

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Moa Apagodu and Ji-Cai Liu, Congruence properties for the trinomial coefficients, arXiv:1907.13547 [math.NT], 2019.
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. 24.

Cf. A025191, A027915, A081673, A092255, A055217, A005773.

a027914 n = sum $ take (n + 1) $ a027907_row n
-- Reinhard Zumkeller, Jan 22 2013

a := n -> simplify((3^n + GegenbauerC(n,-n,-1/2))/2):
seq(a(n), n=0..25); # Peter Luschny, May 12 2016

CoefficientList[ Series[ (1 + x + Sqrt[1 - 2x - 3x^2])/(2 - 4x - 6x^2), {x, 0, 26}], x] (* Robert G. Wilson v, Jul 21 2015 *)
Table[(3^n + Hypergeometric2F1[1/2 - n/2, -n/2, 1, 4])/2, {n, 0, 20}] (* Vladimir Reshetnikov, May 07 2016 *)
f[n_] := Plus @@ Take[ CoefficientList[ Sum[x^k, {k, 0, 2}]^n, x], n +1]; Array[f, 26, 0] (* Robert G. Wilson v, Jan 30 2017 *)

a(n)=sum(i=0,n,polcoeff((1+x+x^2)^n,i,x))

a(n)=sum(i=0,n,sum(j=0,n,sum(k=0,j,if(i+j+k-n,0,(n!/i!/j!/k!)))))

x='x+O('x^99); Vec((1+x+(1-2*x-3*x^2)^(1/2))/(2*(1-2*x-3*x^2))) \\ Altug Alkan, May 12 2016

a(n) = ( 3^n + A002426(n) )/2; lim n -> infinity a(n+1)/a(n) = 3; 3^n < 2*a(n) < 3^(n+1). - Benoit Cloitre, Sep 28 2002
From Benoit Cloitre, Jan 26 2003: (Start)
a(n) = (1/2) *( Sum{k = 0..n} binomial(n,k)*binomial(n-k,k) + 3^n );
a(n) = Sum_{k = 0..n} Sum_{i = 0..k} binomial(n,i)*binomial(n-i,k);
a(n) = 3^n/2*(1+c/sqrt(n)+O(n^-1/2)) where c=0.5... (End)
c = sqrt(3/Pi)/2 = 0.4886025119... - Vaclav Kotesovec, May 07 2016
a(n) = n!*Sum(i+j+k=n, 1/(i!*j!*k!)) 0<=i<=n, 0<=k<=j<=n. - Benoit Cloitre, Mar 23 2004
G.f.: (1+x+sqrt(1-2x-3x^2))/(2(1-2x-3x^2)); a(n) = Sum_{k = 0..n} floor((k+2)/2)*Sum_{i = 0..floor((n-k)/2)} C(n,i)*C(n-i,i+k)* ((k+1)/(i+k+1)). - Paul Barry, Sep 23 2005; corrected Jan 20 2008
D-finite with recurrence: n*a(n) +(-5*n+4)*a(n-1) +3*(n-2)*a(n-2) +9*(n-2)*a(n-3)=0. - R. J. Mathar, Dec 02 2012
G.f.: (1+x+1/G(0))/(2*(1-2*x-3*x^2)), 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, Jul 30 2013
From Peter Bala, Jul 21 2015: (Start)
a(n) = [x^n]( 3*x - 1/(1 - x) )^n.
1 + x*exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + ... is the o.g.f. for A005773. (End)
a(n) = (3^n + GegenbauerC(n,-n,-1/2))/2. - Peter Luschny, May 12 2016

A025564 Triangular array, read by rows: pairwise sums of trinomial array A027907.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 4, 3, 1, 1, 4, 8, 10, 8, 4, 1, 1, 5, 13, 22, 26, 22, 13, 5, 1, 1, 6, 19, 40, 61, 70, 61, 40, 19, 6, 1, 1, 7, 26, 65, 120, 171, 192, 171, 120, 65, 26, 7, 1, 1, 8, 34, 98, 211, 356, 483, 534, 483, 356, 211, 98, 34, 8, 1, 1, 9, 43, 140, 343, 665, 1050, 1373

Offset: 0

Clark Kimberling

Counting the top row as row 0, T(n,k) is the number of strings of nonnegative integers "s(1)s(2)s(3)...s(k)" such that s(1)+s(2)+s(3)+...+s(k) = n and the string does not contain the substring "00". E.g., T(3,5) = 8 because the valid strings are 02010, 01020, 11010, 10110, 10101, 01110, 01101 and 01011. T(4,3) = 13, counting 040, 311, 301, 130, 031, 103, 013, 220, 202, 022, 211, 121 and 112. - Jose Luis Arregui (arregui(AT)unizar.es), Dec 05 2007

			                  1
              1   2   1
          1   3   4   3   1
      1   4   8  10   8   4   1
  1   5  13  22  26  22  13   5   1

Columns include A025565, A025566, A025567, A025568.

Cf. A025177.

T[, 0] = 1; T[1, 1] = 2; T[n, k_] /; 0 <= k <= 2n := T[n, k] = T[n-1, k-2] + T[n-1, k-1] + T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, 2n}] // Flatten (* Jean-François Alcover, Jul 22 2018 *)

{T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, if( n==1, [1,2,1][k+1], if( n==2, [1,3,4,3,1][k+1], T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)))))};

T(n,k)=polcoeff(Ser(polcoeff(Ser((1+y*z)/(1-z*(1+y+y^2)),y),k,y),z),n,z)

{T(n, k) = if( k<0 || k>2*n, 0, if(n==0, 1, polcoeff( (1 + x + x^2)^n, k)+ polcoeff( (1 + x + x^2)^(n-1), k-1)))};

T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 2, 1], [1, 3, 4, 3, 1].

G.f.: (1+yz)/[1-z(1+y+y^2)].

Edited by Ralf Stephan, Jan 09 2005

Edited by Clark Kimberling, Jun 20 2012

A082758 Sum of the squares of the trinomial coefficients (A027907).

Original entry on oeis.org

1, 3, 19, 141, 1107, 8953, 73789, 616227, 5196627, 44152809, 377379369, 3241135527, 27948336381, 241813226151, 2098240353907, 18252025766941, 159114492071763, 1389754816243449, 12159131877715993, 106542797484006471, 934837217271732457, 8212609533895771131

Offset: 0

Emanuele Munarini, May 21 2003

a(n) = T(2*n, 2*n), the coefficient of x^(2*n) in (1+x+x^2)^(2*n), where T is the trinomial triangle A027907; Integral representation: a(n) = (1/Pi) * Integral_{x=-1..1} ((1+2*x)^(2*n)/sqrt(1-x^2)), i.e., a(n) is the moment of order 2n of the random variable 1+2X, where the distribution of X is an arcsin law on the interval (-1,1). - N-E. Fahssi, Jan 22 2008

			G.f. = 1 + 3*x + 19*x^2 + 141*x^3 + 1107*x^4 + 8953*x^5 + 73789*x^6 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 77. (In the integral formula a left bracket is missing for the cosine argument.)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jelena Đokić, Olga Bodroža-Pantić, and Ksenija Doroslovački, A spanning union of cycles in rectangular grid graphs, thick grid cylinders and Moebius strips, Transactions on Combinatorics (2023) Art. 27132.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
Veronika Irvine, Stephen Melczer, and Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018.
StackExchange, The asymptotic behavior of an integral, 2015.

Cf. A002426, A027907, A132303 - A132305, A168592, A273055, A337389.

a := n -> simplify(GegenbauerC(2*n,-2*n,-1/2)):
seq(a(n), n=0..19); # Peter Luschny, May 07 2016

Table[Sum[(-1)^(i)*Binomial[2*n,i]*Binomial[4*n-3*i-1,2*n-3*i],{i,0,2*n/3}],{n, 0,25}] (* Adi Dani, Jul 03 2011 *)
Table[Hypergeometric2F1[1/2-n,-n,1,4], {n,0,19}] (* Peter Luschny, May 15 2016 *)
a[ n_] := SeriesCoefficient[ (1 - 2 x - 3 x^2)^(-1/2), {x, 0, 2 n}]; (* Michael Somos, Jan 08 2017 *)

makelist(sum(binomial(2*n-k, k)*binomial(2*n, k),k,0,n),n,0,40);

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

a(n)=sum(k=0,n,binomial(2*n-k,k)*binomial(2*n,k));

{a(n)=sum(k=0,n,binomial(n,k)^2*binomial(2*n,n)/binomial(2*k,k))} \\ Paul D. Hanna, Sep 29 2012

def A():
    a, b, n = 1, 3, 1
    yield a
    while True:
        yield b
        n += 1
        a, b = b, ((9-9*n)*(4*n-1)*(2*n-3)*a+(4*n-3)*(20*n^2-30*n+7)*b)//(n*(2*n-1)*(4*n-5))
A082758 = A()
print([next(A082758) for  in range(20)]) # _Peter Luschny, May 16 2016

a(n) = Sum_{k=0..2n} T(n, k)^2, where T(n, k) are trinomial coefficients (A027907).
a(n) = Sum_{k=0..n} binomial(2*n-k, k)*binomial(2*n, k). - Benoit Cloitre, Jul 30 2003
G.f.: (1/sqrt(1+2*x-3*x^2) + 1/sqrt(1-2*x-3*x^2))/2 (with interpolated zeros). - Paul Barry, Jan 04 2005
a(n) = Sum_{k=0..n} binomial(2*n,2*k)*binomial(2*k,k) = Sum_{k=0..n} binomial(n+k,2k)*binomial(2*n,n+k). - Paul Barry, Dec 16 2008
a(n) = Sum_{k=0..n} binomial(n,k)^2*binomial(2*n,n)/ binomial(2*k,k). - Paul D. Hanna, Sep 29 2012
Recurrence: n*(2*n-1)*a(n) = (14*n^2+n-12)*a(n-1) + 3*(14*n^2-71*n+78)*a(n-2) - 27*(n-2)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 3^(2*n+1/2)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 14 2012
a(n) = GegenbauerC(2*n, -2*n, -1/2). - Peter Luschny, May 07 2016
From Peter Luschny, May 15 2016: (Start)
a(n) = ((9-9*n)*(4*n-1)*(2*n-3)*a(n-2)+(4*n-3)*(20*n^2-30*n+7)*a(n-1))/(n*(2*n-1)*(4*n-5)) for n>=2.
a(n) = hypergeom([1/2-n, -n], [1], 4). (End)
a(n) = A002426(2*n). - Michael Somos, Jan 08 2017
From Peter Bala, Mar 16 2018: (Start)
a(n) = sqrt(-3)^(2*n)*P(2*n,-1/sqrt(-3)), where P(n,x) is the Legendre polynomial of degree n.
a(n) = 1/C(2*n,n)*Sum_{k = 0..n} C(n,k)*C(n+k,k)*C(2*n+2*k,n+k)*(-3)^(n-k). Cf. A273055. (End)
From Wolfdieter Lang, Apr 19 2018 : (Start)
a(n) = (2/Pi)*Integral_{phi=0..Pi/2} (sin(3*phi)/sin(phi))^(2*n) [Comtet, p. 77, q=3, n=k -> 2*n] = (2/Pi)*Integral_{x=0..2} (x^2 - 1)^(2*n)/sqrt(4-x^2) (with x = 2*cos(phi)). See also the integral of the above comment.
a(n) = 3^(2*n)*Sum_{k=0..2*n} binomial(2*n, k)*binomial(2*k, k)*(-1/3)^k = 3^(2*n)*hypergeometric([-2*n, 1/2], [1], 4/3) = (-3)^n*LegendreP(2*n, 1/sqrt(-3)). (End)
From Peter Bala, Apr 03 2022: (Start)
Conjecture: a(n) = [x^n] ( (1 + x + x^3 + x^4)/(1 - x)^2 )^n.
If the conjecture is true then the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. [added Aug 29 2025: The conjecture is true. The conjecture leads to the double sum representation a(n) = Sum_{j, k} binomial(n, k)*binomial(n, j)*binomial(3*n-3*j-k-1, n-3*j-k), which satisfies the second-order recurrence given above by Peter Luschny, as can be verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger Maple package.]
Calculation suggests that the supercongruence a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) holds for primes p >= 5 and positive integers n and k.
Column 1 of A337389. (End)

A027908 a(n) = T(2*n, n), T given by A027907.

Original entry on oeis.org

1, 2, 10, 50, 266, 1452, 8074, 45474, 258570, 1481108, 8533660, 49402850, 287134346, 1674425300, 9792273690, 57407789550, 337281021450, 1985342102964, 11706001102180, 69124774458092, 408737856117916, 2419833655003752, 14341910428953018, 85087759173024870

Offset: 0

Clark Kimberling

nonn

Central terms of the triangle in A111808. - Reinhard Zumkeller, Aug 17 2005

Number of paths of semilength n starting at (0,0) and ending on the X-axis using steps (1,1), (1,-1) and (1,3). - David Scambler, Jun 21 2013

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A027907, A027913, A143927, A344396, A370159, A370160.

ogf := series( RootOf( (144*x^2+140*x-27)*g^4+(18-12*x)*g^2+8*g+1, g), x=0, 20); # Mark van Hoeij, Nov 16 2011
a := n -> simplify(GegenbauerC(n, -2*n, -1/2)):
seq(a(n), n=0..23); # Peter Luschny, May 09 2016

Table[Binomial[4 n, n] Hypergeometric2F1[-3 n, -n, 1/2 - 2 n, 1/4], {n, 0, 20}] (* or *) Table[GegenbauerC[3 n, -2 n, -1/2] + KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, May 07 2016 *)

makelist(ultraspherical(n,-2*n,-1/2),n,0,12); /* Emanuele Munarini, Oct 18 2016 */

G.f.: -(g^2+g+1)/(3*g^2+g-1) where g = x times the g.f. of A143927. - Mark van Hoeij, Nov 16 2011
a(n) = GegenbauerC(n, -2*n, -1/2). - Peter Luschny, May 09 2016
From Peter Bala, Jan 26 2020: (Start)
a(n) = [x^(2*n)](1 + x^2 + x^4)^(2*n).
a(n) = Sum_{k = 0..floor(n/2)} C(2*n, n-k)*C(n-k, k).
a(n) = C(2*n,n) * hypergeom([-n/2, (1 - n)/2], [n + 1], 4)
Conjectural: a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) for all primes p >= 5 and positive integers n and k. (End)
From Peter Bala, Aug 03 2023: (Start)
P-recursive: 3*n*(13*n - 17)*(3*n - 1)*(3*n - 2)*a(n) = 2*(2*n - 1)*(455*n^3 - 1050*n^2 + 691*n - 120)*a(n-1) + 36*(n - 1)*(13*n - 4)*(2*n - 1)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 2.
exp(Sum_{n >= 0} a(n)*x^n/n) = 1 + 2*x + 7*x^2 + 28*x^3 + 123*x^4 + ... is the g.f. of A143927.
a(n) = 2*A344396(n-1) for n >= 1. (End)

A094531 Array read by rows: right-hand side of triangle A027907 of trinomial coefficients.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 6, 3, 1, 19, 16, 10, 4, 1, 51, 45, 30, 15, 5, 1, 141, 126, 90, 50, 21, 6, 1, 393, 357, 266, 161, 77, 28, 7, 1, 1107, 1016, 784, 504, 266, 112, 36, 8, 1, 3139, 2907, 2304, 1554, 882, 414, 156, 45, 9, 1, 8953, 8350, 6765, 4740, 2850, 1452, 615, 210, 55

Offset: 0

Paul Barry, May 07 2004

Sometimes called a Motzkin triangle, although that name is usually reserved for A026300.
Expand (1+x+x^2)^n and take last (nonzero) coefficient of first row, last two coefficients of second row, etc.
Equals A094531*(1,xc(-x^2)) where c(x) is the g.f. of A000108. - Paul Barry, May 12 2009
Coefficients of Faber polynomials for (1/x+1+x): Fa(n,x) = Sum_{k=0..n} T(n,k)*x^k, g.f.: -log((sqrt(-3*t^2-2*t+1)-t+1)/2-t*x) = Sum_{n>0} Fa(n,x)*t^n/n. - Vladimir Kruchinin, Jul 01 2013

			Triangle begins:
    1;
    1,   1;
    3,   2,   1;
    7,   6,   3,   1;
   19,  16,  10,   4,   1;
   51,  45,  30,  15,   5,   1;
  141, 126,  90,  50,  21,   6,   1;
  393, 357, 266, 161,  77,  28,   7,   1;
  ...

Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Tian-Xiao He and Renzo Sprugnoli, Sequence characterization of Riordan arrays, Discrete Math. (2009) Vol 309, No. 12, 3962-3974.
Ana Luzón, Donatella Merlini, Manuel A. Morón, and Renzo Sprugnoli, Complementary Riordan arrays, Discrete Applied Mathematics, 172 (2014) 75-87.
Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Article 12.3.3, 2012.
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. 23.
Paul Peart and Wen-jin Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
Louis W. Shapiro, Seyoum Getu, Wen-Jin Woan, and Leon C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

Binomial transform is triangle A094527. Row sums are A027914.

Cf. A111808 (row reversed).

T := (n, k) -> simplify(GegenbauerC(n-k, -n, -1/2)):
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, May 12 2016

max = 10; se = Series[ -Log[ (Sqrt[-3*t^2 - 2*t + 1] - t + 1)/2 - t*x], {t, 0, max + 1}, {x, 0, max}]; a[n_, k_] := SeriesCoefficient[se, {t, 0, n}, {x, 0, k}]*n; a[0, 0] = 1; Table[a[n, k], {n, 0, max }, {k, 0, n}] // Flatten  (* Jean-François Alcover, Jul 02 2013, after Vladimir Kruchinin *)
Table[Binomial[n, k] Hypergeometric2F1[(k - n)/2, (k - n + 1)/2, k + 1, 4], {n, 0, 9}, {k, 0, n}] // Flatten (* or *)
Table[If[n == 0, 1, GegenbauerC[n - k, -n, -1/2]], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 12 2016 *)

Riordan array ( 1/sqrt(1-2*x-3*x^2), (1-x-sqrt(1-2*x-3*x^2))/(2*x) ). - N. J. A. Sloane, Jun 02 2005
Product of Riordan arrays (1/(1-x), x/(1-x)) (Pascal's triangle, A007318) and (1/sqrt(1-4x^2), (1-sqrt(1-4*x^2))/(2*x)) (A108044). Inverse is A102587. - Paul Barry, Jul 14 2005
Column k has e.g.f. exp(x)*Bessel_I(k, 2x). - Paul Barry, Jul 14 2005
T(n, k) = Sum_{i=0..n} C(n-k-i, i)*C(n, k+i). - Paul Barry, Nov 04 2005
T(n, k) = Sum_{j=0..n} C(n,j)*C(j,n-k-j). - Paul Barry, Oct 25 2006
From Paul Barry, May 12 2009: (Start)
Production matrix is
  1, 1;
  2, 1, 1;
  0, 1, 1, 1;
  0, 0, 1, 1, 1;
  0, 0, 0, 1, 1, 1; (End)
From Peter Bala, Jun 29 2015: (Start)
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = (1 - x -sqrt(1 - 2*x - 3*x^2))/(2*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan, Example 1.1).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = 1 + x + x^2. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)
From Peter Luschny, May 12 2016: (Start)
T(n,k) = binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+1], 4):
T(n,k) = GegenbauerC(n-k, -n, -1/2). (End)

A168592 G.f.: exp( Sum_{n>=1} A082758(n)*x^n/n ), where A082758(n) = sum of the squares of the trinomial coefficients in row n of triangle A027907.

Original entry on oeis.org

1, 3, 14, 80, 509, 3459, 24579, 180389, 1356743, 10402493, 81004516, 638886082, 5093081983, 40971735401, 332187974718, 2711668091448, 22267979870143, 183830653156341, 1524747465249750, 12700172705956876, 106187411693668179

Offset: 0

Paul D. Hanna, Dec 01 2009

nonn

Number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k >= 0 and two kinds of (1,1). - Alois P. Heinz, Oct 07 2015

Number of pairs of noncrossing paths of length n which start and end together, each taking steps (1,0), (1,1) or (1,-1) (i.e., Motzkin-type). - Nicholas R. Beaton, Jun 17 2024

			G.f.: A(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 509*x^4 + 3459*x^5 + ...
log(A(x)) = 3*x + 19*x^2/2 + 141*x^3/3 + 1107*x^4/4 + 8953*x^5/5 + ... + A082758(n)*x^n/n + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A168590, A168593, A082758, A027907, A168595, A218321, A263316.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
      add(b(x-i, y-1), i=0..x) +add(b(x-1, y-j), j=0..y)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 07 2015
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1, 3, 14, 80][n+1],
      ((10*(n+1))*(16*n^3-20*n^2-n-1) *a(n-1)
      +(-944*n^4+2596*n^3-1924*n^2+236*n+30) *a(n-2)
      +(90*(n-2))*(16*n^3-52*n^2+45*n-6) *a(n-3)
      -(81*(2*n-5))*(n-2)*(n-3)*(4*n-1) *a(n-4))/
      ((n+1)*(4*n-5)*(2*n+1)*(n+2)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 07 2015

(1/x)*InverseSeries[x*(1 - x)^2/((1 + x)^2*(1 - x + x^2)) + O[x]^30, x] // CoefficientList[#, x]& (* Jean-François Alcover, Jun 09 2018 *)

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sum(k=0,2*m, polcoeff((1+x+x^2)^m,k)^2)*x^m/m) +x*O(x^n)),n))}

{a(n)=polcoeff(1/x*serreverse(x*(1-x)^2/((1+x)^2*(1-x+x^2)+x*O(x^n))),n)}

G.f.: A(x) = (1/x)*Series_Reversion[x*(1-x)^2/((1+x)^2*(1-x+x^2))].
G.f.: A(x) satisfies A(x^2) = M(x)*M(-x), where M(x) is the g.f. of A001006. - Alexander Burstein, Oct 03 2017
G.f.: A(x) satisfies A(x^2) = (1-x - sqrt(1-2*x-3*x^2))*(1+x - sqrt(1+2*x-3*x^2))/(4*x^4). - Paul D. Hanna, Oct 05 2017, concluded from formula of Alexander Burstein.

A136590 Triangle of trinomial logarithmic coefficients: A027907(n,k) = Sum_{i=0..k} T(k,i)*n^i/k!.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -4, 3, 1, 0, 6, -13, 6, 1, 0, 24, -10, -25, 10, 1, 0, -240, 394, -135, -35, 15, 1, 0, 720, -2016, 1834, -525, -35, 21, 1, 0, 5040, -5076, -3668, 5089, -1400, -14, 28, 1, 0, -80640, 170064, -110692, 14364, 9849, -3024, 42, 36, 1, 0, 362880, -1155024, 1339020, -672400, 118125, 12873, -5670, 150, 45, 1

Offset: 0

Paul D. Hanna, Jan 10 2008

A027907 is the triangle of trinomial coefficients.

The Bell transform of A136591(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins:
1;
0, 1;
0, 1, 1;
0, -4, 3, 1;
0, 6, -13, 6, 1;
0, 24, -10, -25, 10, 1;
0, -240, 394, -135, -35, 15, 1;
0, 720, -2016, 1834, -525, -35, 21, 1;
0, 5040, -5076, -3668, 5089, -1400, -14, 28, 1;
0, -80640, 170064, -110692, 14364, 9849, -3024, 42, 36, 1;
0, 362880, -1155024, 1339020, -672400, 118125, 12873, -5670, 150, 45, 1; ...
Trinomial coefficients can be calculated as illustrated by:
A027907(4,3) = (T(3,0)*4^0 + T(3,1)*4^1 + T(3,2)*4^2 + T(3,3)*4^3)/3! =
(0 - 4*4 + 3*4^2 + 1*4^3)/3! = 96/6 = 16.

Cf. columns: A136591, A136592, A136593; A136594 (unsigned row sums); A136595 (matrix inverse); A027907, A002426.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> n!*(modp(n+1,3)-modp(n,3)), 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 11;
M = BellMatrix[#!*(Mod[# + 1, 3] - Mod[#, 3])&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

{T(n,k)=n!/k!*polcoeff(log(1+x+x^2 +x*O(x^n))^k,n)}
for(n=0,10, for(k=0,n, print1( T(n,k),", "));print(""))

# uses[bell_matrix from A264428]
bell_matrix(lambda n: A136591(n+1), 10) # Peter Luschny, Jan 18 2016

E.g.f. of column k = log(1 + x + x^2)^k / k! for k>=0.

Central trinomial coefficients: A002426(n) = Sum_{k=0..n} T(n,k)*n^k/n!.

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A111808 Left half of trinomial triangle (A027907), triangle read by rows.

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A025177 Triangular array, read by rows: first differences in n,n direction of trinomial array A027907.

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A024996 Triangular array, read by rows: second differences in n,n direction of trinomial array A027907.

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A027914 T(n,0) + T(n,1) + ... + T(n,n), T given by A027907.

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A025564 Triangular array, read by rows: pairwise sums of trinomial array A027907.

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A082758 Sum of the squares of the trinomial coefficients (A027907).

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