A002426 -id:A002426 - cp's OEIS frontend

A111960 Renewal array for central trinomial numbers A002426.

1, 1, 1, 3, 2, 1, 7, 7, 3, 1, 19, 20, 12, 4, 1, 51, 61, 40, 18, 5, 1, 141, 182, 135, 68, 25, 6, 1, 393, 547, 441, 251, 105, 33, 7, 1, 1107, 1640, 1428, 888, 420, 152, 42, 8, 1, 3139, 4921, 4572, 3076, 1596, 654, 210, 52, 9, 1, 8953, 14762, 14535, 10456, 5880, 2652, 966, 280, 63, 10, 1

Offset: 0

Paul Barry, Aug 23 2005

Also the convolution triangle of A002426. - Peter Luschny, Oct 06 2022

			Triangle T(n,k) begins:
   1;
   1,  1;
   3,  2,  1;
   7,  7,  3,  1;
  19, 20, 12,  4, 1;
  51, 61, 40, 18, 5, 1;
  ...
From _Paul Barry_, May 12 2009: (Start)
Production matrix is
  1, 1,
  2, 1, 1,
  0, 2, 1, 1,
  -2, 0, 2, 1, 1,
  0, -2, 0, 2, 1, 1,
  4, 0, -2, 0, 2, 1, 1. (End)

Row sums are A111961.
Diagonal sums are A111962.
Inverse is A111963.
Factors as A007318*A111959.
Column k=0 gives A002426.
Cf. A026325.

# Uses function PMatrix from A357368. Adds a row and column above and to the left.
PMatrix(10, n -> A002426(n - 1)); # Peter Luschny, Oct 06 2022

Factors as (1/(1-x), x/(1-x))*(1/sqrt(1-4x^2), x/sqrt(1-4x^2)).
From Paul Barry, May 12 2009: (Start)
Equals ((1-x^2)/(1+x+x^2),x/(1+x+x^2))^{-1}*(1,x/(1-x^2))=A094531*(1,x/(1-x^2)).
Riordan array (1/sqrt(1-2x-3x^2), x/sqrt(1-2x-3x^2));
T(n,k) = Sum_{j=0..n} C(n,j)*C((j-1)/2,(j-k)/2)*2^(j-k)*(1+(-1)^(j-k))/2.
G.f.: 1/(1-xy-x-2x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-... (continued fraction). (End)

A113304 Primes that divide some central trinomial coefficient, A002426.

Original entry on oeis.org

3, 7, 17, 19, 41, 43, 47, 73, 107, 109, 113, 131, 173, 179, 191, 193, 199, 233, 269, 277, 281, 283, 293, 307, 311, 347, 373, 383, 401, 409, 419, 421, 439, 443, 457, 467, 503, 509, 521, 547, 563, 569, 593, 613, 617, 631, 653, 673, 691, 701, 709, 719, 739, 743

Offset: 1

T. D. Noe, Oct 24 2005

nonn

For primes less than 10^6, the density of these primes is near 0.3925.

G. C. Greubel, Table of n, a(n) for n = 1..1100

Cf. A113302 (number of k for which prime(n) divides T(k)), A113303 (least k such that prime(n) divides T(k)).

nn=1000; a=b=1; t=Join[{1}, Table[c=((2n-1)b+3(n-1)a)/n; a=b; b=c; c, {n, 2, nn}]]; pLst={}; Do[p=Prime[n]; k=1; While[k0, k++ ]; If[k

A113305 Primes that do not divide any central trinomial coefficient, A002426.

Original entry on oeis.org

2, 5, 11, 13, 23, 29, 31, 37, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 127, 137, 139, 149, 151, 157, 163, 167, 181, 197, 211, 223, 227, 229, 239, 241, 251, 257, 263, 271, 313, 317, 331, 337, 349, 353, 359, 367, 379, 389, 397, 431, 433, 449, 461, 463, 479

Offset: 1

T. D. Noe, Oct 24 2005

nonn

For primes less than 10^6, the density of these primes is near 0.6075.

G. C. Greubel, Table of n, a(n) for n = 1..1600
Nadav Kohen, Uniform Recurrence in the Motzkin Numbers and Related Sequences mod p, arXiv:2403.00149 [math.CO], 2024.
Narad Rampersad and Jeffrey Shallit, Congruence properties of combinatorial sequences via Walnut and the Rowland-Yassawi-Zeilberger automaton, arXiv:2110.06244 [math.CO], 2021.

Cf. A113302 (number of k for which prime(n) divides T(k)), A113303 (least k such that prime(n) divides T(k)).

nn=1000; a=b=1; t=Join[{1}, Table[c=((2n-1)b+3(n-1)a)/n; a=b; b=c; c, {n, 2, nn}]]; pLst={}; Do[p=Prime[n]; k=1; While[k0, k++ ]; If[k==p, AppendTo[pLst, p]], {n, PrimePi[nn]}]; pLst

A168599 G.f.: exp( Sum_{n>=1} A002426(n)^n * x^n/n ), where A002426(n) is the central trinomial coefficients.

Original entry on oeis.org

1, 1, 5, 119, 32707, 69038213, 1309743837515, 206848589180297555, 281897548265847120670891, 3287603007740009094151486257065, 330891681467139744269091005122077348971

Offset: 0

Paul D. Hanna, Dec 01 2009

nonn

Compare to: exp( Sum_{n>=1} A002426(n)*x^n/n ) = g.f. of the Motzkin numbers (A001006).

			G.f.: A(x) = 1 + x + 5*x^2 + 119*x^3 + 32707*x^4 +...
log(A(x)) = x + 9*x^2/2 + 343*x^3/3 + 130321*x^4/4 +...+ A002426(n)^n*x^n/n +...

G. C. Greubel, Table of n, a(n) for n = 0..45

Cf. A001006, A002426, A168598, A225328.

m:=30;
A002426:= func< n | (&+[ Binomial(n, k)*Binomial(k, n-k): k in [0..n]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( Exp( (&+[A002426(j)^j*x^j/j: j in [1..m+2]]) ) )); // G. C. Greubel, Mar 16 2021

m:=30;
A002426:= n-> add( binomial(n, k)*binomial(k, n-k), k=0..n );
S := series( exp(add(A002426(j)^j*x^j/j, j = 1..m+2)), x, m+1);
seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Mar 16 2021

A002426[n_] := GegenbauerC[n, -n, -1/2];
With[{m=30}, CoefficientList[Series[Exp[Sum[A002426[j]^j*x^j/j, {j, m+2}]], {x, 0, m}], x]] (* G. C. Greubel, Mar 16 2021 *)

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

m=30
def A002426(n): return sum( binomial(n, k)*binomial(k, n-k) for k in (0..n) )
def A168598_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp( sum( A002426(j)^j*x^j/j for j in [1..m+2])) ).list()
A168598_list(m) # G. C. Greubel, Mar 16 2021

A102445 Number of prime divisors (counted with multiplicity) of the central trinomial coefficients (A002426).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 4, 2, 2, 3, 2, 2, 4, 3, 4, 6, 3, 2, 3, 3, 5, 6, 6, 4, 9, 3, 3, 2, 3, 3, 4, 5, 3, 5, 4, 2, 3, 3, 4, 2, 7, 5, 7, 7, 5, 5, 6, 6, 4, 5, 8, 9, 4, 5, 6, 3, 3, 7, 6, 8, 7, 7, 4, 5, 4, 4, 7, 7, 9, 11, 5, 8, 7, 7, 6, 7, 7, 8, 12, 4, 7, 6, 6, 4, 8, 7, 4, 10, 7, 7, 6, 6, 7, 5, 5, 6, 8, 7, 9, 10, 5, 7

Offset: 1

Jonathan Vos Post, Feb 21 2005

nonn

First occurrence of k: 1,2,5,11,8,22,17,42,52,26,89,71,80,....

Amiram Eldar, Table of n, a(n) for n = 1..218
Eric Weisstein's World of Mathematics, Trinomial Coefficient..

Cf. A001222, A002426.

bigomega[n_Integer] := Plus @@ Last /@ FactorInteger[n]; tn[n_] := Sum[Binomial[n, k]*Binomial[n - k, k], {k, 0, n/2}]; Table[bigomega[tn[n]], {n, 103}] (* Robert G. Wilson v, Feb 21 2005 *)

a(n) = A001222(A002426(n)). - Amiram Eldar, Feb 06 2020

Edited and extended by Robert G. Wilson v, Feb 21 2005

A135091 A007318 * triangle M, where M = A002426 * 0^(n-k), 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 9, 7, 1, 4, 18, 28, 19, 1, 5, 30, 70, 95, 51, 1, 6, 45, 140, 285, 306, 141, 1, 7, 63, 245, 665, 1071, 987, 393, 1, 8, 84, 392, 1330, 2856, 3948, 3144, 1107, 1, 9, 108, 588, 2394, 6426, 11844, 14148, 9963, 3139

Offset: 0

Gary W. Adamson, Nov 18 2007

Right border = A002426.
Row sums = A000984: (1, 2, 6, 20, 70, 252, ...).
The n-th row of this triangle lists the coefficients of the polynomial: p := (1/Pi)*Integral_{s=0..Pi} (1 + t - 2*t*cos(s))^n; Pi / 1 | n p := ---- | (1 + t - 2 t cos(s)) ds Pi | / 0 for example n=5 then 4 2 3 p = 19 t + 18 t + 28 t + 4 t + 1. - Theodore Kolokolnikov, Oct 09 2010

			First few rows of the triangle:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  9,   7;
  1, 4, 18,  28,  19;
  1, 5, 30,  70,  95,   51;
  1, 6, 45, 140, 285,  306, 141;
  1, 7, 63, 245, 665, 1071, 987, 393;
  ...

Cf. A002426, A000984, A098473.

A007318 * triangle M, where M = A002426 * 0^(n-k), 0 <= k <= n; i.e., M = an infinite lower triangular matrix with A002426 as the right border and the rest zeros.
O.g.f. appears to be (1/sqrt(1-t*(1-x)))*1/sqrt(1-t*(1+3*x)) = 1 + (1+x)*t + (1 + 2*x + 3*x^2)*t^2 + ....
See A098473.

A168598 G.f.: exp( Sum_{n>=1} A002426(n)^2*x^n/n ), where A002426(n) is the central trinomial coefficients.

Original entry on oeis.org

1, 1, 5, 21, 119, 703, 4515, 30227, 210274, 1503930, 11008198, 82099262, 622013122, 4775754930, 37089503826, 290914775618, 2301706690657, 18351027768401, 147308337621061, 1189704370416949, 9661185599013209, 78844977025403657

Offset: 0

Paul D. Hanna, Dec 01 2009

nonn

Compare to: exp( Sum_{n>=1} A002426(n)*x^n/n ) = g.f. of the Motzkin numbers (A001006).

			G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 119*x^4 + 703*x^5 +...
log(A(x)) = x + 9*x^2/2 + 49*x^3/3 + 361*x^4/4 + 2601*x^5/5 + 19881*x^6/6 +...+ A002426(n)^2*x^n/n +...

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A001006, A002426, A168597, A168599.

m:=30;
A002426:= func< n | (&+[ Binomial(n, k)*Binomial(k, n-k): k in [0..n]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( Exp( (&+[A002426(j)^2*x^j/j: j in [1..m+2]]) ) )); // G. C. Greubel, Mar 16 2021

A002426[n_]:= GegenbauerC[n, -n, -1/2];
With[{m=30}, CoefficientList[Series[Exp[Sum[A002426[j]^2*x^j/j, {j, m+2}]], {x, 0, m}], x]] (* G. C. Greubel, Mar 16 2021 *)

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

m=30
def A002426(n): return sum( binomial(n, k)*binomial(k, n-k) for k in (0..n) )
def A168598_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp( sum( A002426(j)^2*x^j/j for j in [1..m+2])) ).list()
A168598_list(m) # G. C. Greubel, Mar 16 2021

A225602 a(n) = A002426(n^2), where A002426 is the central trinomial coefficients.

Original entry on oeis.org

1, 1, 19, 3139, 5196627, 82176836301, 12159131877715993, 16639279789182494873661, 209099036316263774148543463251, 24017537903429183163390175566336055657, 25134265191388162956642519120384003897467908119, 239089990313305548946878880624659134220897530949847409821

Offset: 0

Paul D. Hanna, Aug 03 2013

nonn

			L.g.f.: L(x) = x + 19*x^2/2 + 3139*x^3/3 + 5196627*x^4/4 + 82176836301*x^5/5 + ...
where exponentiation is an integer series:
exp(L(x)) = 1 + x + 10*x^2 + 1056*x^3 + 1300253*x^4 + 16436676927*x^5 + ... + A225604(n)*x^n + ...

G. C. Greubel, Table of n, a(n) for n = 0..45

Cf. A225604, A002426.

Table[Sum[Binomial[n^2, k]*Binomial[n^2 - k, k], {k, 0, Floor[n^2/2]}], {n,0,50}] (* G. C. Greubel, Feb 27 2017 *)

{a(n)=sum(k=0, n^2\2, binomial(n^2, k)*binomial(n^2-k, k))}
for(n=0, 20, print1(a(n), ", "))

Logarithmic derivative of A225604 (ignoring the initial term of this sequence).

a(n) = Sum_{k=0..floor(n^2/2)} binomial(n^2, k) * binomial(n^2-k, k).

A386649 Product of first n central trinomial coefficients (A002426) for n > 0 with a(0) = 1.

Original entry on oeis.org

1, 1, 3, 21, 399, 20349, 2869209, 1127599137, 1248252244659, 3918263795984601, 35080215765450132753, 899912775031092255512709, 66403663756769266442027284401, 14140062564030204365431731967633341, 8713488333644640745496899895218790824407

Offset: 0

Paul D. Hanna, Aug 08 2025

nonn

Conjecture: a(n) = A214589(n) - 2 for n >= 1, where A214589(n) is the number of n X n X n triangular 0..2 arrays with every horizontal row having the same average value.

			The central trinomial coefficients A002426(n) = [x^n] (1 + x + x^2)^n for n >= 0 begin [1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, ...], where a(n) equals the product of the terms A002426(0) through A002426(n).

Paul D. Hanna, Table of n, a(n) for n = 0..70

Cf. A214589, A386650, A342177, A003046, A002426.

Table[Product[3^k * Hypergeometric2F1[1/2, -k, 1, 4/3], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 09 2025 *)

{a(n) = prod(k=0,n, polcoef((1 + x + x^2)^k, k) )}
for(n=0,15,print1(a(n),", "))

a(n) = Product_{k=0..n} A002426(k) for n >= 0.

a(n) ~ c * 3^((n-1)*(n+3)/2) * exp(n/2) / (2^(n - 3/4) * Pi^(n/2 - 1/4) * n^(n/2 + 7/16)), where c = 1.123782729130753266489882099159237662230713685736... - Vaclav Kotesovec, Aug 09 2025

A092690 Row sums of triangle A092689, which is related to the central trinomial coefficients (A002426).

Original entry on oeis.org

1, 2, 7, 22, 71, 226, 717, 2262, 7107, 22246, 69413, 215986, 670441, 2076686, 6420403, 19816362, 61070499, 187953174, 577742469, 1773918642, 5441141589, 16674016758, 51052484343, 156188410098, 477487110429, 1458741494826

Offset: 0

Paul D. Hanna, Mar 04 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A002426, A092683, A092686, A092689.

With[{nn = 50}, CoefficientList[Series[Exp[x]*((1 + x)*BesselI[0, 2*x] + x*BesselI[1, 2*x]), {x,0,nn}], x] Range[0, nn]!] (* G. C. Greubel, Feb 27 2017 *)

{T(n,k)=if(n<0 || k>n,0, if(n==0 && k==0,1, if(n==1 && k<=1,1, if(k==n-1,T(n-1,0), if(k==n,T(n,0), 2*T(n-1,k)+T(n-1,k+1))))))} a(n)=sum(k=0,n,T(n,k))

E.g.f.: a(n) = n!* [x^n] exp(x)*((1+x)*BesselI(0, 2*x)+x*BesselI(1, 2*x)). - Peter Luschny, Aug 25 2012

cp's OEIS Frontend

A111960 Renewal array for central trinomial numbers A002426.

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A113304 Primes that divide some central trinomial coefficient, A002426.

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A113305 Primes that do not divide any central trinomial coefficient, A002426.

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A168599 G.f.: exp( Sum_{n>=1} A002426(n)^n * x^n/n ), where A002426(n) is the central trinomial coefficients.

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A102445 Number of prime divisors (counted with multiplicity) of the central trinomial coefficients (A002426).

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A135091 A007318 * triangle M, where M = A002426 * 0^(n-k), 0<=k<=n.

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A168598 G.f.: exp( Sum_{n>=1} A002426(n)^2*x^n/n ), where A002426(n) is the central trinomial coefficients.

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A225602 a(n) = A002426(n^2), where A002426 is the central trinomial coefficients.

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