A002426 -id:A002426 - cp's OEIS frontend

A007971 INVERTi transform of central trinomial coefficients (A002426).

0, 1, 2, 2, 4, 8, 18, 42, 102, 254, 646, 1670, 4376, 11596, 31022, 83670, 227268, 621144, 1706934, 4713558, 13072764, 36398568, 101704038, 285095118, 801526446, 2259520830, 6385455594, 18086805002, 51339636952, 146015545604

Offset: 0

David Dumas (dumas(AT)TCNJ.EDU)

nonn

Number of paths of a walk on the integers, allowing steps of size 0, +1, and -1, which return to the starting point for the first time at time n. [David P. Sanders (dps(AT)fciencias.unam.mx), May 04 2009]

			G.f. = x + 2*x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 18*x^6 + 42*x^7 + 102*x^8 + 254*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.

Cf. A001006, A002426, A005043.

Cf. A025227.

CoefficientList[Series[1-Sqrt[1-2x-3x^2],{x,0,40}],x] (* Harvey P. Dale, Dec 17 2012 *)
a[1]:=1;a[2]:=2;a[n_]:=a[n]=1/2 Sum[a[k] a[n-k],{k,1,n-1}];
Join[{0},Map[a,Range[24]]] (* Oliver Seipel, Nov 03 2024, after Schröder 1870 *)

x='x+O('x^50); concat([0], Vec(1 - sqrt(1 - 2*x - 3*x^2))) \\ G. C. Greubel, Feb 26 2017

A002426(n) = Sum_{i=1..n} a(i)*A002426(n-i), n>0. - Michael Somos, Jun 14 2000
G.f.: 1 - sqrt(1 - 2*x - 3*x^2). - Michael Somos, Jun 14 2000
a(0)=0, a(1)=1, a(2)=2, then a(n) = (1/2) *(a(1)*a(n-1)+a(2)*a(n-2)+....+a(n-1)*a(1)). - Benoit Cloitre, Oct 24 2003
a(n) = 2^(1-n)*Sum_{k=1..n} (binomial(k,n-k)*A000108(k-1)*3^(n-k)), n>0. - Vladimir Kruchinin, Feb 05 2011
G.f.: 1-sqrt(1-2*x-3*(x^2)) =  x/G(0) ; G(k) = 1-2*x/(1+x/(1+x/(1-2*x/(1-x/(2-x/G(k+1)))))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011
a(n+2) = 2 * A001006(n). - Michael Somos, Jun 14 2000
For n>1, a(n) = 2 * (A005043(n-1) + A005043(n-2)). - Ralf Stephan, Jul 06 2003
0 = a(n) * (9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1) * (-3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2) * (a(n+2) + a(n+3)) for all n>0. - Michael Somos, Jan 25 2014
n*a(n) + (-2*n+3)*a(n-1) + *(-n+3)*a(n-2) = 0. - R. J. Mathar, Sep 06 2016

Name corrected by Michael Somos, Mar 23 2012

A242170 Least prime divisor of T(n) which does not divide any T(k) with k < n, or 1 if such a primitive prime divisor of T(n) does not exist, where T(n) is the n-th central trinomial coefficient given by A002426.

Original entry on oeis.org

1, 3, 7, 19, 17, 47, 131, 41, 43, 1279, 503, 113, 2917, 569, 198623, 14083, 26693, 201611, 42998951, 41931041, 52635749, 1296973, 169097, 1451, 1304394227, 107, 233, 173, 2062225210273, 719, 191, 31551555041, 6301, 563, 3769, 967, 9539, 5073466546857451, 4542977, 9739

Offset: 1

Zhi-Wei Sun, May 05 2014

nonn

Conjecture: (i) a(n) > 1 for all n > 1.

(ii) For any integer n > 3, the n-th Motzkin number M(n) given by A001006 has a prime divisor which does not divide any M(k) with k < n.

			a(11) = 503 since T(11) = 3*17*503 with the prime divisor 503 dividing none of T(1),...,T(10), but 3 divides T(2) = 3 and 17 divides T(5) = 51.

Zhi-Wei Sun, Table of n, a(n) for n = 1..168

Cf. A000040, A001006, A002426, A242169, A242171, A242173.

T[n_]:=Sum[Binomial[n,2k]*Binomial[2k,k],{k,0,n/2}]
f[n_]:=FactorInteger[T[n]]
p[n_]:=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
Do[If[T[n]<2,Goto[cc]];Do[Do[If[Mod[T[i],Part[p[n],k]]==0,Goto[aa]],{i,1,n-1}];
Print[n," ",Part[p[n],k]];Goto[bb];Label[aa];Continue,{k,1,Length[p[n]]}];
Label[cc];Print[n," ",1];Label[bb];Continue,{n,1,40}]

A097893 Partial sums of the central trinomial coefficients (A002426).

Original entry on oeis.org

1, 2, 5, 12, 31, 82, 223, 616, 1723, 4862, 13815, 39468, 113257, 326198, 942425, 2730032, 7926659, 23061590, 67214399, 196211252, 573590621, 1678941350, 4920076877, 14433305000, 42381641381, 124558477682, 366371703833

Offset: 0

Emeric Deutsch, Sep 03 2004

a(n) is the number of peaks at odd height in all Motzkin paths of length n+2. Example: a(2)=5 counts the peaks shown between parentheses in the 9 Motzkin paths of length 4: HHHH, HH(UD), H(UD)H, HUHD, (UD)HH, (UD)(UD), UHDH, UHHD and UUDD.
Binomial transform of 1,1,2,2,6,6,20,20,70,70...... (A000984 doubled). It would appear that the Hankel transform of this sequence is a signed version of A128055, with sign pattern given by s(n)=(2/3-sqrt(3)/3)cos(5*Pi*n/6)-sin(5*Pi*n/6)/3+(sqrt(3)/3+2/3)*cos(Pi*n/6)-sin(Pi*n/6)/3-cos(Pi*n/2)/3+sin(Pi*n/2)/3. - Paul Barry, Jan 03 2008
Define triangle T(n,1) = T(n,n) = 1 and T(r,c) = T(r,c-1) + T(r-1,c-1) + T(r-2,c-1). Then the sum of the terms in row(n) is a(n+1). - J. M. Bergot, Apr 30 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Moa Apagodu and Doron Zeilberger, Using the "Freshman's Dream" to Prove Combinatorial Congruences, arXiv:1606.03351 [math.CO], 2016. Also Amer. Math. Monthly. 124 (2017), 597-608.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 14.
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
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. 25.

Cf. A002426, A025178.

a097893 n = a097893_list !! n
a097893_list = scanl1 (+) a002426_list
-- Reinhard Zumkeller, Jan 22 2013

ser:=series(1/(1-z)/sqrt(1-2*z-3*z^2),z=0,32): 1,seq(coeff(ser,z^n),n=1..31);
a := n -> (n+1)*hypergeom([1/2,(1-n)/2,-n/2],[1,3/2],4):
seq(simplify(a(n)), n=0..26); # Peter Luschny, Oct 29 2015

Table[ Sum[ Binomial[n, k]*Binomial[k, n-k], {k, 0, n}], {n, 0, 26}] // Accumulate (* Jean-François Alcover, Jul 10 2013 *)
CoefficientList[Series[1/((1-x)*Sqrt[1-2*x-3*x^2]), {x, 0, 50}], x] (* G. C. Greubel, Dec 21 2017 *)

a(n)=sum(i=0,n,sum(j=0,i,binomial(i,i-j)*binomial(j,i-j)))

vector(30, n, n--; sum(k=0, n\2, binomial(n+1, 2*k+1)* binomial(2*k, k))) \\ Altug Alkan, Oct 29 2015

x='x+O('x^30); Vec(1/((1-x)*sqrt(1-2*x-3*x^2))) \\ G. C. Greubel, Dec 21 2017

from math import comb
def A097893(n): return sum(comb(n+1,(k<<1)|1)*comb(k<<1,k) for k in range((n>>1)+1)) # Chai Wah Wu, Aug 14 2025

G.f.: 1/((1-z)*sqrt(1-2*z-3*z^2)).
a(n) = Sum_{0<=j<=i<=n} C(i, i-j)*C(j, i-j). - Benoit Cloitre, Oct 23 2004
a(n) = sum_{k=0..n} Sum_{j=0..n-k} C(k,j)C(n-k,j)C(2j,j). - Paul Barry, Jan 03 2008
Logarithm g.f. atan(x*M(x)), M(x) - o.g.f. for Motzkin numbers (A001006). - Vladimir Kruchinin_, Aug 11 2010
D-finite with recurrence -n*a(n) +(3*n-1)*a(n-1) +(n-2)*a(n-2) +3*(1-n)*a(n-3)=0. - R. J. Mathar, Nov 09 2012 [Since A002426(n) = a(n) - a(n-1), this third-order recurrence follows easily from the second-order recurrence given in A002426. - Peter Bala, Oct 28 2015]
G.f.:  G(0)/(1-x), 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
a(n) ~ 3^(n+3/2)/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 23 2013
a(n) = Sum_{k = 0..floor(n/2)} binomial(n + 1,2*k + 1) *binomial(2*k,k). Cf. A025178. - Peter Bala, Oct 28 2015
a(n) = (n+1)*hypergeom([1/2,(1-n)/2,-n/2],[1,3/2],4). - Peter Luschny, Oct 29 2015
a(n) = (n+1)*Sum_{k=0..floor(n/2)} multinomial(n;n-2*k,k,k)/(2*k+1). - Chai Wah Wu, Aug 14 2025

A277640 a(n) is the integer r with |r| < prime(n)/2 such that (T(prime(n)^2)-T(prime(n)))/prime(n)^4 == r (mod prime(n)), where T(k) denotes the central trinomial coefficient A002426(k).

Original entry on oeis.org

-2, 1, -3, -1, 7, -1, 6, 4, -15, -15, -13, 1, -23, 1, 8, -15, -22, 13, -33, 27, 25, 11, -17, 24, -32, -53, 31, 42, -19, 18, -35, 55, -5, 38, -29, 76, 34, 44, -71, -21, -13, 16, 46, 70, 92, 70, -39, 88, -84, -118, -120, 64, 107, 111, -56, 124, -13, -23

Offset: 3

Zhi-Wei Sun, Oct 25 2016

sign

Conjecture: (i) For any prime p > 3 and positive integer n, the number (T(p*n)-T(n))/(p*n)^2 is always a p-adic integer.
(ii) For any prime p > 3 and positive integer k, we have (T(p^k)-T(p^(k-1)))/p^(2k) == 1/6*(p^k/3)*B_{p-2}(1/3) (mod p), where (p^k/3) denotes the Legendre symbol and B_{p-2}(x) is the Bernoulli polynomial of degree p-2.
For any prime p > 3, the author has proved that (T(p*n)-T(n))/(p^2*n) is a p-adic integer for each positive integer n, and that T(p) == 1 + p^2/6*(p/3)*B_{p-2}(1/3) (mod p^3).

			a(3) = -2 since (T(prime(3)^2)-T(prime(3)))/prime(3)^4 = (T(25)-T(5))/5^4 = (82176836301-51)/5^4 = 131482938 is congruent to -2 modulo prime(3) = 5 with |-2| < 5/2.

Hao Pan and Zhi-Wei Sun, Supercongruences for central trinomial coefficients, arXiv:2012.05121 [math.NT], 2020.
Zhi-Wei Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57(2014), no.7, 1375-1400.
Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.

Cf. A000040, A002426, A245089, A277860.

T[n_]:=T[n]=Sum[Binomial[n,2k]Binomial[2k,k],{k,0,n/2}]
rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
Do[Print[n," ",rMod[(T[Prime[n]^2]-T[Prime[n]])/Prime[n]^4,Prime[n]]],{n,3,60}]

A025178 First differences of the central trinomial coefficients A002426.

Original entry on oeis.org

0, 2, 4, 12, 32, 90, 252, 714, 2032, 5814, 16700, 48136, 139152, 403286, 1171380, 3409020, 9938304, 29017878, 84844044, 248382516, 727971360, 2135784798, 6272092596, 18435108258, 54228499920, 159636389850, 470256930052, 1386170197704

Offset: 1

Clark Kimberling

Previous name was: "a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0 = s(n), |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n), where T is the array defined in A025177."

Note that n-1 divides a(n) for n>=2. - T. D. Noe, Mar 16 2005

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001006, A002426, A005717, A007971, A025177, A079309, A097893, A105696, A162551.

a := n -> 2*(n-1)*hypergeom([1-n/2, 3/2-n/2], [2], 4):
seq(simplify(a(n)), n=1..28); # Peter Luschny, Oct 29 2015

Rest[Differences[CoefficientList[Series[x/Sqrt[1-2x-3x^2],{x,0,30}],x]]] (* Harvey P. Dale, Aug 22 2011 *)
Differences[Table[Hypergeometric2F1[(1-n)/2,1-n/2,1,4],{n,1,29}]] (* Peter Luschny, Nov 03 2015 *)

a(n) = sum(k=1, n\2, binomial(n-1,2*k-1)*binomial(2*k,k)); \\ Altug Alkan, Oct 29 2015

def a():
    b, c, n = 0, 2, 2
    yield b
    while True:
        yield c
        b, c = c, ((2*n-1)*c+3*(n-1)*b)*n//((n+1)*(n-1))
        n += 1
A025178 = a()
print([next(A025178) for  in (1..20)]) # _Peter Luschny, Nov 04 2015

a(n) = T(n,n) for n>=1, where T is the array defined in A025177.
a(n) = A002426(n+1) - A002426(n). - Benoit Cloitre, Nov 02 2002
a(n) is asymptotic to c*3^n/sqrt(n) with c around 1.02... - Benoit Cloitre, Nov 02 2002
a(n) = 2*(n-1)*A001006(n-2). - M. F. Hasler, Oct 24 2011
a(n) = 2*A005717(n-1). - R. J. Mathar, Jul 09 2012
E.g.f. Integral(Integral(2*exp(x)*((1-1/x)*BesselI(1,2*x) + 2*BesselI(0,2*x)))). - Sergei N. Gladkovskii, Aug 16 2012
G.f.: -1/x + (1/x-1)/sqrt(1-2*x-3*x^2). - Sergei N. Gladkovskii, Aug 16 2012
D-finite with recurrence: a(n) = ((2+n)*a(n-2)+3*(3-n)*a(n-3)+3*(n-1)*a(n-1))/n, a(0)=1, a(1)=0, a(2)=2. - Sergei N. Gladkovskii, Aug 16 2012 [adapted to new offset by Peter Luschny, Nov 04 2015]
G.f.: (1-x)/x^2*G(0) - 1/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 06 2013
From Peter Bala, Oct 28 2015: (Start)
a(n) = Sum_{k = 0..floor(n/2)} binomial(n-1,2*k-1)*binomial(2*k,k). Cf. A097893.
n*(n-2)*a(n) = (2*n-3)*(n-1)*a(n-1) + 3*(n-1)*(n-2)*a(n-2) with a(1) = 0, a(2) = 2. (End)
From Peter Luschny, Oct 29 2015: (Start)
a(n) = 2*(n-1)*hypergeom([1-n/2,3/2-n/2],[2],4).
a(n) = (n-1)!*[x^(n-1)](2*exp(x)*BesselI(1,2*x)).
a(n) = (n-1)*A007971(n) for n>=2.
A105696(n) = a(n-1) + a(n) for n>=2.
A162551(n-2) = (1/2)*Sum_{k=1..n} binomial(n,k)*a(k) for n>=2.
A079309(n) = (1/2)*Sum_{k=1..2*n} (-1)^k*binomial(2*n,k)*a(k) for n>=1.
(End)

New name based on a comment by T. D. Noe, Mar 16 2005, offset set to 1 and a(1) = 0 prepended by Peter Luschny, Nov 04 2015

A163771 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 10, 14, 20, 19, 26, 36, 50, 70, 51, 70, 96, 132, 182, 252, 141, 192, 262, 358, 490, 672, 924, 393, 534, 726, 988, 1346, 1836, 2508, 3432, 1107, 1500, 2034, 2760, 3748, 5094, 6930, 9438, 12870

Offset: 0

Peter Luschny, Aug 05 2009

Triangle read by rows. For n >= 0, k >= 0 let T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i (A056040).

This is also the square array of central binomial coefficients A000984 in column 0 and higher (first: A051924, second, etc.) differences in subsequent columns, read by antidiagonals. - M. F. Hasler, Nov 15 2019

			Triangle begins
    1;
    1,   2;
    3,   4,   6;
    7,  10,  14,  20;
   19,  26,  36,  50,  70;
   51,  70,  96, 132, 182, 252;
  141, 192, 262, 358, 490, 672, 924;
From _M. F. Hasler_, Nov 15 2019: (Start)
The square array having central binomial coefficients A000984 in column 0 and higher differences in subsequent columns (col. 1 = A051924) starts:
     1   1    3    7    19    51 ...
     2   4   10   26    70   192 ...
     6  14   36   96   262   726 ...
    20  50  132  358   988  2760 ...
    70 182  490 1346  3748 10540 ...
   252 672 1836 5094 14288 40404 ...
  (...)
Read by falling antidiagonals this yields the same sequence. (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Row sums are A163774. Cf. A056040, A163650, A163771, A163772, A002426, A000984.

For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
a := n -> DiffTria(k->swing(2*k),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

A168597 Squares of the central trinomial coefficients (A002426).

Original entry on oeis.org

1, 1, 9, 49, 361, 2601, 19881, 154449, 1225449, 9853321, 80156209, 658076409, 5444816521, 45343869481, 379735715529, 3195538786449, 27004932177129, 229066136374761, 1949470542590481, 16640188083903609, 142415188146838161, 1221800234100831441, 10504959504381567729

Offset: 0

Paul D. Hanna, Dec 01 2009

Ignoring initial term, a(n) equals the logarithmic derivative of A168598.
Partial sums of A007987. Hence, a(n) is the number of irreducible words of length at most 2n in the free group with generators x,y such that the total degree of x and the total degree of y both equal zero. - Max Alekseyev, Jun 05 2011
The number of ways a king, starting at the origin of an infinite chessboard, can return to the origin in n moves, where leaving the king where it is counts as a move. Cf. A094061. - Peter Bala, Feb 14 2017

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

Cf. A002426, A133053, A168598, A243949, A094061.

a := n -> (-1)^n*hypergeom([1/2,-n],[1],4)*hypergeom([1/2-n/2,-n/2],[1], 4): seq(simplify(a(n)),n=0..20); # Peter Luschny, Nov 10 2014

Table[(-1)^n*Hypergeometric2F1[1/2, -n, 1, 4] * Hypergeometric2F1[(1 - n)/2, -n/2, 1, 4], {n, 0, 50}] (* G. C. Greubel, Feb 26 2017 *)
CoefficientList[Series[(2 EllipticK[(16 x)/(1 + 3 x)^2])/(Pi (1 + 3 x)), {x, 0, 28}], x, 26]  (* After Mark van Hoeij, Peter Luschny, May 13 2025 *)

{a(n)=polcoeff((1+x+x^2 +x*O(x^n))^n,n)^2}
for(n=0, 20, print1(a(n), ", "))

/* Using AGM: */
{a(n)=polcoeff( 1 / agm(1+3*x, sqrt((1+3*x)^2 - 16*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 04 2014

a(n) = A002426(n)^2.
G.f.: hypergeom([1/12, 5/12],[1],1728*x^4*(x-1)*(9*x-1)*(3*x+1)^2/(81*x^4-36*x^3-26*x^2-4*x+1)^3)/(81*x^4-36*x^3-26*x^2-4*x+1)^(1/4). - Mark van Hoeij, May 07 2013
G.f.: 1 / AGM(1+3*x, sqrt((1-x)*(1-9*x))), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean. - Paul D. Hanna, Sep 04 2014
G.f.: 1 / AGM((1-x)*(1-3*x), (1+x)*(1+3*x)) = Sum_{n>=0} a(n)*x^(2*n). - Paul D. Hanna, Oct 04 2014
a(n) = (-1)^n*hypergeom([1/2,-n],[1],4)*hypergeom([(1-n)/2,-n/2],[1],4). - Peter Luschny, Nov 10 2014
a(n) ~ 3^(2*n+1) / (4*Pi*n). - Vaclav Kotesovec, Sep 28 2019
From Peter Bala, Feb 08 2022: (Start)
a(n) = Sum_{k = 0..n} (-3)^(n-k)*binomial(2*k,k)*binomial(n,k)* binomial(n+k,k).
n^2*(2*n-3)*a(n)= (7*n^2-14*n+6)*(2*n-1)*a(n-1) + 3*(7*n^2-14*n+6)*(2*n-3)*a(n-2) - 27*(2*n-1)*(n-2)^2*a(n-3) with a(0) = 1, a(1) = 1 and a(2) = 9.
G.f.: A(x) = Sum_{n >= 0} binomial(2*n,n)^2*x^n/(1 + 3*x)^(2*n+1).
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 primes p >= 5 and positive integers n and k. (End)
G.f.: hypergeom([1/2, 1/2],[1],16*x/(1+3*x)^2)/(1+3*x). - Mark van Hoeij, May 13 2025

A248133 Least positive integer m such that m + n divides T(m) + T(n), where T(.) is given by A002426.

Original entry on oeis.org

1, 3, 1, 1, 7, 2, 2, 2, 1, 1, 7, 4, 37, 145, 35, 1, 25, 16, 5, 16, 1, 1, 18, 19, 3, 11, 41, 1, 7, 2, 48, 415, 1, 2, 15, 7, 13, 34, 97, 1, 27, 18, 56, 22, 1, 1, 5, 26, 22, 36, 18, 1, 117, 52, 376, 11, 1, 1, 23, 26

Offset: 1

Zhi-Wei Sun, Oct 02 2014

nonn

Conjecture: a(n) exists for any n > 0. Moreover, a(n) <= n^2 - n + 1 except for n = 274.

Note that a(274) = 188847 > 2*274^2.

			a(5) = 7 since 5 + 7 divides T(5) + T(7) = 51 + 393 = 444 = 12*37.
a(2539) = 643425 since 2539 + 643425 = 645964 divides T(2539) + T(643425).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A002426, A247824, A247937, A247940, A248124, A248125, A248136, A248137, A248139, A248142.

T[n_]:=Sum[Binomial[n,2k]Binomial[2k,k],{k,0,n/2}]
Do[m=1;Label[aa];If[Mod[T[m]+T[n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]

A113302 Number of k such that prime(n) divides T(k), the central trinomial coefficient A002426(k), with 0

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 0, 0, 4, 2, 3, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 3, 4, 0, 2, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 3, 0, 0, 4, 2, 2, 4, 0, 0, 3, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 6, 2, 2, 0, 0, 2, 0, 4, 2, 0

Offset: 1

T. D. Noe, Oct 24 2005

nonn

For primes less than 10^6, a(n) <= 10. Is 10 the largest possible value? When a(n)=0, prime(n) is in A113305. When a(n)>0, prime(n) is in A113304.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. 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}]]; Table[p=Prime[i]; cnt=0; Do[If[Mod[t[[j]], p]==0, cnt++ ], {j, p}]; cnt, {i, PrimePi[nn]}]

A113303 Least number k such that prime(n) divides T(k), the central trinomial coefficient A002426(k), or 0 if there is no such k.

Original entry on oeis.org

0, 2, 0, 3, 0, 0, 5, 4, 0, 0, 0, 0, 8, 9, 6, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 26, 47, 12, 0, 7, 0, 0, 0, 0, 0, 0, 0, 28, 44, 0, 31, 51, 0, 61, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 89, 0, 90, 135, 98, 48, 122, 120, 0, 0, 0, 0, 121, 0, 0, 0, 0, 112, 0, 91, 0, 0, 55, 63, 133, 48, 0, 0, 126, 179, 0, 78

Offset: 1

T. D. Noe, Oct 24 2005

nonn

When a(n)=0, prime(n) is in A113305. When a(n)>0, prime(n) is in A113304.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A113302 (number of k for which 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}]]; Table[p=Prime[i]; k=1; While[k0, k++ ]; If[k==nn, 0, k], {i, PrimePi[nn]}]

cp's OEIS Frontend

A007971 INVERTi transform of central trinomial coefficients (A002426).

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A242170 Least prime divisor of T(n) which does not divide any T(k) with k < n, or 1 if such a primitive prime divisor of T(n) does not exist, where T(n) is the n-th central trinomial coefficient given by A002426.

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A097893 Partial sums of the central trinomial coefficients (A002426).

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A277640 a(n) is the integer r with |r| < prime(n)/2 such that (T(prime(n)^2)-T(prime(n)))/prime(n)^4 == r (mod prime(n)), where T(k) denotes the central trinomial coefficient A002426(k).

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A025178 First differences of the central trinomial coefficients A002426.

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A163771 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).

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A168597 Squares of the central trinomial coefficients (A002426).

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