A002426 -id:A002426 - cp's OEIS frontend

A181147 a(n) = Sum_{k=0..n-1} (105k+44)C(2k,k)^2T(k)(-1)^(n-1-k)/(2nC(2n,n)), where T(k) (k=0,1,2,...) are central trinomial coefficients given by A002426.

11, 23, 224, 1747, 16754, 162392, 1651206, 17126327, 181182446, 1943132842, 21080299228, 230802972664, 2546569337336, 28280754214358, 315824396838386, 3544003431783795, 39936833763112790, 451718158386620678

Offset: 1

Zhi-Wei Sun, Jan 24 2011

nonn

On Jan 22 2011, Zhi-Wei Sun conjectured that a(n) is a positive integer for every n=1,2,3,... Let p > 3 be a prime. He also conjectured that a(p) == 5 + 6*(p/3)*(2-3^(p-1)) (mod p^2). Another conjecture of his states that Sum_{k=0..p-1} (-1)^k*binomial(2k,k)^2*T(k) is congruent to b(p) modulo p^2, where b(p)=0 if (p/15)=-1, b(p) = 4x^2-2p if p == 1,4 (mod 15) and p = x^2+15y^2 with x,y integers, and b(p) = 20x^2-2p if p == 2,8 (mod 15) and p=5x^2+3y^2 with x,y integers.

			For n=2 we have a(2) = (44*1^2*T(0)(-1) + (105+44)*2^2*T(1))/(2*2*binomial(4,2)) = 23.

D. S. McNeil, Table of n, a(n) for n = 1..900
Zhi-Wei Sun, Open Conjectures on Congruences, preprint, arXiv:0911.5665 [math.NT], 2009-2011.
Zhi-Wei Sun, On sums related to central binomial and trinomial coefficients, preprint, arXiv:1101.0600 [math.NT], 2011-2014.

Cf. A000984, A002426, A173774.

T:= n-> `if`(n=0, 1, coeff ((x^2+x+1)^n, x, n)):
a:= n-> add ((105*k+44) *binomial (2*k, k)^2 *T(k)*(-1)^(n-1-k),
             k=0..n-1)/ (2*n*binomial (2*n,  n)):
seq (a(n), n=1..30);

T[k_]:=If[k>0,Coefficient[(x^2+x+1)^k,x^k],1]
A[n_]:=Sum[(105k+44)Binomial[2k,k]^2*T[k](-1)^(n-1-k),{k,0,n-1}]/(2n*Binomial[2n,n])
Table[A[n],{n,1,50}]

A186037 a(n) = log_2((1+A002426(n))/numerator((1+A002426(n))/2^n)).

Original entry on oeis.org

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

Offset: 0

Paul Barry, Feb 11 2011

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

b[n_] := If[n < 0, 0, 3^n Hypergeometric2F1[1/2, -n, 1, 4/3]];
a[n_] := Log[2, (1 + b[n])/Numerator[(1 + b[n])/2^n]]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Feb 27 2017 *)

A192670 Floor-Sqrt transform of central trinomial coefficients (A002426).

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 11, 19, 33, 56, 94, 160, 271, 461, 785, 1337, 2279, 3890, 6644, 11357, 19426, 33246, 56930, 97535, 167177, 286665, 491745, 843842, 1448530, 2487302, 4272238, 7340067, 12614059, 21682694, 37279415, 64108676, 110268453, 189700207, 326408942, 561733093, 966869803

Offset: 0

Emanuele Munarini, Jul 07 2011

nonn

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

FSFromSeries[f_,x_,n_] := Map[Floor[Sqrt[#]]&,CoefficientList[Series[f,{x,0,n}],x]]
FSFromSeries[1/Sqrt[1-2x-3x^2],x,100]

a(n) = floor(sqrt(trinomial(2*n,n))).

A212848 Least prime factor of n-th central trinomial coefficient (A002426).

Original entry on oeis.org

1, 1, 3, 7, 19, 3, 3, 3, 3, 43, 7, 3, 113, 73, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 7, 17, 3, 719, 7, 3, 3, 3, 3, 967, 9539, 3, 17, 47, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 19

Offset: 0

Jonathan Vos Post, May 28 2012

A002426(n) is prime for n = 2, 3, 4, no more through 10^5. A002426 is semiprime iff A102445(n) = 2 (as is the case for n = 5, 6, 7, 9, 10, 12, 13).

			a(9) = 43 because A002426(9) = 3139 = 43 * 73.

Robert Israel, Table of n, a(n) for n = 0..729

Cf. A000040, A002426, A020639, A102445, A212791.

A002426:= gfun:-rectoproc({(n+2)*a(n+2)-(2*n+3)*a(n+1)-3*(n+1)*a(n) = 0, a(0)=1, a(1)=1},a(n),remember):
lpf:= proc(n) local F;
    F:= map(proc(t) if t[1]::integer then t[1] else NULL fi end proc,
       ifactors(n, easy)[2]);
    if nops(F) > 0 then min(F)
    else min(numtheory:-factorset(n))
    fi
end proc:
lpf(1):= 1:
map(lpf @ A002426, [$0..100]); # Robert Israel, Jun 20 2017

a = b = 1; t = Join[{a, b}, Table[c = ((2 n - 1) b + 3 (n - 1) a)/n; a = b; b = c; c, {n, 2, 100}]]; Table[FactorInteger[n][[1, 1]], {n, t}] (* T. D. Noe, May 30 2012 *)

a(n) = my(x=polcoeff((1 + x + x^2)^n, n)); if (x==1, 1, vecmin(factor(x)[,1])); \\ Michel Marcus, Jun 20 2017

a(n) = A020639(A002426(n)).

A216586 G.f.: exp( Sum_{n>=1} A002426(n)/2 * A002426(n) * x^n/n ), where A002426 is the central binomial coefficients and A002426 is the central trinomial coefficients.

Original entry on oeis.org

1, 1, 5, 28, 202, 1579, 13375, 118858, 1098458, 10453452, 101872926, 1012109860, 10218226307, 104570617520, 1082633236498, 11321654913838, 119438468577559, 1269787015989428, 13592294300856138, 146390465351654178, 1585337895099162317, 17253991887494062080

Offset: 0

Paul D. Hanna, Sep 09 2012

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 28*x^3 + 202*x^4 + 1579*x^5 + 13375*x^6 +...
such that
log(A(x)) = 1*1*x + 3*3*x^2/2 + 10*7*x^3/3 + 35*19*x^4/4 + 126*51*x^5/5 + 462*141*x^6/6 +...+ A001700(n)*A002426(n)*x^n/n +...

Cf. A216585, A002426, A001700, A000984.

{a(n)=polcoeff(exp(sum(m=1,n+1,binomial(2*m,m)/2*polcoeff((1+x+x^2)^m,m)*x^m/m+x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))

Self-convolution yields A216585.

A225604 G.f.: exp( Sum_{n>=1} A002426(n^2) * x^n/n ), where A002426 is the central trinomial coefficients.

Original entry on oeis.org

1, 1, 10, 1056, 1300253, 16436676927, 2026538428535847, 2377041996570919354629, 26137381916593225072659360863, 2668615348740645885804068311893052895, 2513426521807431879643802805359800329740903335, 21735453667359385540995804455408000917620356989063370267

Offset: 0

Paul D. Hanna, Aug 03 2013

nonn

			G.f.: A(x) = A(x) = 1 + x + 10*x^2 + 1056*x^3 + 1300253*x^4 + 16436676927*x^5 +...
where
log(A(x)) = x + 19*x^2/2 + 3139*x^3/3 + 5196627*x^4/4 + 82176836301*x^5/5 +...+ A225602(n)*x^n/n +...

Cf. A225602, A002426.

{A002426(n)=sum(k=0,n, binomial(n, k)*binomial(k, n-k))}
{a(n)=polcoeff(exp(sum(m=1,n+1,A002426(m^2)*x^m/m) +x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

Logarithmic derivative yields A225602.

A231373 G.f. A(x) satisfies: A(x-x^2-x^3) = 1/sqrt(1-2x-3x^2), which is the g.f. the central trinomial coefficients (A002426).

Original entry on oeis.org

1, 1, 4, 16, 71, 327, 1550, 7490, 36720, 182028, 910330, 4585318, 23233722, 118315318, 605088690, 3105994302, 15994906965, 82602799485, 427662046960, 2219130114108, 11538302709769, 60102637378353, 313591732265662, 1638671208390738, 8574718477933404, 44926247350136232

Offset: 0

Paul D. Hanna, Nov 08 2013

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 71*x^4 + 327*x^5 + 1550*x^6 +...
where A(x-x^2-x^3)^2 = 1/(1-2*x-3*x^2):
A(x-x^2-x^3) = 1 + x + 3*x^2 + 7*x^3 + 19*x^4 + 51*x^5 + 141*x^6 + 393*x^7 + 1107*x^8 +...+ A002426(n)*x^n +...
The square of the g.f. begins (cf. A038112):
A(x)^2 = 1 + 2*x + 9*x^2 + 40*x^3 + 190*x^4 + 924*x^5 + 4578*x^6 +...
such that A(x)^2 = d/dx x*G(x) where G(x) is the g.f. of A001002:
G(x) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 154*x^5 + 654*x^6 +...
and satisfies G(x-x^2-x^3) = 1/(1-x-x^2).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A002426, A038112, A001002.

CoefficientList[Series[Sqrt[D[InverseSeries[Series[x - x^2 - x^3, {x, 0, 30}], x], x]], {x, 0, 30}], x] (* Vaclav Kotesovec, Mar 31 2014 *)

{a(n)=local(G=serreverse(x-x^2-x^3+x^2*O(x^n)),A);A=sqrt(deriv(G));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} \\ = d^n/dx^n F
{a(n)=local(A2=x); A2=1+sum(m=1, n+1, Dx(m, x^(2*m)*(1+x +x*O(x^n))^m/m!)); polcoeff(sqrt(A2), n)}
for(n=0,30,print1(a(n),", "))

Self-convolution yields A038112.
G.f. A(x) satisfies:
(1) A(x) = sqrt( Sum_{n>=0} d^n/dx^n x^(2*n)*(1+x)^n/n! ).
(2) A(x) = sqrt((1+x)*(5-27*x)*A(x)^6 - 1)/2, from a formula by Mark van Hoeij in A038112.
(3) A(x) = sqrt( d/dx x*G(x) ) where G(x) = Series_Reversion(x-x^2-x^3)/x is the g.f. of A001002.
(4) A(x) = 1/sqrt(1 - 2*x*G(x) - 3*x^2*G(x)^2) where G(x) = Series_Reversion(x-x^2-x^3)/x is the g.f. of A001002.
Sum_{k=0..n} a(k)*a(n-k) = Sum_{k=0..n} C(n+k, k)*C(k, n-k), from a formula by Paul Barry in A038112.
Recurrence: 25*(n-2)*(n-1)*n*a(n) = 110*(n-2)*(n-1)*(2*n-3)*a(n-1) - (n-2)*(214*n^2 - 856*n + 717)*a(n-2) - 33*(2*n-5)*(18*n^2 - 90*n + 113)*a(n-3) - 81*(n-3)*(3*n-11)*(3*n-7)*a(n-4). - Vaclav Kotesovec, Nov 10 2013
a(n) ~ 3^(3/4) * GAMMA(3/4) * (27/5)^n / (2*10^(1/4)*Pi*n^(3/4)). - Vaclav Kotesovec, Dec 29 2013

A329475 a(n) = Sum_{k=0..n} C(n,k)^2T(k)T(n-k), where T(k) = A002426(k) is the coefficient of x^k in the expansion of (x^2+x+1)^k.

Original entry on oeis.org

1, 2, 10, 68, 586, 5252, 49204, 475400, 4723786, 47937812, 494786260, 5177188040, 54794164660, 585565913480, 6309889976680, 68484312535568, 747985368753226, 8214968193003860, 90669516557975524, 1005156080857529768, 11187435500257898836, 124964856185950621832

Offset: 0

Zhi-Wei Sun, Nov 13 2019

nonn

The author introduced this sequence in arXiv:1911.05456 and made the following conjecture.
Conjecture: Let p be an odd prime and let S = Sum_{k=0..p-1}a(k)/(-4)^k. If p == 1 (mod 12) and p = x^2 + 9*y^2 with x and y integers, then  S == 4*x^2-2*p (mod p^2). If p == 5 (mod 12) and p = x^2 + y^2 with x == y (mod 3), then S == 4*x*y (mod p^2). If p == 3 (mod 4), then S == 0 (mod p^2).
Note that if p > 3 is a prime, then a(p-1) == Sum_{k=0..p-1} T(k)*T(p-1-k) == Legendre(p/3)*Sum_{k=0..p-1}T(k)^2/(-3)^k == 1 (mod p) by (1.7) and (2.3) of the author's 2014 paper in Sci. China Math.

			a(1) = 2 since Sum_{k=0,1} C(1,k)^2*T(k)*T(1-k) = C(1,0)^2*T(0)*T(1) + C(1,1)^2*T(1)*T(0) = 2*T(0)*T(1) = 2*1*1 = 2.

Seiichi Manyama, Table of n, a(n) for n = 0..931 (terms 0..150 from Zhi-Wei Sun)
Zhi-Wei Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57(2014), no.7, 1375-1400.
Zhi-Wei Sun, On sums related to central binomial and trinomial coefficients, in: M. B. Nathanson (ed.), Combinatorial and Additive Number Theory: CANT 2011 and 2012, Springer Proc. in Math. & Stat., Vol. 101, Springer, New York, 2014, pp. 257-312. Also available from arXiv:1101.0600 [math.NT], 2011-2014.
Zhi-Wei Sun, New series for powers of Pi and related congruences, Electron. Res. Arch. 28 (2020), no. 3, 1273--1342.
Zhi-Wei Sun, On central trinomial coefficients, Question 491563 at MathOverflow, April 23, 2025.

Cf. A002426, A002895.

T[0]=1; T[1]=1; T[n_]:=T[n]=((2n-1)T[n-1]+3*(n-1)*T[n-2])/n;
a[n_]:=a[n]=Sum[Binomial[n,k]^2*T[k]*T[n-k],{k,0,n}];
Table[a[n],{n,0,21}]

a(n) ~ (3/2)*12^n/(n*Pi)^(3/2) as n tends to the infinity.

A179100 a(n) = (1/n) * Sum_{k=0..n-1} (8k+5) T_k^2, where T_0, T_1, ... are central trinomial coefficients given by A002426.

Original entry on oeis.org

5, 9, 69, 407, 2997, 22005, 169389, 1325889, 10573677, 85386881, 697013325, 5739021051, 47599593941, 397234035333, 3332690347437, 28089543969855, 237711099004461, 2018856328439841, 17200553934626253, 146966002696538271

Offset: 1

Zhi-Wei Sun, Jun 29 2010

nonn

On Jun 17 2010, Zhi-Wei Sun conjectured that a(n) is an integer for every n=1,2,3,... and that a(p) == 3(p/3) (mod p) for any prime p, where (p/3) is the Legendre symbol. He also observed that Sum_{k=0..n-1} (2k+1) T_k*3^{n-1-k} = n * Sum_{k=0..n-1} C(n-1,k)*(-1)^(n-1-k)*(k+1)*C(2k,k).

			For n=3 we have a(3) = (5*T_0^2 + 13*T_1^2 + 21*T_2^2)/3 = (5 + 13 + 21*9)/3 = 69.

Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers, preprint, arXiv:1006.2776 [math.NT], 2010-2011.

Cf. A002426, A179089, A178808, A178790, A178791, A173774.

TT[n_]:=Sum[Binomial[n,2k]Binomial[2k,k],{k,0,Floor[n/2]}] SS[n_]:=Sum[(8k+5)*TT[k]^2,{k,0,n-1}]/n Table[SS[n],{n,1,50}]

A308482 Composites c such that T_{c-1} == (c/3)*3^(c-1) (mod c), where T_i denotes the i-th central trinomial coefficient (A002426) and (/) denotes the Kronecker symbol.

Original entry on oeis.org

4, 9, 20, 25, 27, 40, 49, 80, 81, 121, 169, 189, 243, 272, 289, 361, 369, 400, 416, 470, 529, 544, 567, 729, 841, 961, 1071, 1323, 1369, 1539, 1681, 1849, 2000, 2187, 2209, 2809, 2889, 3213, 3481, 3721, 4489, 4617, 5041, 5329, 6241, 6561, 6889, 7749, 7921, 8667

Offset: 1

Felix Fröhlich, May 30 2019

nonn

Composites satisfying a weaker version of an analog to a congruence satisfied by all primes > 3 (cf. Cao, Sun, 2015, Theorem 1.1 (i); cf. Sun, 2011, Remark to Conjecture A69).

Up to 9000, 189 is the only composite satisfying the congruence modulo c^2. Do any other such composites exist?

Hui-Qin Cao and Zhi-Wei Sun, Some congruences involving binomial coefficients, Colloquium Mathematicum 139 (2015), 127-136, arXiv:1006.3069 [math.NT], 2010-2015.
Zhi-Wei Sun, Open Conjectures on Congruences, arXiv:0911.5665 [math.NT], 2011.

Cf. A002426.

aQ[n_] := CompositeQ[n] && Divisible[3^(n-1)*(Hypergeometric2F1[1/2, 1-n, 1, 4/3] - JacobiSymbol[n,3]) ,n]; Select[Range[1000], aQ] (* Amiram Eldar, Jul 10 2019 *)

t(n) = sum(k=0, floor(n/2), binomial(n, k)*binomial(n-k, k))
is(n) = Mod(t(n-1), n)==kronecker(n, 3)*3^(n-1)
forcomposite(c=1, , if(is(c), print1(c, ", ")))

a(45)-a(50) from Amiram Eldar, Jul 10 2019

cp's OEIS Frontend

A181147 a(n) = Sum_{k=0..n-1} (105k+44)*C(2k,k)^2*T(k)*(-1)^(n-1-k)/(2n*C(2n,n)), where T(k) (k=0,1,2,...) are central trinomial coefficients given by A002426.

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A186037 a(n) = log_2((1+A002426(n))/numerator((1+A002426(n))/2^n)).

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A192670 Floor-Sqrt transform of central trinomial coefficients (A002426).

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A212848 Least prime factor of n-th central trinomial coefficient (A002426).

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A216586 G.f.: exp( Sum_{n>=1} A002426(n)/2 * A002426(n) * x^n/n ), where A002426 is the central binomial coefficients and A002426 is the central trinomial coefficients.

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A225604 G.f.: exp( Sum_{n>=1} A002426(n^2) * x^n/n ), where A002426 is the central trinomial coefficients.

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A231373 G.f. A(x) satisfies: A(x-x^2-x^3) = 1/sqrt(1-2*x-3*x^2), which is the g.f. the central trinomial coefficients (A002426).

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A329475 a(n) = Sum_{k=0..n} C(n,k)^2*T(k)*T(n-k), where T(k) = A002426(k) is the coefficient of x^k in the expansion of (x^2+x+1)^k.

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A181147 a(n) = Sum_{k=0..n-1} (105k+44)C(2k,k)^2T(k)(-1)^(n-1-k)/(2nC(2n,n)), where T(k) (k=0,1,2,...) are central trinomial coefficients given by A002426.

A231373 G.f. A(x) satisfies: A(x-x^2-x^3) = 1/sqrt(1-2x-3x^2), which is the g.f. the central trinomial coefficients (A002426).

A329475 a(n) = Sum_{k=0..n} C(n,k)^2T(k)T(n-k), where T(k) = A002426(k) is the coefficient of x^k in the expansion of (x^2+x+1)^k.