A002426 -id:A002426 - cp's OEIS frontend

A179089 a(n) = (1/n^2) * Sum_{k=0..n-1} (2k+1)*T_k^2(-3)^(n-1-k), where T_0, T_1, ... are central trinomial coefficients given by A002426.

1, 0, 5, 13, 105, 576, 4005, 27000, 193193, 1402672, 10433709, 78807785, 603996745, 4683970032, 36702939429, 290184446349, 2312460578025, 18556825469040, 149842592021997, 1216719520281045, 9929612901775761, 81406058258856240

Offset: 1

Zhi-Wei Sun, Jun 29 2010

nonn

On Jun 28 2010, Zhi-Wei Sun conjectured that a(n) is an integer for every n=1,2,3,... and that a(p) = (1+p/3)/2 (mod p) for any prime p, where (p/3) is the Legendre symbol. In contrast, he showed that Sum_{k=0..n-1} (2k+1)*T_k^2*3^(n-1-k) = n*T_n*T_{n-1} for all n=1,2,3,...

The formula for a(n) in the formula section implies that a(n) is an integer. - Mark van Hoeij, Nov 13 2022

			For n = 4 we have a(4) = (T_0^2(-3)^3 + 3*T_1^2(-3)^2 + 5*T_2^2(-3) + 7*T_3^2)/4^2 = (-27 + 27 - 5*27 + 7^3)/16 = 13.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers, preprint, arXiv:1006.2776 [math.NT], 2010-2011.

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

A002426 := n -> simplify(GegenbauerC(n, -n, -1/2)); seq( (A002426(n)+A002426(n-1))*(3*A002426(n-1)-A002426(n))/4, n=1..20); # Mark van Hoeij, Nov 13 2022

TT[n_]:=Sum[Binomial[n,2k]Binomial[2k,k],{k,0,Floor[n/2]}] SS[n_]:=Sum[(2k+1)*TT[k]^2*(-3)^(n-1-k),{k,0,n-1}]/n^2 Table[SS[n],{n,1,50}]

G.f.: Integral(hypergeom([1/2, 3/2], [2], 16*x/(1 + 3*x)^2)/(1 + 3*x)^2). - Mark van Hoeij, Nov 10 2022

a(n) = (A002426(n)+A002426(n-1))*(3*A002426(n-1)-A002426(n))/4. - Mark van Hoeij, Nov 13 2022

A186038 a(n) = log_3(A002426(n)/numerator(A002426(n)/3^n)).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 3, 0, 0, 1, 0, 0, 1, 2, 2, 4, 1, 1, 2, 1, 1, 2, 2, 2, 5, 0, 0, 1, 0, 0, 1, 1, 1, 3, 0, 0, 1, 0, 0, 1, 5, 3, 4, 2, 2, 3, 2, 2, 3, 3, 3, 6, 1, 1, 2, 1, 1, 2, 2, 2, 4, 1, 1, 2, 1, 1, 2, 3, 3, 6, 2, 2, 3, 2, 2, 3, 3, 3, 7

Offset: 0

Paul Barry, Feb 11 2011

(-1)^a(n) = A186039(n).

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

A186038 := proc(n)
    local m,ml ;
    m := A002426(n) ;
    ml := %/3^n ;
    m/numer(ml) ;
    ilog[3](%) ;
end proc: # R. J. Mathar, Feb 13 2015

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

A216584 a(n) = A002426(n)*A000984(n); product of central trinomial coefficients and central binomial coefficients.

Original entry on oeis.org

1, 2, 18, 140, 1330, 12852, 130284, 1348776, 14247090, 152618180, 1654120468, 18096447096, 199536967084, 2214714164600, 24720932068200, 277289164574640, 3123590583844530, 35318969120870820, 400692715550057700, 4559427798654821400, 52020436064931914580

Offset: 0

Paul D. Hanna, Sep 08 2012

nonn

			L.g.f.: L(x) = 2*x + 18*x^2/2 + 140*x^3/3 + 1330*x^4/4 + 12852*x^5/5 + 130284*x^6/6 + ...
where
exp(L(x)) = 1 + 2*x + 11*x^2 + 66*x^3 + 485*x^4 + 3842*x^5 + 32712*x^6 + ... + A216585(n)*x^n/n + ...
The central trinomial coefficients (A002426) begin:
[1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, ...];
The central binomial coefficients (A000984) begin:
[1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, ...].

G. C. Greubel, Table of n, a(n) for n = 0..925

Cf. A216585, A151341, A168595, A002426, A000984.

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

{a(n) = polcoeff((1+x+x^2)^n,n) * polcoeff((1+2*x+x^2)^n,n)}

{a(n)=binomial(2*n,n)*sum(k=0,n\2,binomial(n,2*k)*binomial(2*k,k))}
for(n=0,21,print1(a(n),", "))

a(n) = binomial(2*n, n) * Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k, k).
Logarithmic derivative of A216585, after ignoring initial term a(0).
a(n) = [x^n*y^n] ( 1 + (x + y)^2 + (x + y)^4 )^n. - Peter Bala, Feb 17 2020
G.f.: hypergeom([1/2, 1/2],[1],16*x/(1+4*x))/sqrt(1+4*x). - Mark van Hoeij, May 13 2025

A216585 G.f.: exp( Sum_{n>=1} A000984(n)A002426(n)x^n/n ), where A000984 is the central binomial coefficients and A002426 is the central trinomial coefficients.

Original entry on oeis.org

1, 2, 11, 66, 485, 3842, 32712, 291568, 2697610, 25679316, 250190125, 2484270622, 25062816127, 256275246582, 2650947762450, 27697861115740, 291943603838698, 3101066786857876, 33167191013319532, 356924515784037128, 3862299973917286526, 42003704374124712172

Offset: 0

Paul D. Hanna, Sep 09 2012

nonn

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 66*x^3 + 485*x^4 + 3842*x^5 + 32712*x^6 +...
such that
log(A(x)) = 2*1*x + 6*3*x^2/2 + 20*7*x^3/3 + 70*19*x^4/4 + 252*51*x^5/5 + 924*141*x^6/6 +...+ A000984(n)*A002426(n)*x^n/n +...

Cf. A216584, A216586, A002426, A000984.

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

Logarithmic derivative yields A216584.

A225328 a(n) = A002426(n)^n, where A002426 is the central trinomial coefficients.

Original entry on oeis.org

1, 1, 9, 343, 130321, 345025251, 7858047974841, 1447930954097073657, 2255178731296086753063201, 29588424532574699588724679418659, 3308916781795356089160906125431831800049, 3166064605712293355286523525163381509588445189997

Offset: 0

Paul D. Hanna, Aug 03 2013

nonn

Logarithmic derivative of A168599 (upon ignoring the initial term, a(0), of this sequence).

			L.g.f.: L(x) = x + 9*x^2/2 + 343*x^3/3 + 130321*x^4/4 + 345025251*x^5/5 + ...
where exponentiation is an integer series:
exp(L(x)) = 1 + x + 5*x^2 + 119*x^3 + 32707*x^4 + 69038213*x^5 + 1309743837515*x^6 + ... + A168599(n)*x^n + ...

G. C. Greubel, Table of n, a(n) for n = 0..46

Cf. A168599, A002426.

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

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

L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=0} A168599(n)*x^n ).

A059728 a(0)=3; thereafter, a(n) = A002426(n+1) + Fibonacci(n-1)*(Fibonacci(n-1) + 1).

Original entry on oeis.org

3, 3, 9, 21, 57, 153, 423, 1179, 3321, 9415, 26843, 76869, 220951, 637107, 1842129, 5339133, 15507641, 45127965, 131548859, 384059009, 1122835671, 3286907517, 9633053985, 28262033613, 82998088607, 243963263943, 717698981853, 2112976735749

Offset: 0

N. J. A. Sloane, Feb 09 2001

nonn

			a(6) = F_5*(F_5+1) + A002426(7) = 30 + 393 = 423.

G. C. Greubel, Table of n, a(n) for n = 0..1000
G. E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
L. Euler, (E326) Observationes analyticae, also reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 54.

P:=PolynomialRing(Integers()); [Max(Coefficients((1+x+x^2)^(n+1)))+(Fibonacci(n-1)*(Fibonacci(n-1)+1)): n in [0..30]]; // Vincenzo Librandi, Dec 24 2016

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

A080896 Expansion of the exponential series exp( x * T(x) ) = exp( x / sqrt(1 - 2x - 3x^2) ), where T(x) is the ordinary generating series of the central trinomial coefficients (A002426).

Original entry on oeis.org

1, 1, 3, 25, 265, 3741, 64051, 1298053, 30295665, 800411545, 23601417571, 768200763441, 27352316065273, 1057402991121205, 44102326806885075, 1973793512480683741, 94345589402816289121, 4796647490592139950513

Offset: 0

Emanuele Munarini, Mar 31 2003

G. C. Greubel, Table of n, a(n) for n = 0..379 (terms 0..100 from T. D. Noe)

Cf. A002426.

With[{nn=20},CoefficientList[Series[Exp[x/Sqrt[1-2 x-3x^2]],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Sep 08 2011 *)

x='x + O('x^50); Vec(serlaplace(exp(x/sqrt(1 - 2*x - 3*x^2)))) \\ G. C. Greubel, Feb 27 2017

E.g.f.: exp(x/sqrt(1 - 2*x - 3*x^2)).

A087178 ((1/2)/p^2)*(T(p)-1) where p runs through the prime >= 5 and T(k) is the k-th central trinomial number (A002426).

Original entry on oeis.org

1, 4, 106, 630, 26185, 178666, 8991709, 3678166249, 28031525940, 13143231877116, 824038411943548, 6585233816779471, 427212999940414402, 230740449590738871715, 128696758387000610268418, 1065763722815793191814814, 614675485079469646826034034

Offset: 1

Benoit Cloitre, Oct 19 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..317

Cf. A002426.

f[0] = f[1] = 1; f[n_] := f[n] = ((2*n-1)*f[n-1] + 3*(n-1)*f[n-2])/n; Table[(f[p]-1)/(2*p^2), {p, Prime[Range[3, 20]]}] (* Amiram Eldar, Apr 21 2025 *)

T(n) == 1 (mod n^2) iff n is prime.

A152227 Eigentriangle, row sums = A002426.

Original entry on oeis.org

1, 2, 1, 2, 2, 3, 4, 2, 6, 7, 8, 4, 6, 14, 19, 18, 8, 12, 14, 38, 51, 42, 18, 24, 28, 38, 102, 141, 102, 42, 54, 56, 76, 102, 282, 393254, 102, 126, 126, 152, 204, 282, 786, 1107, 646, 254, 306, 2944, 342, 408, 564, 786, 2214, 3139

Offset: 1

Gary W. Adamson, Nov 29 2008

Row sums = A002426 starting with offset 1: (1, 3, 7, 19, 51, 141, 393,...).
Right border = A002426, left border = A007971
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
2, 1;
2, 2, 3;
4, 2, 6, 7;
8, 4, 6, 14, 19;
18, 8, 12, 14, 38, 51;
42, 18, 24, 28, 38, 102, 141;
102, 42, 54, 56, 76, 102, 282, 393;
254, 102, 126, 126, 152, 204, 282, 786, 1107;
646, 254, 306, 394, 342, 408, 564, 786, 2214, 3139;
...
Row 3 = (4, 2, 6, 7) = termwise products of (4, 2, 2, 1) and (1, 1, 3, 7)

Triangle read by rows, M*Q. M = an infinite lower triangular matrix with A007971 in every column: (1, 2, 2, 4, 8, 18, 42,...); and Q = a matrix with A002426 as the main diagonal and the rest zeros.

A162533 a(n) = Sum_{k=0..n} binomial(n,2k)*A002426(k).

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 68, 176, 454, 1174, 3052, 7976, 20932, 55108, 145448, 384704, 1019462, 2706214, 7194956, 19155896, 51065260, 136284236, 364097912, 973654240, 2605983772, 6980545276, 18712478072, 50196568144, 134739960904, 361892443592, 972537193168

Offset: 0

Paul Barry, Jul 05 2009

Hankel transform is (-1)^binomial(n,2)*(-2)^A128054(n) (see A128055).

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

Cf. A027826.

b[n_] := If[n < 0, 0, 3^n Hypergeometric2F1[1/2, -n, 1, 4/3]]; Table[Sum[Binomial[n, 2*k]*b[k], {k, 0, n}], {n, 0, 50}] (* or *) CoefficientList[Series[(1-x)/sqrt(1-4*x+4*x^2-4*x^4), {x, 0, 50}], x] (* G. C. Greubel, Feb 27 2017 *)

x='x+O('x^50); Vec((1-x)/sqrt(1-4*x+4*x^2-4*x^4)) \\ G. C. Greubel, Feb 27 2017

G.f.: (1-x)/((1-x)^2-x^2-2x^4/((1-x)^2-x^2-x^4/((1-x)^2-x^2-x^4/(1-... (continued fraction).
G.f.: (1-x)/sqrt(1-4*x+4*x^2-4*x^4) = (1-x)/sqrt((1-2*x)^2-4*x^4) = (1-x)/sqrt((1-x-2*x^2)*(1-x+2*x^2)). - Paul Barry, Oct 13 2009
Conjecture: n*a(n) + (4-5*n)*a(n-1) + 2*(4*n-7)*a(n-2) + 4*(3-n)*a(n-3) + 4*(2-n)*a(n-4) + 4*(n-4)*a(n-5) = 0. - R. J. Mathar, Nov 16 2011
a(n) ~ 3^(1/4) * (1 + sqrt(3))^(n + 1/2) / (2^(3/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jun 08 2019

cp's OEIS Frontend

A179089 a(n) = (1/n^2) * Sum_{k=0..n-1} (2k+1)*T_k^2(-3)^(n-1-k), where T_0, T_1, ... are central trinomial coefficients given by A002426.

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A186038 a(n) = log_3(A002426(n)/numerator(A002426(n)/3^n)).

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Mathematica

A216584 a(n) = A002426(n)*A000984(n); product of central trinomial coefficients and central binomial coefficients.

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A216585 G.f.: exp( Sum_{n>=1} A000984(n)*A002426(n)*x^n/n ), where A000984 is the central binomial coefficients and A002426 is the central trinomial coefficients.

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A225328 a(n) = A002426(n)^n, where A002426 is the central trinomial coefficients.

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A059728 a(0)=3; thereafter, a(n) = A002426(n+1) + Fibonacci(n-1)*(Fibonacci(n-1) + 1).

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Magma

Mathematica

A080896 Expansion of the exponential series exp( x * T(x) ) = exp( x / sqrt(1 - 2*x - 3*x^2) ), where T(x) is the ordinary generating series of the central trinomial coefficients (A002426).

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A087178 ((1/2)/p^2)*(T(p)-1) where p runs through the prime >= 5 and T(k) is the k-th central trinomial number (A002426).

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A216585 G.f.: exp( Sum_{n>=1} A000984(n)A002426(n)x^n/n ), where A000984 is the central binomial coefficients and A002426 is the central trinomial coefficients.

A080896 Expansion of the exponential series exp( x * T(x) ) = exp( x / sqrt(1 - 2x - 3x^2) ), where T(x) is the ordinary generating series of the central trinomial coefficients (A002426).