A268555 -id:A268555 - cp's OEIS frontend

A081798 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2*k,k).

1, 7, 115, 2371, 54091, 1307377, 32803219, 844910395, 22188235867, 591446519797, 15953338537885, 434479441772845, 11927609772412075, 329653844941016785, 9163407745486783435, 255982736410338609931, 7181987671728091545787

Offset: 0

Emanuele Munarini, Apr 23 2003

a(n) is also a generalization of Delannoy numbers to 3D; i.e. the number of walks from (0,0,0) to (n,n,n) in a 3D square lattice where each step is in the direction of one of (1,0,0), (0,1,0), (0,0,1) and (1,1,1). - Theodore Kolokolnikov, Jul 04 2010

Diagonal of the rational function 1/(1 - x - y - z - x*y*z). - Gheorghe Coserea, Jul 06 2016

Alois P. Heinz, Table of n, a(n) for n = 0..500
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
E. W. Weisstein, in MathWorld: Multinomial Coefficient.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Column k = 3 of A229142. Cf. A001850, A082488, A082459, A229049, A229674, A229675, A229676, A229677.

Related to diagonal of rational functions: A268545-A268555.

w := proc(i,j,k) option remember; if i=0 and j=0 and k = 0 then 1; elif i<0 or j<0 or k<0 then 0 else w(i-1,j,k)+w(i,j-1,k)+w(i,j,k-1)+w(i-1,j-1,k-1); end: end: for k from 0 to 10 do lprint(w(k,k,k)):end: # Theodore Kolokolnikov, Jul 04 2010
# second Maple program:
a:= proc(n) option remember; `if`(n<3, 51*n^2-45*n+1,
     ((3*n-2)*(30*n^2-50*n+13)*a(n-1)+(3*n-1)*(n-2)^2*a(n-3)
     -(9*n^3-30*n^2+29*n-6)*a(n-2))/(n^2*(3*n-4)))
    end:
seq(a(n), n=0..20); # Alois P. Heinz, Sep 22 2013

f[n_] := Sum[ Binomial[n, k] Binomial[n + k, k] Binomial[n + 2k, k], {k, 0, n}]; Array[f, 17, 0] (* Robert G. Wilson v *)
CoefficientList[Series[HypergeometricPFQ[{1/3, 2/3}, {1}, 27*x/(1 - x)^3]/(1 - x), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 07 2016 *)

makelist(sum(binomial(n,k)*binomial(n+k,k)*binomial(n+2*k,k),k,0,n),n,0,12);

{a(n)=polcoeff(sum(m=0,n,(3*m)!/m!^3*x^m/(1-x+x*O(x^n))^(3*m+1)), n)} \\ Paul D. Hanna, Sep 22 2013

a(n) = sum(k = 0, n, binomial(n, k) * binomial(n+k, k) * binomial(n+2*k, k)); \\ Michel Marcus, Jan 14 2016

a(n) = w(n,n,n) where w(i,j,k)=w(i-1,j,k)+w(i,j-1,k)+w(i,j,k-1)+w(i-1,j-1,k-1) and where w(0,0,0)=1 and w(i,j,k)=0 if one of i,j,k is strictly negative. - Theodore Kolokolnikov, Jul 04 2010
G.f.: hypergeom([1/3, 2/3],[1],27*x/(1-x)^3)/(1-x). - Mark van Hoeij, Oct 24 2011
G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^n / (1-x)^(3*n+1). - Paul D. Hanna, Sep 22 2013
a(n) ~ c*d^n/(Pi*n), where d = (3*(292 + 4*sqrt(5))^(2/3) + 132 + 20*(292 + 4*sqrt(5))^(1/3)) / (2*(292 + 4*sqrt(5))^(1/3)) = 29.900786688498085... is the root of the equation -1 + 3*d - 30*d^2 + d^3 = 0 and c = 1/(2*sqrt(((81 - 27*sqrt(5))/2)^(1/3) + 3*((3 + sqrt(5))/2)^(1/3) - 6)) = 0.8959908650405192232... is the root of the equation -1 - 72*c^2 - 1296*c^4 + 1728*c^6 = 0. - Vaclav Kotesovec, Sep 23 2013, updated Jul 07 2016
From Peter Bala, Jan 13 2016: (Start)
a(n) = Sum_{k = 0..n} multinomial(n + 2*k, k, k, k, n - k). Cf. A001850(n) = Sum_{k = 0..n} multinomial(n + k, k, k, n - k).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 42*x^3 + 639*x^4 + 11571*x^5  + ... appears to have integer coefficients. (End)
Conjecture: n^2*(3*n-4)*a(n) -(3*n-2)*(30*n^2-50*n+13)*a(n-1) +(9*n^3-30*n^2+29*n-6)*a(n-2) -(3*n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Apr 15 2016
Conjecture: (n^2)*a(n) +(-28*n^2+24*n-3)*a(n-1) +3*(-19*n^2+78*n-77)*a(n-2) +(5*n-12)*(n-3)*a(n-3) -2*(n-3)^2*a(n-4)=0. - R. J. Mathar, Apr 15 2016
0 = (2*x+1)*(x^3-3*x^2+30*x-1)*x*y'' + (6*x^4-8*x^3+51*x^2+60*x-1)*y' + (x-1)*(2*x^2+2*x-7)*y, where y is g.f. - Gheorghe Coserea, Jul 06 2016

A082488 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2k,k) C(n+3*k,k).

Original entry on oeis.org

1, 25, 2641, 392641, 67982041, 12838867105, 2564949195985, 533008982952625, 114035552691160585, 24950692835328410305, 5557138347370070346601, 1255741805437716400557625, 287180884347761929741524361, 66343186345544102086872515761

Offset: 0

Emanuele Munarini, Apr 28 2003

Diagonal of the rational function 1/(1-(x + y + z + w + x*y*z*w)). - Gheorghe Coserea, Jul 15 2016

			G.f.: A(x) = 1 + 25*x + 2641*x^2 + 392641*x^3 + 67982041*x^4 + 12838867105*x^5 +...
where
A(x) = 1/(1-x) + (4!/1!^4)*x/(1-x)^5 + (8!/2!^4)*x^2/(1-x)^9 + (12!/3!^4)*x^3/(1-x)^13 + (16!/4!^4)*x^4/(1-x)^17 + (20!/5!^4)*x^5/(1-x)^21 +... [Hanna]
Equivalently,
A(x) = 1/(1-x) + 24*x/(1-x)^5 + 2520*x^2/(1-x)^9 + 369600*x^3/(1-x)^13 + 63063000*x^4/(1-x)^17 + 11732745024*x^5/(1-x)^21 +...+ A008977(n)*x^n/(1-x)^(4*n+1) +...

Alois P. Heinz, Table of n, a(n) for n = 0..400
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A081798.
Column k = 4 of A229142.
Cf. A008977, A001850, A082489, A229049, A229050.
Related to diagonal of rational functions: A268545-A268555.

End

[&+[Binomial(n, k)*Binomial(n+k, k)*Binomial(n+2*k, k)*Binomial(n+3*k, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Oct 16 2018

with(combinat):
a:= n-> add(multinomial(n+3*k, n-k, k$4), k=0..n):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 23 2013

Table[Sum[Binomial[n,k]*Binomial[n+k,k]*Binomial[n+2*k,k]* Binomial[n+3*k,k], {k, 0, n}],{n,0,20}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (4*m)!/m!^4*x^m/(1-x+x*O(x^n))^(4*m+1)), n)}
for(n=0, 15, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(n, k)*binomial(n+k, k)*binomial(n+2*k, k)*binomial(n+3*k, k))}
for(n=0, 15, print1(a(n), ", "))

G.f.: Sum_{n>=0} (4*n)!/n!^4 * x^n / (1-x)^(4*n+1). - Paul D. Hanna, Sep 22 2013
Recurrence: n^3*(2*n-3)*(4*n-9)*(4*n-5)*a(n) = (4*n-9)*(4*n-3)*(520*n^4 - 1820*n^3 + 2109*n^2 - 905*n + 121)*a(n-1) - (192*n^6 - 1536*n^5 + 4748*n^4 - 7050*n^3 + 5065*n^2 - 1563*n + 171)*a(n-2) + (4*n-1)*(32*n^5 - 296*n^4 + 1040*n^3 - 1689*n^2 + 1209*n - 279)*a(n-3) - (n-3)^3*(2*n-1)*(4*n-5)*(4*n-1)*a(n-4). - Vaclav Kotesovec, Sep 23 2013
a(n) ~ c*d^n/(Pi^(3/2)*n^(3/2)), where d = 65 + 46*sqrt(2) + 2*sqrt(2*(1055 + 746*sqrt(2))) = 259.976980158726979... is the maximal positive root of the equation 1 - 4*d + 6*d^2 - 260*d^3 + d^4 = 0 and c = sqrt(8 + 5*sqrt(2) + sqrt(14*(11 + 8*sqrt(2))))/8 = 0.71529801573844067904424114047445568721... - Vaclav Kotesovec, Sep 23 2013, updated Jul 16 2016
G.f.: hypergeom([1/8, 3/8],[1],256*x/(1-x)^4)^2/(1-x). - Mark van Hoeij, Sep 23 2013
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 13*x^2 + 893*x^3 + 99125*x^4 + 13706093*x^5 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016
0 = x^2*(3*x+1)^2*(1-260*x+6*x^2-4*x^3+x^4)*y''' + 3*x*(3*x+1)*(1-390*x-378*x^2+8*x^3-15*x^4+6*x^5)*y'' + (1-836*x+133*x^2+768*x^3-69*x^4-60*x^5+63*x^6)*y' + (-25+397*x-378*x^2-6*x^3+3*x^4+9*x^5)*y, where y is the g.f. - Gheorghe Coserea, Jul 15 2016

A206178 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 2^k.

Original entry on oeis.org

1, 3, 21, 171, 1521, 14283, 138909, 1385163, 14072193, 145039923, 1512191781, 15914734443, 168802010001, 1802247516891, 19350710547021, 208783189719531, 2262263134211073, 24604815145831011, 268499713118585781, 2938736789722114731, 32250788066104022961

Offset: 0

Paul D. Hanna, Feb 04 2012

nonn

Ignoring initial term, equals the logarithmic derivative of A206177.
Compare to Franel numbers: A000172(n) = Sum_{k=0..n} binomial(n,k)^3.
Diagonal of rational functions 1/(1 - x*y + y*z + 2*x*z - 3*x*y*z), 1/(1 + y + z + x*y + y*z + 2*x*z + 3*x*y*z), 1/(1 - x + 2*z + x*y - y*z - 2*x*z + 3*x*y*z), 1/(1 - x - y - z + x*y + y*z + x*z - 3*x*y*z), 1/(1 - x + y + 2*z - x*y + 2*y*z - 2*x*z - 3*x*y*z). - Gheorghe Coserea, Jul 03 2018

			L.g.f.: L(x) = 3*x + 21*x^2/2 + 171*x^3/3 + 1521*x^4/4 + 14283*x^5/5 +...
Exponentiation equals the g.f. of A206177:
exp(L(x)) = 1 + 3*x + 15*x^2 + 93*x^3 + 657*x^4 + 5067*x^5 + 41579*x^6 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012.

Cf. A206177, A000172, A206180, A216483, A216636.

Related to diagonal of rational functions: A268545-A268555.

Flatten[{1,RecurrenceTable[{(n+3)^2*(3*n+4)*a[n+3]-3*(9*n^3+57*n^2+116*n+74)*a[n+2]-3*(27*n^3+144*n^2+252*n+145)*a[n+1]-27*(3*n+7)*(n+1)^2*a[n]==0, a[1]==3, a[2]==21, a[3]==171},a,{n,1,20}]}] (* Vaclav Kotesovec, Sep 11 2012 *)
Table[HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -2], {n, 0, 20}] (* Jean-François Alcover, Oct 25 2019 *)

PARI
```
{a(n)=sum(k=0,n,binomial(n,k)^3*2^k)}
```

A206178 = lambda n: hypergeometric([-n,-n,-n], [1,1], -2)
[Integer(A206178(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 23 2014

a(2*3^n) == 3 (mod 9) for n>=0; a(n) == 0 (mod 9) if n/2 > 1 is not a power of 3.
Recurrence: (n+3)^2*(3*n+4)*a(n+3) - 3*(9*n^3+57*n^2+116*n+74)*a(n+2) - 3*(27*n^3+144*n^2+252*n+145)*a(n+1) - 27*(3*n+7)*(n+1)^2*a(n) = 0. - Vaclav Kotesovec, Sep 11 2012
a(n) ~ (1 + 2^(1/3))^(3*n + 2) / (2^(4/3)*sqrt(3)*Pi*n). - Vaclav Kotesovec, Sep 19 2012, simplified Apr 24 2025
G.f.: hypergeom([1/3, 2/3],[1],54*x^2/(1-3*x)^3)/(1-3*x). - Mark van Hoeij, May 02 2013
a(n) = hypergeom([-n,-n,-n],[1,1], -2). - Peter Luschny, Sep 23 2014
G.f. y=A(x) satisfies: 0 = x*(3*x + 2)*(27*x^3 + 27*x^2 + 9*x - 1)*y'' + (243*x^4 + 378*x^3 + 189*x^2 + 36*x - 2)*y' + 3*(x + 1)*(27*x^2 + 12*x + 2)*y. - Gheorghe Coserea, Jul 01 2018

Minor edits by Vaclav Kotesovec, Mar 31 2014

A275051 Expansion of 3F2([1/9, 4/9, 5/9], [1/3,1], 729*x).

Original entry on oeis.org

1, 60, 20475, 9373650, 4881796920, 2734407111744, 1605040007778900, 973419698810097000, 604759111060745718000, 382741738086972337402560, 245810413547242455520545552, 159759730493918131135425965280, 104861901534978616465850670348000

Offset: 0

Gheorghe Coserea, Jul 19 2016

nonn

"One may consider the following conjecture: all the irreducible factors of the minimal order linear differential operator annihilating a diagonal of a rational function should be homomorphic to their adjoint (possibly on an algebraic extension). [...]

"If our conjecture above was correct, this would be a way to show that the series cannot be the diagonal of a rational function." (See Boukraa link.)

			1 + 60*x + 20475*x^2 + 9373650*x^3 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..200
A. Bostan, S. Boukraa, G. Christol, S. Hassani, and J-M. Maillard Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity, arXiv:1211.6031 [math-ph], 2012.
S. Boukraa, S. Hassani, J-M. Maillard, and J-A. Weil, Differential algebra on lattice Green functions and Calabi-Yau operators (unabridged version), arXiv:1311.2470 [math-ph], 2013.

Cf. A268545-A268555.

CoefficientList[Series[HypergeometricPFQ[{1/9, 4/9, 5/9}, {1/3,1}, 729*x], {x, 0, 15}], x] (* Vaclav Kotesovec, Jul 28 2016 *)
a[n_] := FullSimplify[(729^n Cos[Pi/18] Gamma[1/3] Gamma[1/9 + n] Gamma[4/9 + n] Gamma[5/9 + n])/(Pi Gamma[1/9] Gamma[1/3 + n] n!^2)] (* Benedict W. J. Irwin, Aug 05 2016 *)

\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 21; x = 'x + O('x^N);
Vec(hypergeom_sym([1/9, 4/9, 5/9], [1/3,1], 729*x, N))

my(x = 'x + O('x^20)); Vec(hypergeom([1/9, 4/9, 5/9], [1/3,1], 729*x)) \\ Michel Marcus, Apr 11 2023

G.f.: hypergeom([1/9, 4/9, 5/9], [1/3,1], 729*x).
From Vaclav Kotesovec, Jul 28 2016: (Start)
Recurrence: n^2*(3*n - 2)*a(n) = 3*(9*n - 8)*(9*n - 5)*(9*n - 4)*a(n-1).
a(n) ~ Gamma(1/3) * cos(Pi/18) * 3^(6*n) / (Pi * Gamma(1/9) * n^(11/9)).
(End)
a(n) = 729^n*cos(Pi/18)*Gamma(1/3)*Gamma(1/9+n)*Gamma(4/9+n)*Gamma(5/9+n) /(Pi*Gamma(1/9)*Gamma(1/3+n)*n!^2). - Benedict W. J. Irwin, Aug 05 2016
From Karol A. Penson, Apr 11 2023: (Start)
a(n) = Integral_{x=0..729} x^n*W(x), where
W(x) = W1(x) + W2(x) + W3(x), and
W1(x) = (2*cos(Pi/18)*3^(1/3)*2^(4/9)*sqrt(Pi)*Gamma(13/18)*hypergeom([1/9, 1/9, 7/9], [5/9, 2/3], x/729))/(9*Gamma(2/3)^2*Gamma(1/9)*Gamma(8/9)^2*x^(8/9));
W2(x) = cos(Pi/18)*2^(1/9)*Gamma(2/9)*Gamma(1/18)*hypergeom([4/9, 4/9, 10/9], [8/9, 4/3], x/729)/(162*Gamma(2/3)*Gamma(1/9)*Pi^(3/2)*x^(5/9));
W3(x) = cos(Pi/18)*3^(1/6)*2^(4/9)*Gamma(5/18)*Gamma(-1/18)*hypergeom([5/9, 5/9, 11/9], [10/9, 13/9], x/729)/(324*Gamma(2/3)*Gamma(1/9)*Pi*Gamma(4/9)*x^(4/9)).
This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with the singularity x^(-4/9), and for x > 0 is monotonically decreasing to zero at x = 729. At x = 729  the first derivative of W(x) is infinite. (End)

A275054 G.f.: 3F2([1/9, 2/9, 8/9], [2/3,1], 729 x).

Original entry on oeis.org

1, 24, 6732, 2771340, 1342525275, 711891288108, 399866544799722, 233750557331494632, 140707672445849703480, 86621407014527646518400, 54278825541246092520182592, 34504174655166790354911360048, 22195631874904018057471849288020, 14421008706115620277976088538033200

Offset: 0

Gheorghe Coserea, Jul 20 2016

nonn

"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

			1 + 24*x + 6732*x^2 + 2771340*x^3 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..200
A. Bostan, S. Boukraa, G. Christol, S. Hassani, J-M. Maillard Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity, arXiv:1211.6031 [math-ph], 2012.

Cf. A268545-A268555.

CoefficientList[Series[HypergeometricPFQ[{1/9, 2/9, 8/9}, {2/3, 1}, 729 x], {x, 0, 13}], x] (* Michael De Vlieger, Jul 26 2016 *)
a[n_] := FullSimplify[(729^n Gamma[2/3] Gamma[1/9 + n] Gamma[2/9 + n] Gamma[8/9 + n] Sin[Pi/9])/(Pi (n!)^2 Gamma[2/9] Gamma[2/3 + n])] (* Benedict W. J. Irwin, Aug 05 2016 *)

\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([1/9, 2/9, 8/9], [2/3,1], 729*x, N))

G.f.: hypergeom([1/9, 2/9, 8/9], [2/3,1], 729*x).
From Vaclav Kotesovec, Jul 28 2016: (Start)
Recurrence: n^2*(3*n - 1)*a(n) = 3*(9*n - 8)*(9*n - 7)*(9*n - 1)*a(n-1).
a(n) ~ 2 * sin(Pi/9) * 3^(6*n - 1/2) / (Gamma(1/3) * Gamma(2/9) * n^(13/9)).
(End)
a(n) = 729^n*Gamma(2/3)*Gamma(1/9+n)*Gamma(2/9+n)*Gamma(8/9+n)*Sin(Pi/9)/(Pi*(n!)^2*Gamma(2/9)*Gamma(2/3+n)). - Benedict W. J. Irwin, Aug 05 2016

A184423 a(n) = (2n)!(3*n)!/n!^5.

Original entry on oeis.org

1, 12, 540, 33600, 2425500, 190702512, 15849497664, 1369618398720, 121821136479900, 11079206239530000, 1025579963180813040, 96310511463483233280, 9152842704012278107200, 878622906816654279840000

Offset: 0

Paul D. Hanna, Jan 13 2011

Denoted by h_3[n] by T. Piezas III. He also gives formulas for 1/Pi such as 1/Pi = 2 * Sum_{n>=0} a(n) * (-1)^n * (51*n + 7) / (12^3)^(n + 1/2). - Michael Somos, May 31 2012
Diagonal of the rational function R(x,y,z,w) = 1/(1-(w*y+w*z+x+y+z)). - Gheorghe Coserea, Jul 15 2016
From Peter Bala, Jun 28 2023: (Start)
The supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for primes p >= 5 and positive integers n and r. This follows from Meštrović equation 39, since a(n) = binomial(3*n,n) * binomial(2*n,n)^2.
Inductively define a family of sequences {a(i,n) : n >= 0}, i >= 1, by setting a(1,n) = a(n) and, for i >= 2, a(i,n) = [x^n] ( exp(Sum_{k >= 1} a(i-1,k)*x^k/k) )^n. We conjecture that the sequences {a(i,n) : n >= 0}, i >= 2, also satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for primes p >= 5 and positive integers n and r. Cf. A362730 and A362732. (End)

			G.f.: A(x) = 1 + 12*x + 540*x^2 + 33600*x^3 + 2425500*x^4 +...
G.f. of A184424 equals A(x)^(1/2):
A(x)^(1/2) = 1 + 6*x + 252*x^2 + 15288*x^3 + 1089270*x^4 + 84963060*x^5 +...+ [(3^n/n!^2)*Product_{k=1..n} (6*k-4)*(6*k-5)]*x^n +...

Gheorghe Coserea, Table of n, a(n) for n = 0..200
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012), arXiv:1111.3057 [math.NT], 2011.
Titus Piezas III, 0013: Article 3 (Pi Formulas and the Monster Group).

Cf. A184424, A275047.

Related to diagonal of rational functions: A268545-A268555.

Table[((2n)!(3n)!)/(n!)^5,{n,0,20}] (* Harvey P. Dale, Dec 18 2018 *)

PARI
```
{a(n)=(3*n)!*(2*n)!/n!^5}
```

{a(n)=polcoeff(sum(m=0,n,3^m*prod(k=1,m,(6*k-4)*(6*k-5))/m!^2*x^m+x*O(x^n))^2,n)}

Self-convolution of A184424:
a(n) = Sum_{k=0..n} A184424(k)*A184424(n-k) where A184424(n) = (3^n/n!^2)*Product_{k=1..n} (6*k-4)*(6*k-5).
a(n) = 6 * (2*n - 1) * (3*n - 1) * (3*n - 2) / n^3 * a(n-1) if n>0. - Michael Somos, May 31 2012
0 = (x^2-108*x^3)*y''' + (3*x-486*x^2)*y''+ (1-348*x)*y' - 12*y, where y is g.f. - Gheorghe Coserea, Jul 15 2016
a(n) ~ 3^(1/2)/(2*Pi^(3/2)) * n^(-3/2) * 108^n. - Ilya Gutkovskiy, Jul 15 2016
a(n) = C(2*n,n)^2 * C(3*n,n) = ( [x^n](1 + x)^(2*n) )^2 * ( [x^n](1 + x)^(3*n) ) = [x^n]( F(x)^(12*n) ), where [x^n] is the coefficient extraction operator and where F(x) = 1 + x + 11*x^2 + 350*x^3  + 15293*x^4  + 794433*x^5  + 45958617*x^6 + ... appears to have integral coefficients. Cf. A000897 and A001451. - Peter Bala, Dec 30 2019

A243949 Squares of the central Delannoy numbers: a(n) = A001850(n)^2.

Original entry on oeis.org

1, 9, 169, 3969, 103041, 2832489, 80802121, 2365752321, 70611901441, 2139090528969, 65568745087209, 2029206892664961, 63300531617048961, 1987912809986437161, 62787371136571152009, 1992942254830520803329, 63531842302018973818881, 2033004661359005674887561

Offset: 0

Paul D. Hanna, Aug 17 2014

In general, we have the binomial identity:
if b(n) = Sum_{k=0..n} t^k * C(2*k, k) * C(n+k, n-k), then b(n)^2 = Sum_{k=0..n} (t^2+t)^k * C(2*k, k)^2 * C(n+k, n-k), where the g.f. of b(n) is 1/sqrt(1 - (4*t+2)*x + x^2), and the g.f. of b(n)^2 is 1 / AGM(1-x, sqrt((1+x)^2 - (4*t+2)^2*x)), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.
Note that the g.f. of A001850 is 1/sqrt(1 - 6*x + x^2).
Limit_{n -> oo} a(n+1)/a(n) = (3 + 2*sqrt(2))^2 = 17 + 12*sqrt(2).
From Gheorghe Coserea, Jul 05 2016: (Start)
Diagonal of the rational function 1/(1 - x - y - z - x*y + x*z - y*z - x*y*z).
Annihilating differential operator: x*(x-1)*(x+1)*(x^2-34*x+1)*Dx^2 + (3*x^4-66*x^3-70*x^2+70*x-1)*Dx + x^3-7*x^2-35*x+9.
(End).
The sequence b(n) mentioned above is the sequence of shifted Legendre polynomials P(n,2*t + 1) (see A063007). See Zudilin for a g.f. for the sequence b(n)^2. - Peter Bala, Mar 02 2017

			G.f.: A(x) = 1 + 9*x + 169*x^2 + 3969*x^3 + 103041*x^4 + 2832489*x^5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..500
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"
W. Zudilin, A generating function of the squares of Legendre polynomials, arXiv:1210.2493v2 [math.CA], 2012.

Sequences of the form LegendreP(n, 2*m+1)^2: A000012 (m=0), this sequence (m=1), A243943 (m=2), A243944 (m=3), A243007 (m=4).
Related to diagonal of rational functions: A268545 - A268555.
Cf. A001850, A243945, A245925.

[Evaluate(LegendrePolynomial(n), 3)^2 : n in [0..40]]; // G. C. Greubel, May 17 2023

Table[Sum[2^k *Binomial[2*k, k]^2 *Binomial[n+k, n-k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 18 2014 *)
a[n_]:= HypergeometricPFQ[{1/2, -n, n+1}, {1, 1}, -8];
Table[a[n], {n, 0, 17}] (* Peter Luschny, Mar 14 2018 *)
LegendreP[Range[0, 30], 3]^2 (* G. C. Greubel, May 17 2023 *)

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

{a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 36*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

from math import comb
def A243949(n): return sum(comb(n,k)*comb(n+k,k) for k in range(n+1))**2 # Chai Wah Wu, Mar 23 2023

[gen_legendre_P(n,0,3)^2 for n in range(41)] # G. C. Greubel, May 17 2023

G.f.: 1 / AGM(1-x, sqrt(1-34*x+x^2)). - Paul D. Hanna, Aug 30 2014
a(n) = Sum_{k=0..n} 2^k * C(2*k, k)^2 * C(n+k, n-k).
a(n)^(1/2) = Sum_{k=0..n} C(2*k, k) * C(n+k, n-k).
Recurrence: n^2*(2*n-3)*a(n) = (2*n-1)*(35*n^2 - 70*n + 26)*a(n-1) - (2*n-3)*(35*n^2 - 70*n + 26)*a(n-2) + (n-2)^2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Aug 18 2014
a(n) ~ (4 + 3*sqrt(2)) * (3 + 2*sqrt(2))^(2*n) / (8*Pi*n). - Vaclav Kotesovec, Aug 18 2014
From Gheorghe Coserea, Jul 05 2016: (Start)
G.f.: hypergeom([1/12, 5/12],[1],27648*x^4*(x^2-34*x+1)*(x-1)^2/(1-36*x+134*x^2-36*x^3+x^4)^3)/(1-36*x+134*x^2-36*x^3+x^4)^(1/4).
0 = x*(x-1)*(x+1)*(x^2-34*x+1)*y'' + (3*x^4-66*x^3-70*x^2+70*x-1)*y' + (x^3-7*x^2-35*x+9)*y, where y is g.f.
(End)
a(n) = Sum_{k = 0..n} 4^k*binomial(n+k,2*k)^2*binomial(2*k,k). - Peter Bala, Mar 02 2017
a(n) = hypergeom([1/2, -n, n + 1], [1, 1], -8). - Peter Luschny, Mar 14 2018
G.f.: Sum_{n >= 0} (2^n)*binomial(2*n,n)^2 *x^n/(1-x)^(2*n+1). - Peter Bala, Feb 07 2022

A078678 Number of binary strings with n 1's and n 0's avoiding zigzags, that is avoiding the substrings 101 and 010.

Original entry on oeis.org

1, 2, 4, 8, 18, 42, 100, 242, 592, 1460, 3624, 9042, 22656, 56970, 143688, 363348, 920886, 2338566, 5949148, 15157874, 38674978, 98803052, 252701484, 646990518, 1658066668, 4252908542, 10917422860, 28046438252, 72099983802, 185469011130, 477383400300

Offset: 0

Emanuele Munarini, Dec 17 2002

nonn

Also number of Grand Dyck paths of length 2*n with no zigzags, that is, with no factors UDU or DUD. - Emanuele Munarini, Jul 07 2011

			For n = 2 : 0011, 0110, 1001, 1100.
For n = 3 : 000111, 011001, 100011, 110001, 001110, 011100, 100110, 111000.

Vincenzo Librandi, Table of n, a(n) for n = 0..220
Andrei Asinowski and Cyril Banderier, On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16.
T. Doslic, Seven lattice paths to log-convexity, Acta Appl. Mathem. 110 (3) (2010) 1373-139, eq 4.
Emanuele Munarini and N. Z. Salvi, Binary strings without zigzags, Séminaire Lotharingien de Combinatoire, B49h (2004), 15 pp.
R. Pemantle and M. C. Wilson, Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, arXiv:math/0512548 [math.CO], 2007.

Cf. A078679, A128588.
Cf. A003440.
Main diagonal of array A099172.
Related to diagonal of rational functions: A268545-A268555.

a:= proc(n) option remember; `if`(n<5, [1, 2, 4, 8, 18][n+1],
     (2*n*a(n-1)+(n-2)*a(n-2)+(2*n-8)*a(n-3)-(n-4)*a(n-4))/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 13 2020

Table[SeriesCoefficient[Series[Sqrt[(1 + x + x^2)/(1 - 3 x + x^2)], {x, 0, n}], n], {n, 0, 40}]

a(n):=coeff(taylor((1+x+x^2)/sqrt(1-2*x-x^2-2*x^3+x^4),x,0,n),x,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, Jul 07 2011 */

my(x='x+O('x^99)); Vec(((1+x+x^2)/(1-3*x+x^2))^(1/2)) \\ Altug Alkan, Jul 18 2016

G.f.: sqrt( ( 1 + x + x^2 ) / ( 1 - 3*x + x^2 ) ).
a(n) = Sum_{k=0..n+floor(n/2)} binomial( n - k + 2*floor(k/3), floor(k/3) )^2.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^2*( 2*n^2 - 6*n*k + 6*k^2 )/(n-k)^2, n > 0.
a(n) ~ 2 * ((3+sqrt(5))/2)^n / (5^(1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 21 2014
a(n) = [x^n y^n](1+x*y+x^2*y^2)/(1-x-y+x*y-x^2*y^2). - Gheorghe Coserea, Jul 18 2016
D-finite with recurrence: n*a(n) -2*n*a(n-1) +(-n+2)*a(n-2) +2*(-n+4)*a(n-3) +(n-4)*a(n-4)=0. [Doslic] - R. J. Mathar, Jun 21 2018

A124435 Number of effective multiple alignments of three equal-length sequences.

Original entry on oeis.org

1, 5, 67, 1109, 20251, 391355, 7847155, 161476565, 3387271675, 72114452255, 1553475100717, 33786532319435, 740681494769659, 16346552430326123, 362830907979309067, 8093356178498583509, 181311959402343288955, 4077310062938894133623, 91999289732199733092601

Offset: 0

Lee A. Newberg, Dec 15 2006

This counts effective alignments rather than standard alignments, so that for example the following two alignments are equivalent:
   -A A-
   -T T-
   C- -C
See Dress, Morgenstern and Stoye for more information.

			a(1) = 5 because the five alignments are
  A--   A-   A-   A-   A
  -C-   C-   -C   -C   C
  --T   -T   T-   -T   T

Robert Israel, Table of n, a(n) for n = 0..720
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
A. Dress, B. Morgenstern and J. Stoye, On the number of standard and of effective multiple alignments, Applied Mathematics Letters, Vol. 11, No. 4, 1998, pp. 43-49.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A268545 - A268555, A000172, A318108, A318109.

G := series( hypergeom([1/3, 2/3],[1],27*x/(1+x)^3)/(1+x), x=0, 31);
seq(coeff(G,x,i),i=0..30);  # Mark van Hoeij, Dec 20 2013

a[n_] := Sum[(-1)^(n-k) Binomial[n, k] Binomial[n+2k, n] Binomial[2k, k], {k, 0, n}];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Sep 18 2018, after Wadim Zudilin *)

diag(expr, N=22, var=variables(expr)) = {
  my(a = vector(N));
  for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));
  for (n = 1, N, a[n] = expr;
    for (k = 1, #var, a[n] = polcoef(a[n], n-1)));
  return(a);
};
x='x; y='y; z='z; diag(1/(1 - x - y - z + x*y*z), 19)

\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 20; x = 'x + O('x^N);
Vec(hypergeom([1/3, 2/3],[1],27*x/(1+x)^3, N)/(1+x)) \\ Gheorghe Coserea, Jul 06 2016

The recurrence is three-dimensional with the order of the three parameters immaterial. That is, a(i,j,k)=a(i,k,j)=a(j,i,k)=a(j,k,i)=a(k,i,j)=a(k,j,i). a(i, j, 0) = (i+j)! / i! / j! a(i, j, k) = a(i-1,j,k) + a(i,j-1,k) + a(i,j,k-1) - a(i-1,j-1,k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*binomial(n+2*k,n)*binomial(2*k,k). - Wadim Zudilin, Nov 26 2015
Diagonal of 1/(1 - x - y - z + x*y*z). - Mark van Hoeij, Dec 20 2013
G.f.: hypergeom([1/3, 2/3],[1],27*x/(1+x)^3)/(1+x). - Mark van Hoeij, Dec 20 2013
(3*n-1)*(n+1)^2*a(n+1)-(3*n+1)*(24*n^2+8*n-5)*a(n)+(9*n^3-3*n^2-4*n+2)*a(n-1)+(3*n+2)*(n-1)^2*a(n-2)=0. - Robert Israel, Nov 26 2015
0 = (2*x-1)*(x^3+3*x^2-24*x+1)*x*y'' + (6*x^4+8*x^3-57*x^2+48*x-1)*y' + (x+1)*(2*x^2-2*x+5)*y, where y is g.f. - Gheorghe Coserea, Jul 06 2016
From Peter Bala, Mar 16 2023: (Start)
(3*n - 4)*n^2*a(n) = (3*n - 2)*(24*n^2 - 40*n + 11)*a(n-1) - (9*n^3 - 30*n^2 + 29*n - 6)*a(n-2) - (3*n - 1)*(n - 2)^2*a(n-3) with a(0) = 1, a(1) = 5 and a(2) = 67.
Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for positive integers n and r and all primes p >= 5. (End)

More terms from Mark van Hoeij, Dec 21 2013

A268549 Diagonal of (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - z) * (1 - v - w)).

Original entry on oeis.org

1, 12, 648, 50400, 4630500, 468087984, 50345463168, 5655718328832, 656151696743400, 78036148295820000, 9465472643689782720, 1166663950520357802240, 145719568153188579382560, 18405635030728188793200000

Offset: 0

N. J. A. Sloane, Feb 29 2016

nonn

"The corresponding (order-three) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension." (see A. Bostan link) - Gheorghe Coserea, Aug 15 2016

			1 + 12*x + 648*x^2 + 50400*x^3 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..100
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015, Eq. (30).
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A004987, A268545-A268555.

A268549 := proc(n)
    (1-9*x*y)/(1-3*y-2*x+3*y^2+9*x^2*y)/(1-u-z)/(1-v-w) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,n) ;
    coeftayl(%,z=0,n) ;
    coeftayl(%,u=0,n) ;
    coeftayl(%,v=0,n) ;
    coeftayl(%,w=0,n) ;
end proc:
seq(A268549(n),n=0..40) ; # R. J. Mathar, Mar 11 2016
series(hypergeom([1/3, 1/2, 1/2], [1, 1], 144*x), x=0, 14); # Gheorghe Coserea, Aug 15 2016

FullSimplify[Table[3^(2*n)*(2*n)!^2*Gamma[n + 1/3]/(Gamma[1/3]*(n!)^5), {n, 0, 15}]] (* Vaclav Kotesovec, Jul 01 2016 *)

a(n) = [(xyzuvw)^n] (1 - 9*x*y)/((1 - 3*y - 2*x + 3*y^2 + 9*x^2*y) * (1 - u - z) * (1 - v - w)).
D-finite with recurrence: n^3*a(n) -12*(3*n-2)*(-1+2*n)^2*a(n-1)=0. - R. J. Mathar, Mar 11 2016 [follows from the hypergeometric g.f. below - Georg Fischer, Jul 30 2022]
From Vaclav Kotesovec, Jul 01 2016: (Start)
a(n) = 3^(2*n) * (2*n)!^2 * Gamma(n + 1/3) / (Gamma(1/3) * (n!)^5).
a(n) ~ 12^(2*n)/(Gamma(1/3)*Pi*n^(5/3)).
(End)
From Gheorghe Coserea, Aug 16 2016: (Start)
a(n) = [(xyzuv)^n] 1/((1 - x + 3*y - 27*x*y^3 - 27*x*y^2 - 9*x*y + 3*y^2) * (1 - u - v - u*z - v*z)).
G.f.: hypergeom([1/3, 1/2, 1/2], [1, 1], 144*x).
(End)

cp's OEIS Frontend

A081798 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2*k,k).

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A082488 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2*k,k) * C(n+3*k,k).

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A206178 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 2^k.

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A275051 Expansion of 3F2([1/9, 4/9, 5/9], [1/3,1], 729*x).

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A275054 G.f.: 3F2([1/9, 2/9, 8/9], [2/3,1], 729 x).

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A184423 a(n) = (2*n)!*(3*n)!/n!^5.

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A082488 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2k,k) C(n+3*k,k).

A184423 a(n) = (2n)!(3*n)!/n!^5.