A274786
Diagonal of the rational function 1/(1 - (wxz + wy + wz + xy + xz + y + z)).
Original entry on oeis.org
1, 6, 114, 2940, 87570, 2835756, 96982116, 3446781624, 126047377170, 4712189770860, 179275447715364, 6918537571788024, 270178056420497316, 10656693484898995800, 423937118582497715400, 16989669600664370275440, 685277433339552643145490, 27797911234749454227812460, 1133299570662800455270517700
Offset: 0
- Gheorghe Coserea, Table of n, a(n) for n = 0..200
- 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.
- Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
- Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"
-
a[n_] := Sum[(-1)^j Binomial[2n, j] Binomial[j, n]^3, {j, n, 2n}];
(* or much faster *)
a[0] = 1; a[1] = 6; a[n_] := a[n] = (2*(2*n - 1)*(11*n^2 - 11*n + 3)*a[n - 1] + 4*(n - 1)*(2*n - 3)*(2*n - 1)*a[n - 2])/n^3;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 01 2017, after Vaclav Kotesovec *)
-
a(n) = sum(j=n, 2*n, (-1)^(j)*binomial(2*n, 2*n - j)*binomial(j, n)^3);
-
my(x='x, y='y, z='z, w='w);
R = 1/(1-(w*x*z+w*y+w*z+x*y+x*z+y+z));
diag(n, expr, var) = {
my(a = vector(n));
for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
for (k = 1, n, a[k] = expr;
for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
return(a);
};
diag(18, R, [x,y,z,w])
A275046
Number of binary strings with n zeros and n ones avoiding the substrings 10101101 and 1110101.
Original entry on oeis.org
1, 2, 6, 20, 70, 245, 874, 3164, 11577, 42694, 158431, 590873, 2212797, 8315535, 31341163, 118423810, 448455754, 1701534151, 6467049185, 24617030774, 93834205107, 358116770601, 1368283768753, 5233261657558, 20034371696497, 76763164565117, 294357181436313, 1129575035419485
Offset: 0
For n = 5 there are binomial(10,5) = 252 binary strings with 5 zeros and 5 ones; seven out of this 252 binary strings contain as substrings w1=10101101 or w2=1110101, i.e.
0123456789
----------
1 0001110101 contains w2 at offset 3
2 0010101101 contains w1 at offset 2
3 0011101010 contains w2 at offset 2
4 0101011010 contains w1 at offset 1
5 0111010100 contains w2 at offset 1
6 1010110100 contains w1 at offset 0
7 1110101000 contains w2 at offset 0
Therefore a(5) = 252 - 7 = 245.
-
a[n_] := SeriesCoefficient[(1 + x^2 y^3 + x^2 y^4 + x^3 y^4 - x^3 y^6) / (1 - x - y + x^2 y^3 - x^3 y^3 - x^4 y^4 - x^3 y^6 + x^4 y^6), {x, 0, n}, {y, 0, n}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Aug 20 2018 *)
-
r1 = (1+x^2*y^3+x^2*y^4+x^3*y^4-x^3*y^6);
r2 = (1-x-y+x^2*y^3-x^3*y^3-x^4*y^4-x^3*y^6+x^4*y^6);
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] = polcoeff(a[n], n-1)));
return(a);
};
diag(r1/r2, 28)
F = (x + 1)*(4*x^20 + 8*x^19 - 23*x^18 - 63*x^17 - 62*x^16 - 26*x^15 + 43*x^14 + 11*x^13 + 182*x^12 + 56*x^11 - x^10 + 203*x^9 - 66*x^8 - 154*x^7 + 286*x^6 - 368*x^5 + 233*x^4 - 75*x^3 - 8*x^2 + 20*x - 4)*(y^4 - y^3) - (12*x^17 + 48*x^16 + 72*x^15 + 49*x^14 - 23*x^13 - 57*x^12 - 91*x^11 - 137*x^10 - 84*x^9 - 34*x^8 - 91*x^7 + 62*x^6 + 24*x^5 - 34*x^4 + 41*x^3 - 10*x^2 - 3*x - 3)*y^2 + (x^15 + 4*x^14 + 6*x^13 + 3*x^12 - 6*x^11 - 11*x^10 - 11*x^9 - 8*x^8 - 3*x^7 + 12*x^6 + 11*x^4 + 5*x^3 - 6*x^2 - 4)*y - x^4 + x + 1;
\\ test: y = Ser(diag(r1/r2, 100)); 0 == subst(F, 'y, y)
-
x='x; y='y; t='t;
seq(N) = {
my(Fx = substvec(F, [x, y], [t, x]), y0 = 1 + O('t^N), y1=0, n=1);
while (n++,
y1 = y0 - subst(Fx, 'x, y0)/subst(deriv(Fx, 'x), 'x, y0);
if (y1 == y0, break()); y0 = y1); Vec(y0);
};
seq(28)
\\ Gheorghe Coserea, Jul 18 2018
A268542
The diagonal of the rational function 1/(1 - x - y - x y - x z - y z).
Original entry on oeis.org
1, 4, 42, 520, 7090, 102144, 1525776, 23380368, 365130810, 5786380600, 92774019052, 1501646797248, 24498046138384, 402329384914240, 6645072333486720, 110293868867458080, 1838511122725436250, 30762545845461663240
Offset: 0
-
A268542 := proc(n)
1/(1-x-y-x*y-x*z-y*z) ;
coeftayl(%,x=0,n) ;
coeftayl(%,y=0,n) ;
coeftayl(%,z=0,n) ;
end proc:
seq(A268542(n),n=0..40) ; # R. J. Mathar, Mar 11 2016
-
gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*x^4*(x + 1)^2*(27*x^2 + 34*x - 2)/(-1 + 16*x + 8*x^2)^3]/(1 - 16*x - 8*x^2)^(1/4);
CoefficientList[gf + O[x]^18, x] (* Jean-François Alcover, Dec 02 2017, after Gheorghe Coserea *)
-
my(x='x, y='y, z='z);
R = 1/(1 - x - y - x*y - x*z - y*z);
diag(n, expr, var) = {
my(a = vector(n));
for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
for (k = 1, n, a[k] = expr;
for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
return(a);
};
diag(10, R, [x, y, z])
-
\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 20; x = 'x + O('x^N);
Vec(hypergeom_sym([1/12, 5/12], [1], 1728*x^4*(x+1)^2*(27*x^2+34*x-2)/(-1+16*x+8*x^2)^3, N)/(1-16*x-8*x^2)^(1/4)) \\ Gheorghe Coserea, Jul 06 2016
A268543
The diagonal of 1/(1 - (y + z + x z + x w + x y w)).
Original entry on oeis.org
1, 8, 156, 3800, 102340, 2919168, 86427264, 2626557648, 81380484900, 2559296511200, 81443222791216, 2616761264496288, 84749038859067856, 2763262653898544000, 90615128199047200800, 2986287891921565639200, 98841887070519004625700
Offset: 0
- Gheorghe Coserea, Table of n, a(n) for n = 0..310
- 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.
- S. Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
- Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"
-
A268543 := proc(n)
1/(1-y-z-x*z-x*w-x*y*w) ;
coeftayl(%,x=0,n) ;
coeftayl(%,y=0,n) ;
coeftayl(%,z=0,n) ;
coeftayl(%,w=0,n) ;
end proc:
seq(A268543(n),n=0..40) ; # R. J. Mathar, Mar 11 2016
#alternative program
with(combinat):
seq(binomial(2*n,n)*add(binomial(n,k)*binomial(2*n+k,k), k = 0..n), n = 0..20); # Peter Bala, Jan 27 2018
-
CoefficientList[Series[HypergeometricPFQ[{1/12, 5/12}, {1}, 1728*x^3*(2 - 71*x + 16*x^2)/(1 - 32*x + 16*x^2)^3]*(1 - 32*x + 16*x^2)^(-1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 05 2016 *)
-
my(x='x, y='y, z='z, w='w);
R = 1/(1 - x - y - z - x*y);
diag(n, expr, var) = {
my(a = vector(n));
for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
for (k = 1, n, a[k] = expr;
for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
return(a);
};
diag(10, R, [x,y,z])
-
\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 20; x = 'x + O('x^N);
Vec(hypergeom_sym([1/12,5/12],[1],1728*x^3*(16*x^2-71*x+2)/(16*x^2-32*x+1)^3, N)/(16*x^2-32*x+1)^(1/4)) \\ Gheorghe Coserea, Jul 03 2016
A268551
Diagonal of 1/(1 - x + y + z + x y + x z - y z + x y z).
Original entry on oeis.org
1, 11, 325, 11711, 465601, 19590491, 855266581, 38319499775, 1750193256961, 81131090245931, 3805404745303525, 180207832513958975, 8601942203526345025, 413358969518738106875, 19977566733574388828725, 970297391859524593324031, 47330511448436249282088961
Offset: 0
-
A268551 := proc(n)
1/(1-x+y+z+x*y+x*z-y*z+x*y*z) ;
coeftayl(%,x=0,n) ;
coeftayl(%,y=0,n) ;
coeftayl(%,z=0,n) ;
end proc:
seq(A268551(n),n=0..40) ; # R. J. Mathar, Mar 10 2016
-
gf = Hypergeometric2F1[1/12, 5/12, 1, 13824*x^3*(x^2 - 52*x + 1)/(x^2 - 46*x + 1)^3/(x + 1)^2]/((x^2 - 46*x + 1)*(x + 1)^2)^(1/4);
CoefficientList[gf + O[x]^40, x] (* Jean-François Alcover, Dec 03 2017, after Gheorghe Coserea *)
-
my(x='x, y='y, z='z);
R = 1/(1 - x + y + z + x*y + x*z - y*z + x*y*z);
diag(n, expr, var) = {
my(a = vector(n));
for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
for (k = 1, n, a[k] = expr;
for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
return(a);
};
diag(10, R, [x, y, z])
-
\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 20; x = 'x + O('x^N);
Vec(hypergeom([1/12, 5/12],[1],13824*x^3*(x^2-52*x+1)/(x^2-46*x+1)^3/(x+1)^2, N)/((x^2-46*x+1)*(x+1)^2)^(1/4)) \\ Gheorghe Coserea, Jul 06 2016
A274665
Diagonal of the rational function 1/(1 - x - y - z + x*y + x*z - y*z).
Original entry on oeis.org
1, 4, 30, 280, 2890, 31584, 358176, 4168560, 49455450, 595480600, 7254787540, 89234708160, 1106335812400, 13808393670400, 173332340911200, 2186551157230560, 27701981424940890, 352297514508697800, 4495418315974868700, 57535568476437651600, 738373616359119126540
Offset: 0
-
a[0] = 1; a[1] = 4; a[n_] := a[n] = ((29*n^2 - 29*n + 8)*a[n-1] - 3*(3*n - 4)*(3*n - 2)*a[n-2])/(2*n^2);
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 01 2017, after Vaclav Kotesovec *)
-
my(x='x, y='y, z='z);
R = 1/(1 - x - y - z + x*y + x*z - y*z);
diag(n, expr, var) = {
my(a = vector(n));
for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
for (k = 1, n, a[k] = expr;
for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
return(a);
};
diag(10, R, [x,y,z])
-
\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 21; x = 'x + O('x^N);
Vec(hypergeom([1/12, 5/12],[1],3456*x^5*(1-31/2*x+28*x^2-27/2*x^3)/(1-16*x+40*x^2)^3, N)/(1-16*x+40*x^2)^(1/4))
A274666
Diagonal of the rational function 1/(1 - x - y + x y - x z - y z - x y z).
Original entry on oeis.org
1, 5, 43, 461, 5491, 69395, 910855, 12274925, 168668035, 2352544535, 33204000853, 473179375355, 6797163712639, 98299113206663, 1429765398030943, 20899401842991341, 306819063154144675, 4521526749077118143, 66858281393757281641, 991598171159871109391
Offset: 0
-
CoefficientList[Series[HypergeometricPFQ[{1/12, 5/12},{1},1728*x^5*(x^2-47*x+3)*(-1+2*x)^2/(1-20*x+78*x^2-44*x^3+x^4)^3]/(1-20*x+78*x^2-44*x^3+x^4)^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 05 2016 *)
-
my(x='x, y='y, z='z);
R = 1 / (1 - x - y + x*y - x*z - y*z - x*y*z);
diag(n, expr, var) = {
my(a = vector(n));
for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
for (k = 1, n, a[k] = expr;
for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
return(a);
};
diag(10, R, [x, y, z])
-
\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 20; x = 'x + O('x^N);
Vec(hypergeom([1/12, 5/12],[1],1728*x^5*(x^2-47*x+3)*(-1+2*x)^2/(1-20*x+78*x^2-44*x^3+x^4)^3, N)/(1-20*x+78*x^2-44*x^3+x^4)^(1/4))
A274667
Diagonal of the rational function 1/(1 - x - y - x y - x z - y z + x y z).
Original entry on oeis.org
1, 3, 31, 339, 4131, 53013, 705139, 9618003, 133672387, 1884947073, 26889061761, 387207732453, 5619687743151, 82101265925409, 1206262382507451, 17809706204128659, 264074421220475427, 3930338612143125849, 58692717332813782501, 879093138034007102289, 13202346737893575996541
Offset: 0
-
CoefficientList[Series[HypergeometricPFQ[{1/12, 5/12},{1},1728*x^4*(x^3-41*x^2-29*x+2)*(1+2*x)^2/(1-12*x-34*x^2-36*x^3+x^4)^3]/(1-12*x-34*x^2-36*x^3+x^4)^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 05 2016 *)
-
my(x='x, y='y, z='z);
R = 1/(1 - x - y - x*y - x*z - y*z + x*y*z);
diag(n, expr, var) = {
my(a = vector(n));
for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
for (k = 1, n, a[k] = expr;
for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
return(a);
};
diag(10, R, [x, y, z])
-
\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 22; x = 'x + O('x^N);
Vec(hypergeom([1/12, 5/12],[1],1728*x^4*(x^3-41*x^2-29*x+2)*(1+2*x)^2/(1-12*x-34*x^2-36*x^3+x^4)^3, N)/(1-12*x-34*x^2-36*x^3+x^4)^(1/4))
A274669
Diagonal of the rational function 1/(1 - x - y - z + x y - x z - y z).
Original entry on oeis.org
1, 8, 138, 2960, 70090, 1756608, 45678864, 1219013664, 33162009210, 915589703600, 25578044554348, 721420319128704, 20509529725235824, 586986330979489280, 16895932626393943680, 488743896405192037440, 14198840150264907505050, 414069243091986225102480, 12115901803035178006468500
Offset: 0
-
gf = Hypergeometric2F1[1/12, 5/12, 1, -1728*x^4*(27*x^2 + 92*x - 3)*(x - 2)^2/(1 - 32*x + 88*x^2)^3]/(1 - 32*x + 88*x^2)^(1/4);
CoefficientList[gf + O[x]^20, x] (* Jean-François Alcover, Dec 01 2017 *)
-
my(x='x, y='y, z='z);
R = 1/(1 - x - y - z + x*y - x*z - y*z);
diag(n, expr, var) = {
my(a = vector(n));
for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
for (k = 1, n, a[k] = expr;
for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
return(a);
};
diag(10, R, [x, y, z])
-
\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 20; x = 'x + O('x^N);
Vec(hypergeom([1/12, 5/12],[1],-1728*x^4*(27*x^2+92*x-3)*(x-2)^2/(1-32*x+88*x^2)^3, N)/(1-32*x+88*x^2)^(1/4))
A274670
Diagonal of the rational function 1/(1 - x - y - z - x y + x z - x y z).
Original entry on oeis.org
1, 7, 103, 1891, 38371, 824377, 18379579, 420563731, 9810403267, 232264240957, 5564072675833, 134574852764821, 3280845731941519, 80522277272406613, 1987608338377888483, 49305191067563987731, 1228368016027453079587, 30719511029184435338053, 770839386237255136597501
Offset: 0
-
gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*x^4*(6 - 167*x + 205*x^2 - 9*x^3 + 2*x^4)/(1 - 28*x + 78*x^2 - 4*x^3 + x^4)^3]/(1 - 28*x + 78*x^2 - 4*x^3 + x^4)^(1/4);
CoefficientList[gf + O[x]^20, x] (* Jean-François Alcover, Dec 01 2017 *)
-
\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 20; x = 'x + O('x^N);
Vec(hypergeom([1/12, 5/12],[1],1728*x^4*(6-167*x+205*x^2-9*x^3+2*x^4)/(1-28*x+78*x^2-4*x^3+x^4)^3, N)/(1-28*x+78*x^2-4*x^3+x^4)^(1/4))
-
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] = polcoeff(a[n], n-1)));
return(a);
};
diag(1/(1 - x - y - z - x*y + x*z - x*y*z), 19)
\\ test: diag(1/(1 - x - y - z - x*y + x*z - x*y*z)) == diag(1/(1 - 2*x - y - z - x*y + y*z + x*z + x*y*z))
\\ Gheorghe Coserea, Jul 03 2018
Comments