A361637 -id:A361637 - cp's OEIS frontend

A361703 Constant term in the expansion of (1 + w + x + y + z + 1/(wxy*z))^n.

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 15121, 143641, 1302841, 7579441, 32586841, 113753641, 509068561, 3599319361, 25076993761, 142188273361, 662296228561, 2933770097881, 15581813723281, 99333170493481, 623696622059281, 3466773281312881, 17406784944114721

Offset: 0

Seiichi Manyama, Mar 21 2023

Diagonal of the rational function 1/(1 - (v^5 + w^5 + x^5 + y^5 + z^5 + v*w*x*y*z)). - Seiichi Manyama, Jul 04 2025

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A361675, A361704, A361705.
Cf. A002426, A344560, A361637.
Cf. A082489, A361636.

a(n) = sum(k=0, n\5, (4*k)!/k!^4*binomial(5*k, 4*k)*binomial(n, 5*k));

a(n) = Sum_{k=0..floor(n/5)} (4*k)!/k!^4 * binomial(5*k,4*k) * binomial(n,5*k) = Sum_{k=0..floor(n/5)} (5*k)!/k!^5 * binomial(n,5*k).

a(n) ~ 9 * 6^n / (sqrt(5) * Pi^2 * n^2). - Vaclav Kotesovec, Mar 25 2023

A274783 Diagonal of the rational function 1/(1 - (wxy*z + wxy + wxz + wyz + xyz)).

Original entry on oeis.org

1, 1, 1, 25, 121, 361, 3361, 24361, 116425, 790441, 6060121, 36888721, 238815721, 1760983225, 11968188961, 79763351305, 570661612585, 4040282139625, 27901708614985, 198090585115105, 1420583920034161, 10056659775872161, 71730482491962361, 517012699162717825, 3713833648541268121

Offset: 0

Gheorghe Coserea, Jul 13 2016

Diagonal of the rational function 1/(1 - (x^3 + y^3 + z^3 + w^3 + x*y*z*w)). - Seiichi Manyama, Jul 04 2025

Seiichi Manyama, Table of n, a(n) for n = 0..1000
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"

Cf. A268545-A268555.
Cf. A001850, A208425, A361636.
Cf. A082488, A274785, A361637.

with(combinat):
seq(add((n+k)!/(k!^4*(n-3*k)!), k = 0..floor(n/3)), n = 0..20); # Peter Bala, Jan 27 2018

my(x='x, y='y, z='z, w='w);
R = 1/(1-(w*x*y*z+w*x*y+w*x*z+w*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(20, R, [x,y,z,w])

0 = x^2*(x+3)^2*(x^4 - 260*x^3 + 6*x^2 - 4*x + 1)*y''' + 3*x*(x+3)*(2*x^5 - 381*x^4 - 1944*x^3 + 34*x^2 - 18*x + 3)*y'' + (7*x^6 - 764*x^5 - 9101*x^4 - 27264*x^3 + 381*x^2 - 132*x + 9)*y' + (x^5 - 13*x^4 - 246*x^3 - 5946*x^2 + 69*x - 9)*y, where y is the g.f.
a(n) = Sum_{k = 0..floor(n/3)} (n+k)!/(k!^4*(n-3*k)!) = Sum_{k = 0..floor(n/3)} binomial(n,3*k)*binomial(n+k,k)*(3*k)!/k!^3 (apply Eger, Theorem 3 to the set of column vectors S = {[1,1,1,1], [1,1,0,1], [1,0,1,1], [0,1,1,1], [1,1,1,0]}). - Peter Bala, Jan 27 2018
G.f.: Sum_{k>=0} (4*k)!/k!^4 * x^(3*k)/(1-x)^(4*k+1). - Seiichi Manyama, Mar 19 2023
From Vaclav Kotesovec, Mar 19 2023: (Start)
Recurrence: n^3*(2*n - 5)*(4*n - 11)*(4*n - 7)*a(n) = (4*n - 11)*(32*n^5 - 184*n^4 + 368*n^3 - 327*n^2 + 147*n - 27)*a(n-1) - (192*n^6 - 1920*n^5 + 7628*n^4 - 15366*n^3 + 16567*n^2 - 9117*n + 2025)*a(n-2) + (4*n - 9)*(4*n - 3)*(520*n^4 - 4420*n^3 + 13809*n^2 - 18769*n + 9367)*a(n-3) - (n-3)^3*(2*n - 3)*(4*n - 7)*(4*n - 3)*a(n-4).
a(n) ~ sqrt(9/8 + 3/(32*sqrt(2)) + sqrt(1085/32 + 161/(2*sqrt(2)))/8) * (1 + 2*sqrt(2) + 2*sqrt(2*(2*sqrt(2) - 1)))^n / (Pi^(3/2) * n^(3/2)). (End)

A361657 Constant term in the expansion of (1 + x^2 + y^2 + 1/(x*y))^n.

Original entry on oeis.org

1, 1, 1, 1, 13, 61, 181, 421, 1261, 5293, 21421, 73261, 232321, 789361, 2954953, 11127481, 39961741, 139908301, 499315501, 1835933293, 6792310153, 24827506873, 90058277233, 328509505633, 1210097040769, 4473191880961, 16495696956961, 60721903812961

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

Cf. A002426, A361658.

Cf. A000897, A344560, A361637.

Table[n!*Sum[1/(k!^2*(2*k)!*(n - 4*k)!), {k, 0, n/4}], {n, 0, 30}] (* Vaclav Kotesovec, Mar 20 2023 *)

a(n) = n!*sum(k=0, n\4, 1/(k!^2*(2*k)!*(n-4*k)!));

a(n) = n! * Sum_{k=0..floor(n/4)} 1/(k!^2 * (2*k)! * (n-4*k)!) = Sum_{k=0..floor(n/4)} binomial(n,4*k) * A000897(k).
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: (n-2)*n^2*a(n) = (4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) - (n-1)*(6*n^2 - 18*n + 13)*a(n-2) + 4*(n-2)^2*(n-1)*a(n-3) + 63*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ (1 + 2*sqrt(2))^(n+1) / (4*Pi*n). (End)

A361658 Constant term in the expansion of (1 + x^3 + y^3 + z^3 + 1/(xyz))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 121, 841, 3361, 10081, 25201, 55441, 194041, 1287001, 7927921, 38438401, 152312161, 516079201, 1627691521, 5745472321, 25999820401, 133086258481, 651284938921, 2860955078521, 11312609403481, 42039298455001, 158864460354601, 658342633033801

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

Cf. A002426, A361657.

Cf. A001421, A361637.

Table[n!*Sum[1/(k!^3 * (3*k)! * (n-6*k)!), {k, 0, n/6}], {n, 0, 30}] (* Vaclav Kotesovec, Mar 20 2023 *)

a(n) = n!*sum(k=0, n\6, 1/(k!^3*(3*k)!*(n-6*k)!));

a(n) = n! * Sum_{k=0..floor(n/6)} 1/(k!^3 * (3*k)! * (n-6*k)!) = Sum_{k=0..floor(n/6)} binomial(n,6*k) * A001421(k).
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: (n-4)*(n-2)*n^3*a(n) = (6*n^5 - 45*n^4 + 112*n^3 - 123*n^2 + 68*n - 15)*a(n-1) - 3*(n-1)*(5*n^4 - 40*n^3 + 111*n^2 - 132*n + 59)*a(n-2) + 2*(n-2)*(n-1)*(10*n^3 - 75*n^2 + 181*n - 144)*a(n-3) - (n-3)*(n-2)*(n-1)*(15*n^2 - 90*n + 133)*a(n-4) + 3*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 7)*a(n-5) + 1727*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).
a(n) ~ (1 + 2*sqrt(3))^(n + 3/2) / (4 * 3^(1/4) * Pi^(3/2) * n^(3/2)). (End)

A274785 Diagonal of the rational function 1/(1-(wxy*z + wxz + wy + xy + z)).

Original entry on oeis.org

1, 1, 25, 121, 2881, 23521, 484681, 5223625, 97949041, 1243490161, 22061635465, 309799010665, 5331441539425, 79799232449665, 1352284119871465, 21095036702450281, 355125946871044561, 5694209222592780625, 95705961654403180201, 1563714140278617173641, 26311422169994777663761

Offset: 0

Gheorghe Coserea, Jul 13 2016

Diagonal of the rational function 1/(1 - (x^2 + y^2 + z^2 + w^2 + x*y*z*w)). - Seiichi Manyama, Jul 04 2025

Seiichi Manyama, Table of n, a(n) for n = 0..801
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.
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"

Cf. A268545-A268555.

Cf. A082488, A274783, A361637.

seq(add(binomial(n+2*k, 2*k)*binomial(n, 2*k)*binomial(2*k, k)^2, k = 0..floor(n/2)), n = 0..20); # Peter Bala, Jan 27 2018

Table[Sum[Binomial[n + 2*k, 2*k]*Binomial[n, 2*k]*Binomial[2*k, k]^2, {k, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 17 2023 *)

my(x='x, y='y, z='z, w='w);
R = 1/(1-(w*x*y*z+w*x*z+w*y+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(12, R, [x,y,z,w])

a(n) = sum(k=0, n\2, binomial(n + 2*k,2*k) * binomial(n,2*k) * binomial(2*k,k)^2) \\ Andrew Howroyd, Mar 18 2023

0 = (-x^2+2*x^3+257*x^4+508*x^5+257*x^6+2*x^7-x^8)*y''' + (-3*x+15*x^2+1524*x^3+2286*x^4+789*x^5+3*x^6-6*x^7)*y'' + (-1+16*x+1687*x^2+1168*x^3+217*x^4-8*x^5-7*x^6)*y' + (1+183*x-178*x^2-2*x^3-3*x^4-x^5)*y, where y is the g.f.
a(n) = Sum_{k = 0..floor(n/2)} C(n + 2*k,2*k)*C(n,2*k)*C(2*k,k)^2 (apply Eger, Theorem 3 to the set of column vectors S = {[0,0,1,0], [1,1,0,0], [0,1,0,1], [1,0,1,1],[1,1,1,1]}). - Peter Bala, Jan 27 2018
n^3*(n - 2)*(2*n - 5)*a(n) = (2*n - 5)*(2*n - 1)*(2*n^3 - 6*n^2 + 4*n - 1)*a(n-1) + (2*n - 3)*(250*n^4 - 1500*n^3 + 3066*n^2 - 2448*n + 629)*a(n-2) + (2*n - 5)*(2*n - 1)*(2*n^3 - 12*n^2 + 22*n - 11)*a(n-3) - (2*n - 1)*(n - 1)*(n - 3)^3*a(n-4). - Peter Bala, Mar 17 2023
a(n) ~ 5^(1/4) * phi^(6*n + 3) / (2^(5/2) * Pi^(3/2) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 17 2023

A361673 Constant term in the expansion of (1 + xy + yz + zx + 1/(xy*z))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 61, 361, 1261, 3361, 7561, 34021, 235621, 1294921, 5482621, 19039021, 65345281, 286147681, 1511480881, 7688794681, 34337600281, 138221512741, 554603041441, 2454508134541, 11874549049441, 57412094595241, 261925516443361, 1134301869703861

Offset: 0

Seiichi Manyama, Mar 20 2023

nonn

Also constant term in the expansion of (1 + x^2 + y^2 + z^2 + 1/(x*y*z))^n.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A344560, A361675.
Cf. A361637, A361658.
Cf. A001460.

Table[n! * Sum[1/(k!^3 * (2*k)! * (n-5*k)!), {k,0,n/5}], {n,0,20}] (* Vaclav Kotesovec, Mar 22 2023 *)

a(n) = n!*sum(k=0, n\5, 1/(k!^3*(2*k)!*(n-5*k)!));

a(n) = n! * Sum_{k=0..floor(n/5)} 1/(k!^3 * (2*k)! * (n-5*k)!) = Sum_{k=0..floor(n/5)} binomial(n,5*k) * A001460(k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: 2*n^3*(2*n - 5)*a(n) = 2*(10*n^4 - 40*n^3 + 50*n^2 - 30*n + 7)*a(n-1) - 10*(n-1)*(4*n^3 - 18*n^2 + 26*n - 13)*a(n-2) + 40*(n-2)^3*(n-1)*a(n-3) - 10*(n-3)*(n-2)*(n-1)*(2*n - 5)*a(n-4) + 3129*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ sqrt(c) * (1 + 5/2^(2/5))^n / (Pi^(3/2) * n^(3/2)), where c = 3.154712586460560795509193778252140601572145506226776094640234924884123818... is the real root of the equation -30634915689 + 95407210000*c - 127160000000*c^2 + 79846400000*c^3 - 25600000000*c^4 + 3276800000*c^5 = 0. (End)

A361678 Constant term in the expansion of (1 + w + x + y + z + 1/(xyz) + 1/(wyz) + 1/(wxz) + 1/(wxy))^n.

Original entry on oeis.org

1, 1, 1, 1, 97, 481, 1441, 3361, 77281, 647137, 3195361, 11674081, 116286721, 1147935361, 7611379777, 37451144641, 263670781921, 2456043418081, 19073086806241, 115319128034017, 748239468100417, 6179458007222977, 50636218964639617, 350400618132423937

Offset: 0

Seiichi Manyama, Mar 20 2023

nonn

Cf. A201805, A361677.

Cf. A361637.

Table[Sum[(4*k)!/k!^4 * Binomial[4*k,k] * Binomial[n,4*k], {k,0,n/4}], {n,0,25}] (* Vaclav Kotesovec, Mar 22 2023 *)

a(n) = sum(k=0, n\4, (4*k)!/k!^4*binomial(4*k, k)*binomial(n, 4*k));

a(n) = Sum_{k=0..floor(n/4)} (4*k)!/k!^4 * binomial(4*k,k) * binomial(n,4*k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: 3*n^4*(3*n - 8)*(3*n - 4)*a(n) = 3*(63*n^6 - 405*n^5 + 1015*n^4 - 1355*n^3 + 1049*n^2 - 439*n + 77)*a(n-1) - 3*(n-1)*(189*n^5 - 1485*n^4 + 4685*n^3 - 7575*n^2 + 6313*n - 2163)*a(n-2) + 3*(n-2)*(n-1)*(315*n^4 - 2610*n^3 + 8285*n^2 - 12030*n + 6749)*a(n-3) + (n-3)*(n-2)*(n-1)*(64591*n^3 - 385926*n^2 + 701651*n - 375786)*a(n-4) - 3*(n-4)*(n-3)*(n-2)*(n-1)*(65347*n^2 - 326519*n + 391384)*a(n-5) + 3*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(65473*n - 196383)*a(n-6) - 65509*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7).
a(n) ~ sqrt(65563/12288 + 3*sqrt(3)/8 + sqrt(3/8 + 65563/(4096*sqrt(3)))) * (1 + 16/3^(3/4))^n / (Pi^2 * n^2). (End)

A361701 Constant term in the expansion of (1 + x^4 + y^4 + z^4 + 1/(xyz))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 211, 1681, 7561, 25201, 69301, 166321, 360361, 990991, 5405401, 34834801, 187867681, 833709241, 3153281041, 10491944401, 31945216801, 97323704941, 345845431471, 1529597398561, 7451402805001, 35092646589001, 151591791651301

Offset: 0

Seiichi Manyama, Mar 21 2023

Winston de Greef, Table of n, a(n) for n = 0..1611

Cf. A361637, A361658, A361673.

Cf. A361677.

Table[Sum[(3*k)!/k!^3 * Binomial[7*k,3*k] * Binomial[n,7*k], {k,0,n/7}], {n,0,20}] (* Vaclav Kotesovec, Mar 22 2023 *)

a(n) = sum(k=0, n\7, (3*k)!/k!^3*binomial(7*k, 3*k)*binomial(n, 7*k));

a(n) = Sum_{k=0..floor(n/7)} (3*k)!/k!^3 * binomial(7*k,3*k) * binomial(n,7*k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: 8*n^3*(2*n - 7)*(4*n - 21)*(4*n - 7)*a(n) = 8*(224*n^6 - 2688*n^5 + 11550*n^4 - 22736*n^3 + 22666*n^2 - 11746*n + 2475)*a(n-1) - 56*(n-1)*(96*n^5 - 1200*n^4 + 5540*n^3 - 11982*n^2 + 12466*n - 5115)*a(n-2) + 224*(n-2)*(n-1)*(40*n^4 - 480*n^3 + 2065*n^2 - 3822*n + 2607)*a(n-3) - 56*(n-3)*(n-2)*(n-1)*(160*n^3 - 1680*n^2 + 5730*n - 6407)*a(n-4) + 112*(n-4)*(n-3)*(n-2)*(n-1)*(48*n^2 - 384*n + 757)*a(n-5) - 896*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 9)*a(n-6) + 823799*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7).
a(n) ~ sqrt(c) * (1 + 7/2^(8/7))^n / (Pi^(3/2) * n^(3/2)), where c = 3.4855654710461411310762468259332410505173151761420224383969482891017005063... is the real root of the equation -559066901335151399 + 2527163634923732000*c - 5081793740448746496*c^2 + 5406293137205395456*c^3 - 3558495001867452416*c^4 + 1393309590535274496*c^5 - 303305489096114176*c^6 + 28296722014797824*c^7 = 0. (End)

cp's OEIS Frontend

A361703 Constant term in the expansion of (1 + w + x + y + z + 1/(w*x*y*z))^n.

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A274783 Diagonal of the rational function 1/(1 - (w*x*y*z + w*x*y + w*x*z + w*y*z + x*y*z)).

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A361657 Constant term in the expansion of (1 + x^2 + y^2 + 1/(x*y))^n.

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A361658 Constant term in the expansion of (1 + x^3 + y^3 + z^3 + 1/(x*y*z))^n.

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A274785 Diagonal of the rational function 1/(1-(w*x*y*z + w*x*z + w*y + x*y + z)).

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A361673 Constant term in the expansion of (1 + x*y + y*z + z*x + 1/(x*y*z))^n.

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A361678 Constant term in the expansion of (1 + w + x + y + z + 1/(x*y*z) + 1/(w*y*z) + 1/(w*x*z) + 1/(w*x*y))^n.

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A361701 Constant term in the expansion of (1 + x^4 + y^4 + z^4 + 1/(x*y*z))^n.

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A361703 Constant term in the expansion of (1 + w + x + y + z + 1/(wxy*z))^n.

A274783 Diagonal of the rational function 1/(1 - (wxy*z + wxy + wxz + wyz + xyz)).

A361658 Constant term in the expansion of (1 + x^3 + y^3 + z^3 + 1/(xyz))^n.

A274785 Diagonal of the rational function 1/(1-(wxy*z + wxz + wy + xy + z)).

A361673 Constant term in the expansion of (1 + xy + yz + zx + 1/(xy*z))^n.

A361678 Constant term in the expansion of (1 + w + x + y + z + 1/(xyz) + 1/(wyz) + 1/(wxz) + 1/(wxy))^n.

A361701 Constant term in the expansion of (1 + x^4 + y^4 + z^4 + 1/(xyz))^n.