A082488 -id:A082488 - cp's OEIS frontend

A229142 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 13, 1, 1, 1, 25, 115, 63, 1, 1, 1, 121, 2641, 2371, 321, 1, 1, 1, 721, 114121, 392641, 54091, 1683, 1, 1, 1, 5041, 7489441, 169417921, 67982041, 1307377, 8989, 1, 1, 1, 40321, 681120721, 137322405361, 308238414121, 12838867105, 32803219, 48639, 1, 1

Offset: 0

Alois P. Heinz, Sep 23 2013

Column k is the diagonal of the rational function 1 / (1 - Sum_{j=1..k} x_j - Product_{j=1..k} x_j) for k>1. - Seiichi Manyama, Jul 10 2020

			A(1,3) = 3*2+1 = 7:
          (0,1,1)-(0,0,1)
         /       X       \
  (1,1,1)-(1,0,1) (0,1,0)-(0,0,0)
       \ \       X       / /
        \ (1,1,0)-(1,0,0) /
         `---------------´
Square array A(n,k) begins:
  1, 1,    1,       1,           1,               1, ...
  1, 1,    3,       7,          25,             121, ...
  1, 1,   13,     115,        2641,          114121, ...
  1, 1,   63,    2371,      392641,       169417921, ...
  1, 1,  321,   54091,    67982041,    308238414121, ...
  1, 1, 1683, 1307377, 12838867105, 629799991355641, ...

Alois P. Heinz, Antidiagonals n = 0..44, flattened

Columns k=0+1, 2-10 give: A000012, A001850, A081798, A082488, A082489, A229049, A229674, A229675, A229676, A229677.
Rows n=0-1 give: A000012, A038507 (for k>1).
Main diagonal gives: A229267.
Cf. A060854, A227578, A227655, A225094, A210472, A262809, A263159.

with(combinat):
A:= (n,k)-> `if`(k<2, 1, add(multinomial(n+(k-1)*j, n-j, j$k), j=0..n)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

a[, 0] = a[, 1] = 1; a[n_, k_] := Sum[Product[Binomial[n+j*m, m], {j, 0, k-1}], {m, 0, n}]; Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013 *)

A(n,k) = Sum_{j=0..n} multinomial(n+(k-1)*j; n-j, {j}^k) for k>1, A(n,0) = A(n,1) = 1.

G.f. of column k: Sum_{j>=0} (k*j)!/j!^k * x^j / (1-x)^(k*j+1). for k>1. - Seiichi Manyama, Jul 10 2020

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

Original entry on oeis.org

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

A082489 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2k,k) C(n+3k,k) C(n+4*k,k).

Original entry on oeis.org

1, 121, 114121, 169417921, 308238414121, 629799991355641, 1387152264043496161, 3220175519103433952161, 7771784978946238318454761, 19326687177288750280293146161, 49215884415076728067274047737961, 127771596843320597524806463425540481

Offset: 0

Emanuele Munarini, Apr 28 2003

Diagonal of the rational function 1 / (1 - x - y - z - u - v - x*y*z*u*v). - Ilya Gutkovskiy, Apr 22 2025

			G.f.: A(x) = 1 + 121*x + 114121*x^2 + 169417921*x^3 + 308238414121*x^4 +...
where
A(x) = 1/(1-x) + (5!/1!^5)*x/(1-x)^6 + (10!/2!^5)*x^2/(1-x)^11 + (15!/3!^5)*x^3/(1-x)^16 + (20!/4!^5)*x^4/(1-x)^21 + (25!/5!^5)*x^5/(1-x)^26 +... [Hanna]
Equivalently,
A(x) = 1/(1-x) + 120*x/(1-x)^6 + 113400*x^2/(1-x)^11 + 168168000*x^3/(1-x)^16 + 305540235000*x^4/(1-x)^21 + 623360743125120*x^5/(1-x)^26 +...+ A008978(n)*x^n/(1-x)^(5*n+1) +...

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k = 5 of A229142.

Cf. A008978, A001850, A081798, A082488, A229049, A229050.

with(combinat):
a:= n-> add(multinomial(n+4*k, n-k, k$5), 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]*Binomial[n+4*k,k], {k, 0, n}],{n,0,20}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (5*m)!/m!^5*x^m/(1-x+x*O(x^n))^(5*m+1)), n)}
for(n=0, 15, print1(a(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) *binomial(n+3*k, k)*binomial(n+4*k, k))}
for(n=0, 15, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 22 2013

G.f.: Sum_{n>=0} (5*n)!/n!^5 * x^n / (1-x)^(5*n+1). - Paul D. Hanna, Sep 22 2013
Recurrence: n^4*(5*n-16)*(5*n-12)*(5*n-11)*(5*n-8)*(5*n-7)*(5*n-6)*a(n) = (5*n-16)*(5*n-12)*(5*n-11)*(5*n-4)*(78250*n^6 - 422550*n^5 + 885665*n^4 - 906704*n^3 + 468906*n^2 - 114379*n + 10086)*a(n-1) - (5*n-16)*(31250*n^9 - 400000*n^8 + 2154375*n^7 - 6337750*n^6 + 11073100*n^5 - 11721380*n^4 + 7379043*n^3 - 2629646*n^2 + 489456*n - 36000)*a(n-2) + (5*n-1)*(31250*n^9 - 556250*n^8 + 4241875*n^7 - 18056500*n^6 + 46858025*n^5 - 76033760*n^4 + 76116292*n^3 - 44628880*n^2 + 13702848*n - 1693440)*a(n-3) - (5*n-6)*(5*n-2)*(5*n-1)*(625*n^7 - 11375*n^6 + 86025*n^5 - 347305*n^4 + 798274*n^3 - 1025292*n^2 + 661408*n - 156480)*a(n-4) + (n-4)^4*(5*n-11)*(5*n-7)*(5*n-6)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-5). - Vaclav Kotesovec, Sep 23 2013
a(n) ~ c*d^n/n^2, where d = 3129.996806129131084... is the root of the equation -1 + 5*d - 10*d^2 + 10*d^3 - 3130*d^4 +d^5 = 0 and c = 0.05674890286773483081841276583916042181... - Vaclav Kotesovec, Sep 23 2013
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 61*x^2 + 38101*x^3 + 42394381*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016

A361637 Constant term in the expansion of (1 + x + y + z + 1/(xyz))^n.

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 841, 4201, 25705, 118441, 423721, 1628881, 8065201, 41225185, 184416961, 768211081, 3420474121, 16620237001, 79922011465, 364149052705, 1638806098945, 7655390077105, 36739991161105, 174363209490625, 811840219629121, 3790118889635521

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

Diagonal of the rational function 1/(1 - (x^4 + y^4 + z^4 + w^4 + x*y*z*w)). - Seiichi Manyama, Jul 04 2025

Winston de Greef, Table of n, a(n) for n = 0..1429

Cf. A002426, A344560, A361703.
Cf. A008977, A328725.
Cf. A082488, A274783, A274785.

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

a(n) = n! * Sum_{k=0..floor(n/4)} 1/(k!^4 * (n-4*k)!).
G.f.: Sum_{k>=0} (4*k)!/k!^4 * x^(4*k)/(1-x)^(4*k+1).
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: n^3*a(n) = (2*n - 1)*(2*n^2 - 2*n + 1)*a(n-1) - (n-1)*(6*n^2 - 12*n + 7)*a(n-2) + 2*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 255*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 5^(n + 3/2) / (2^(5/2) * Pi^(3/2) * n^(3/2)). (End)

A229049 G.f.: Sum_{n >= 0} (6n)!/n!^6 x^n / (1-x)^(6*n+1).

Original entry on oeis.org

1, 721, 7489441, 137322405361, 3249278494922881, 88913838899388373921, 2672932604015450235911761, 85821373873727899097881489201, 2892941442791065880984595547275841, 101239016762863657789924022384195015281, 3649717311362112161867915447690005522733041

Offset: 0

Paul D. Hanna, Sep 22 2013

Diagonal of the rational function 1 / (1 - x - y - z - u - v - w - x*y*z*u*v*w). - Ilya Gutkovskiy, Apr 22 2025

			G.f.: A(x) = 1 + 721*x + 7489441*x^2 + 137322405361*x^3 + 3249278494922881*x^4 +...
where
A(x) = 1/(1-x) + (6!/1!^6)*x/(1-x)^7 + (12!/2!^6)*x^2/(1-x)^13 + (18!/3!^6)*x^3/(1-x)^19 + (24!/4!^6)*x^4/(1-x)^25 + (30!/5!^6)*x^5/(1-x)^31 +...
Equivalently,
A(x) = 1/(1-x) + 720*x/(1-x)^7 + 7484400*x^2/(1-x)^13 + 137225088000*x^3/(1-x)^19 + 3246670537110000*x^4/(1-x)^25 + 88832646059788350720*x^5/(1-x)^31 +...+ A008979(n)*x^n/(1-x)^(6*n+1) +...

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A008979, A001850, A081798, A082488, A082489, A229050.

Column k = 6 of A229142.

with(combinat):
a:= n-> add(multinomial(n+5*k, n-k, k$6), 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] *Binomial[n+4*k,k]*Binomial[n+5*k,k], {k, 0, n}],{n,0,20}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (6*m)!/m!^6*x^m/(1-x+x*O(x^n))^(6*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) *binomial(n+4*k,k)*binomial(n+5*k,k))}
for(n=0,15,print1(a(n),", "))

a(n) = Sum_{k=0..n} C(n,k) * C(n+k,k) * C(n+2*k,k) * C(n+3*k,k) * C(n+4*k,k) * C(n+5*k,k).
Recurrence: n^5*(2*n - 5)*(2*n - 3)*(3*n - 10)*(3*n - 7)*(3*n - 5)*(3*n - 4)*(6*n - 25)*(6*n - 19)*(6*n - 13)*(6*n - 7)*a(n) = (2*n - 5)*(3*n - 10)*(3*n - 7)*(6*n - 25)*(6*n - 19)*(6*n - 13)*(6*n - 5)*(839916*n^8 - 6159384*n^7 + 18804591*n^6 - 30967129*n^5 + 29803190*n^4 - 16984623*n^3 + 5534242*n^2 - 929843*n + 60482)*a(n-1) - (3*n - 10)*(6*n - 25)*(6*n - 19)*(58320*n^12 - 1030320*n^11 + 7973640*n^10 - 35550360*n^9 + 101096973*n^8 - 191892891*n^7 + 247426961*n^6 - 216687345*n^5 + 127127767*n^4 - 48662719*n^3 + 11593839*n^2 - 1535715*n + 84350)*a(n-2) + 10*(6*n - 25)*(6*n - 1)*(23328*n^13 - 618192*n^12 + 7344432*n^11 - 51616440*n^10 + 238504338*n^9 - 761904909*n^8 + 1722993100*n^7 - 2778206390*n^6 + 3175831572*n^5 - 2526793076*n^4 + 1352618106*n^3 - 459806772*n^2 + 89082095*n - 7435050)*a(n-3) - 5*(3*n - 1)*(6*n - 7)*(6*n - 1)*(11664*n^12 - 369360*n^11 + 5235192*n^10 - 43777800*n^9 + 239670873*n^8 - 901183065*n^7 + 2374616540*n^6 - 4392523494*n^5 + 5622136222*n^4 - 4816276070*n^3 + 2596763070*n^2 - 784074950*n + 100205500)*a(n-4) + (2*n - 1)*(3*n - 4)*(3*n - 1)*(6*n - 13)*(6*n - 7)*(6*n - 1)*(648*n^9 - 19548*n^8 + 256338*n^7 - 1909293*n^6 + 8851093*n^5 - 26285080*n^4 + 49492875*n^3 - 56141750*n^2 + 34024625*n - 8063750)*a(n-5) - (n-5)^5*(2*n - 3)*(2*n - 1)*(3*n - 7)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(6*n - 19)*(6*n - 13)*(6*n - 7)*(6*n - 1)*a(n-6). - Vaclav Kotesovec, Sep 23 2013
a(n) ~ c*d^n/(n^(5/2)), where d = 46661.9996785484656481246... is the root of the equation 1 - 6*d + 15*d^2 - 20*d^3 + 15*d^4 - 46662*d^5 + d^6 = 0 and c = 0.024758197509539176365175770882978221... - Vaclav Kotesovec, Sep 23 2013
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 361*x^2 + 2496841*x^3 + 34333162981*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016

A229050 G.f.: Sum_{n>=0} (n^2)!/n!^n * x^n / (1-x)^(n^2+1).

Original entry on oeis.org

1, 2, 9, 1714, 63079895, 623361815288736, 2670177752844538217570947, 7363615666255986180456959666126927684, 18165723931631174937747337664794705661513150850379149, 53130688706387570972824498004857476332107293478561950967662962585645710

Offset: 0

Paul D. Hanna, Sep 22 2013

nonn

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 1714*x^3 + 63079895*x^4 + 623361815288736*x^5 +...
where
A(x) = 1/(1-x) + x/(1-x)^2 + (4!/2!^2)*x^2/(1-x)^5 + (9!/3!^3)*x^3/(1-x)^10 + (16!/4!^4)*x^4/(1-x)^17 + (25!/5!^5)*x^5/(1-x)^26 +...
Equivalently,
A(x) = 1/(1-x) + x/(1-x)^2 + 6*x^2/(1-x)^5 + 1680*x^3/(1-x)^10 + 63063000*x^4/(1-x)^17 + 623360743125120*x^5/(1-x)^26 +...+ A034841(n)*x^n/(1-x)^(n^2+1) +...
Illustrate formula a(n) = Sum_{k=0..n} Product_{j=0..k-1} C(n+j*k,k) for initial terms:
a(0) = 1;
a(1) = 1 + C(1,1);
a(2) = 1 + C(2,1) + C(2,2)*C(4,2);
a(3) = 1 + C(3,1) + C(3,2)*C(5,2) + C(3,3)*C(6,3)*C(9,3);
a(4) = 1 + C(4,1) + C(4,2)*C(6,2) + C(4,3)*C(7,3)*C(10,3) + C(4,4)*C(8,4)*C(12,4)*C(16,4);
a(5) = 1 + C(5,1) + C(5,2)*C(7,2) + C(5,3)*C(8,3)*C(11,3) + C(5,4)*C(9,4)*C(13,4)*C(17,4) + C(5,5)*C(10,5)*C(15,5)*C(20,5)*C(25,5); ...
which numerically equals:
a(0) = 1;
a(1) = 1 + 1 = 2;
a(2) = 1 + 2 + 1*6 = 9;
a(3) = 1 + 3 + 3*10 + 1*20*84 = 1714;
a(4) = 1 + 4 + 6*15 + 4*35*120 + 1*70*495*1820 = 63079895;
a(5) = 1 + 5 + 10*21 + 10*56*165 + 5*126*715*2380 + 1*252*3003*15504*53130 = 623361815288736; ...

Alois P. Heinz, Table of n, a(n) for n = 0..26

Cf. A229051, A034841; A001850, A081798, A082488, A082489, A229049.

with(combinat):
a:= n-> add(multinomial(n+(k-1)*k, n-k, k$k), k=0..n):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 23 2013

Table[Sum[Product[Binomial[n+j*k,k],{j,0,k-1}],{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (m^2)!/m!^m*x^m/(1-x+x*O(x^n))^(m^2+1)), n)}
for(n=0,15,print1(a(n),", "))

{a(n)=sum(k=0,n,prod(j=0,k-1,binomial(n+j*k,k)))}
for(n=0,15,print1(a(n),", "))

a(n) = Sum_{k=0..n} Product_{j=0..k-1} binomial(n+j*k,k).

a(n) ~ exp(-1/12) * n^(n^2-n/2+1) / (2*Pi)^((n-1)/2). - Vaclav Kotesovec, Sep 23 2013

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)

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

A229051 G.f.: Sum_{n>=0} (n^2)!/n!^n * (2*x)^n / (1-x)^(n^2+1).

Original entry on oeis.org

1, 3, 29, 13567, 1009142769, 19947560933879891, 170891375663413844489045533, 942542805274443250197129297402029958879, 4650425326497533656923054675764068523027405525255181377, 27202912617670436035808496756146798219927348043651854025145948950565355779

Offset: 0

Paul D. Hanna, Sep 22 2013

nonn

			G.f.: A(x) = 1 + 3*x + 29*x^2 + 13567*x^3 + 1009142769*x^4 +...
where
A(x) = 1/(1-x) + (2*x)/(1-x)^2 + (4!/2!^2)*(2*x)^2/(1-x)^5 + (9!/3!^3)*(2*x)^3/(1-x)^10 + (16!/4!^4)*(2*x)^4/(1-x)^17 + (25!/5!^5)*(2*x)^5/(1-x)^26 +...
Equivalently,
A(x) = 1/(1-x) + (2*x)/(1-x)^2 + 6*(2*x)^2/(1-x)^5 + 1680*(2*x)^3/(1-x)^10 + 63063000*(2*x)^4/(1-x)^17 + 623360743125120*(2*x)^5/(1-x)^26 +...+ A034841(n)*(2*x)^n/(1-x)^(n^2+1) +...
Illustrate formula a(n) = Sum_{k=0..n} 2^k * Product_{j=0..k-1} C(n+j*k,k) for initial terms:
a(0) = 1;
a(1) = 1 + 2*C(1,1);
a(2) = 1 + 2*C(2,1) + 4*C(2,2)*C(4,2);
a(3) = 1 + 2*C(3,1) + 4*C(3,2)*C(5,2) + 8*C(3,3)*C(6,3)*C(9,3);
a(4) = 1 + 2*C(4,1) + 4*C(4,2)*C(6,2) + 8*C(4,3)*C(7,3)*C(10,3) + 16*C(4,4)*C(8,4)*C(12,4)*C(16,4);
a(5) = 1 + 2*C(5,1) + 4*C(5,2)*C(7,2) + 8*C(5,3)*C(8,3)*C(11,3) + 16*C(5,4)*C(9,4)*C(13,4)*C(17,4) + 32*C(5,5)*C(10,5)*C(15,5)*C(20,5)*C(25,5); ...
which numerically equals:
a(0) = 1;
a(1) = 1 + 2*1 = 3;
a(2) = 1 + 2*2 + 4*1*6 = 29;
a(3) = 1 + 2*3 + 4*3*10 + 8*1*20*84 = 13567;
a(4) = 1 + 2*4 + 4*6*15 + 8*4*35*120 + 16*1*70*495*1820 = 1009142769;
a(5) = 1 + 2*5 + 4*10*21 + 8*10*56*165 + 16*5*126*715*2380 + 32*1*252*3003*15504*53130 = 19947560933879891; ...

Alois P. Heinz, Table of n, a(n) for n = 0..26

Cf. A229050, A034841; A001850, A081798, A082488, A082489, A229049.

with(combinat):
a:= n-> add(2^k*multinomial(n+(k-1)*k, n-k, k$k), k=0..n):
seq(a(n), n=0..10);  # Alois P. Heinz, Sep 23 2013

Table[Sum[2^k*Product[Binomial[n+j*k,k],{j,0,k-1}],{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (m^2)!/m!^m*(2*x)^m/(1-x+x*O(x^n))^(m^2+1)), n)}
for(n=0,15,print1(a(n),", "))

{a(n)=sum(k=0,n,2^k*prod(j=0,k-1,binomial(n+j*k,k)))}
for(n=0,15,print1(a(n),", "))

a(n) = Sum_{k=0..n} 2^k * Product_{j=0..k-1} binomial(n+j*k,k).

a(n) ~ exp(-1/12) * n^(n^2-n/2+1) * 2^n / (2*Pi)^((n-1)/2). - Vaclav Kotesovec, Sep 23 2013

A336170 a(n) = Sum_{k=0..n} (-1)^(n-k) * (n+3k)!/((n-k)! k!^4).

Original entry on oeis.org

1, 23, 2401, 347279, 58370761, 10693893503, 2071837562929, 417449585719343, 86587926575712937, 18366152017597820303, 3965385492963153556441, 868598410928920193676023, 192552082030654661729957401, 43117650276328970463683450639, 9738695910884616220689842598481

Offset: 0

Seiichi Manyama, Jul 10 2020

nonn

Diagonal of the rational function 1 / (1 - Sum_{k=1..4} x_k + Product_{k=1..4} x_k).

Seiichi Manyama, Table of n, a(n) for n = 0..418

Column k=4 of A336169.

Cf. A082488.

a[n_] := Sum[(-1)^(n - k)*(n + 3*k)!/((n - k)!*k!^4), {k, 0, n}]; Array[a, 15, 0] (* Amiram Eldar, Jul 10 2020 *)

{a(n) = sum(k=0, n, (-1)^(n-k)*(n+3*k)!/((n-k)!*k!^4))}

N=20; x='x+O('x^N); Vec(sum(k=0, N, (4*k)!/k!^4*x^k/(1+x)^(4*k+1)))

G.f.: Sum_{k>=0} (4*k)!/k!^4 * x^k / (1+x)^(4*k+1).

cp's OEIS Frontend

A229142 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

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A082489 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2*k,k) * C(n+3*k,k) * C(n+4*k,k).

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A361637 Constant term in the expansion of (1 + x + y + z + 1/(x*y*z))^n.

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A229049 G.f.: Sum_{n >= 0} (6*n)!/n!^6 * x^n / (1-x)^(6*n+1).

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A229050 G.f.: Sum_{n>=0} (n^2)!/n!^n * x^n / (1-x)^(n^2+1).

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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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A082489 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2k,k) C(n+3k,k) C(n+4*k,k).

A361637 Constant term in the expansion of (1 + x + y + z + 1/(xyz))^n.

A229049 G.f.: Sum_{n >= 0} (6n)!/n!^6 x^n / (1-x)^(6*n+1).

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

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

A336170 a(n) = Sum_{k=0..n} (-1)^(n-k) * (n+3k)!/((n-k)! k!^4).