A082489 -id:A082489 - 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

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

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

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

Original entry on oeis.org

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

A361636 Diagonal of the rational function 1/(1 - vwx*yz (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

Original entry on oeis.org

1, 1, 1, 1, 121, 721, 2521, 6721, 128521, 1277641, 7539841, 32527441, 281835841, 3031468441, 23779315561, 139431015361, 962322302761, 9034098300361, 79726215362761, 569831799431881, 3952559737085401, 32660742079719601, 289694072383115401

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

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

Winston de Greef, Table of n, a(n) for n = 0..1064

Cf. A001850, A208425, A274783.
Cf. A008978.
Cf. A082489, A361703.

Table[Sum[(n + k)!/(k!^5*(n - 4*k)!), {k, 0, n/4}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 19 2023 *)

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

a(n) = Sum_{k=0..floor(n/4)} (n+k)!/(k!^5 * (n-4*k)!).
G.f.: Sum_{k>=0} (5*k)!/k!^5 * x^(4*k)/(1-x)^(5*k+1).
Recurrence: n^4*(5*n - 19)*(5*n - 18)*(5*n - 17)*(5*n - 14)*(5*n - 13)*(5*n - 9)*a(n) = (5*n - 19)*(5*n - 18)*(5*n - 14)*(625*n^7 - 6125*n^6 + 23025*n^5 - 43195*n^4 + 45394*n^3 - 28716*n^2 + 10144*n - 1536)*a(n-1) - (5*n - 19)*(31250*n^9 - 568750*n^8 + 4441875*n^7 - 19516000*n^6 + 53172025*n^5 - 93366740*n^4 + 106140132*n^3 - 75781664*n^2 + 30987264*n - 5529600)*a(n-2) + (5*n - 4)*(31250*n^9 - 725000*n^8 + 7354375*n^7 - 42784750*n^6 + 157237100*n^5 - 378480620*n^4 + 596812963*n^3 - 594970390*n^2 + 340845072*n - 85743360)*a(n-3) + (5*n - 16)*(5*n - 9)*(5*n - 8)*(5*n - 4)*(78000*n^6 - 1450800*n^5 + 11179085*n^4 - 45672814*n^3 + 104341702*n^2 - 126378083*n + 63400710)*a(n-4) + (n-4)^4*(5*n - 14)*(5*n - 13)*(5*n - 12)*(5*n - 9)*(5*n - 8)*(5*n - 4)*a(n-5). - Vaclav Kotesovec, Mar 19 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

A336171 a(n) = Sum_{k=0..n} (-1)^(n-k) * (n+4k)!/((n-k)! k!^5).

Original entry on oeis.org

1, 119, 112681, 166923119, 302857024681, 616967236620839, 1354737230950753441, 3135180238488702264959, 7543003841027749147438441, 18698821633118804601271092959, 47466852090165503045193665276041, 122841260732098480578334554450553679, 323029586700918689286922557725358306721

Offset: 0

Seiichi Manyama, Jul 10 2020

nonn

Diagonal of the rational function 1 / (1 - Sum_{k=1..5} x_k + Product_{k=1..5} x_k).

Column k=5 of A336169.

Cf. A082489.

a[n_] := Sum[(-1)^(n - k)*(n + 4*k)!/((n - k)!*k!^5), {k, 0, n}]; Array[a, 13, 0] (* Amiram Eldar, Jul 10 2020 *)

{a(n) = sum(k=0, n, (-1)^(n-k)*(n+4*k)!/((n-k)!*k!^5))}

N=20; x='x+O('x^N); Vec(sum(k=0, N, (5*k)!/k!^5*x^k/(1+x)^(5*k+1)))

G.f.: Sum_{k>=0} (5*k)!/k!^5 * x^k / (1+x)^(5*k+1).

A385606 Diagonal of the rational function 1/(1 - (v^3 + w^3 + x^3 + y^3 + z^3 + vwxyz)).

Original entry on oeis.org

1, 1, 1, 121, 721, 2521, 120121, 1262521, 7514641, 200655841, 2804296441, 23211542641, 443673670441, 7070369866561, 73192033638361, 1173608444069881, 19482750854113681, 235115468646608881, 3483568444035458401, 57574418930692099801, 769737183831483390601, 11118980118960559362001

Offset: 0

Seiichi Manyama, Jul 04 2025

nonn

Cf. A082489, A361636, A361703, A385607.

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

a(n) = Sum_{k=0..floor(n/3)} (n+2*k)!/(k!^5 * (n-3*k)!).

A385607 Diagonal of the rational function 1/(1 - (v^2 + w^2 + x^2 + y^2 + z^2 + vwxyz)).

Original entry on oeis.org

1, 1, 121, 721, 115921, 1254121, 175667521, 2723150641, 328524651841, 6553910658241, 694593264839761, 16751100559753561, 1592929589394223081, 44555491032952142881, 3872288533662063462481, 121957120480085202781681, 9836937778718128127534881, 341177468192261294809070401

Offset: 0

Seiichi Manyama, Jul 04 2025

nonn

Cf. A082489, A361636, A361703, A385606.

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

a(n) = Sum_{k=0..floor(n/2)} (n+3*k)!/(k!^5 * (n-2*k)!).

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

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A361636 Diagonal of the rational function 1/(1 - v*w*x*y*z * (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

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

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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).

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

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

A361636 Diagonal of the rational function 1/(1 - vwx*yz (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

A336171 a(n) = Sum_{k=0..n} (-1)^(n-k) * (n+4k)!/((n-k)! k!^5).

A385606 Diagonal of the rational function 1/(1 - (v^3 + w^3 + x^3 + y^3 + z^3 + vwxyz)).

A385607 Diagonal of the rational function 1/(1 - (v^2 + w^2 + x^2 + y^2 + z^2 + vwxyz)).